TBA]]>

TBA]]>

Iterative Methods for Linear Systems over two Centuries

Tags: TAG Events Perlenkolloquium, TAG Events Forschung Mathematik]]>

Iterative methods for linear systems were invented for the same reasons as they are used today, namely to reduce computational cost. Gauss states in a letter to his friend Gerling in 1823: "you will in the future hardly eliminate directly, at least not when you have more than two unknowns". Richardson's paper from 1910 was then very influential, and is a model of a modern numerical analysis paper: modeling, discretization, approximate solution of the discrete problem, and a real application. More general vector extrapolation methods were then introduced and studied in the pioneering work of Bresinzki, and they can be shown to be equivalent to Krylov method. It was however the work of Stiefel, Hestenes and Lanczos in the early 1950 that sparked the success story of Krylov methods with the invention of the conjugate gradient method, and there are now many Krylov methods to choose from. For general linear systems, they come in two main classes, the ones that minimize the residual in a Krylov space, and the ones that make the residual orthogonal to it. This will bring us finally to the modern iterative methods for solving partial differential equations, which also come in two main classes: domain decomposition methods and multigrid methods. Domain decomposition methods go back to the alternating Schwarz method invented by Herman Amandus Schwarz in 1869 to close a gap in the proof of Riemann's famous Mapping Theorem. Multigrid goes back to the seminal work by Fedorenko in 1961, with main contributions by Brandt and Hackbusch in the Seventies.

]]>Modular method for solving Diophantine equations]]>

Sharp energy regularity and typical wild solutions of the incompressible Euler equations

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Actions of Cremona groups on CAT(0) cube complexes.

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

In this talk, based on a common work with Christian Urech, we will construct such complexes on which Cremona groups of rank n act. We will then see which kind of results on these groups we can obtain.

]]>Protocol Replicating Social Graphs - Scuttlebutt

Tags: TAG Events DMI, TAG Events Forschung Informatik]]>

Please register via **christian.tschudin@unibas.ch**.

Computer-based formal proof in computer science and mathematics

Tags: TAG Events DMI, TAG Events Forschung Informatik]]>

]]>

Öffentliche Veranstaltung

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Begrüssung:

Gianluca Crippa, Departementsvorsteher

A.M. Ostrowski und die Präsentation des Preises:

Helmut Harbrecht, Präsident der Ostrowski-Stiftung

Laudatio:

Terence Tao, University of California, Los Angeles

Überreichung der Medaille der Ostrowski-Stiftung

Preisvortrag durch den Preisträger Assaf Naor

Im Anschluss an die Zeremonie wird ein Apéro offeriert.

]]>81 Jahre Versicherungsmathematik an der Universität Basel

Tags: TAG Events DMI]]>

Am 6. Februar 1939 wurde Versicherungslehre an der Universität Basel als selbständiges Examensfach eingeführt. Anlässlich des 80jährigen Bestehens wurde im Februar 2019 AlumniActuarialScience gegründet.

In Zusammenarbeit mit dem Masterstudiengang organisiert AlumniActuarialScience exakt ein Jahr später eine 2. Reunion. Drei Actuarial Science-Studentinnen stellen ihre Abschlussarbeiten vor und Prof.em.Dr. Herbert Lüthy gibt einen kurzen Überblick zur Entwicklung der Versicherungsmathematik. Beim anschliessenden Apéro bietet sich Gelegenheit, Studienkolleg*innen und Dozent*innen wiederzusehen sowie andere Aktuar*innen aus der Region kennenzulernen.

Alle Interessierten sind herzlich eingeladen.

**Anmeldung**

Für die Vorträge und den anschliessenden Apéro in der Alten Universität ist keine Anmeldung erforderlich.

Das AlumniEssen findet im Restaurant *Zum Isaak* statt und ist für Mitglieder *AlumniActuarialScience* und solche, die es gerne werden wollen. Bitte anmelden bis am 26. Januar 2020 an info@*clutter*AlumniActuarialScience.ch.

*AlumniActuarialScience ***werden**

Mitglied werden in der Fachalumni Actuarial Science können ehemalige Studierende Actuarial Science/Versicherungsmathematik, Dozierende und Emeriti, SAV-Hörer*innen und alle am Studiengang interessierten Privatpersonen und Institutionen.

Weitere Informationen: AlumniActuarialScience.

]]>Münsterschatz im Virtuellen Museum

Tags: TAG Events DMI]]>

am Donnerstag, den 9. Januar 2020 ab 09:00 Uhr

Tags: TAG Events DMI]]>

Numerical methods for fractional PDEs with applications to spatial statistics

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We propose an approximation supported by a rigorous error analysis which shows different notions of convergence at explicit and sharp rates. We furthermore discuss the computational complexity of the proposed method. Finally, we present several numerical experiments, which attest the theoretical outcomes, as well as a statistical application where we use the method for inference, i.e., for parameter estimation given data, and for spatial prediction.

For further information about the seminar, please visit this webpage.

]]>Fontes-Isopi-Newman processes associated with resistance forms]]>

Optimal errors and phase transitions in high-dimensional generalised linear models]]>

High-dimensional generalized linear models are basic building blocks of current data analysis tools including multilayers neural networks. They arise in signal processing, statistical inference, machine learning, communication theory, and other fields. I will explain how to establish rigorously the intrinsic information-theoretic limitations of inference and learning for a class of randomly generated instances of generalized linear models, thus closing several old conjectures. Examples will be shown where one can delimit regions of parameters for which the optimal error rates are efficiently achievable with currently known algorithms. I will discuss how the proof technique, based on the recently developed adaptive interpolation method, is able to deal with the output nonlinearity and also to some extent with non-separable input distributions. |

Fluctuation of the free energy of Sherrington-Kirkpatrick model with Curie-Weiss interaction: the paramagnetic regime]]>

We consider a spin system containing pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was considered by Baik and Lee and Baik et al. which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to Baik and Lee in the “paramagnetic regime” when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitable linear spectral statistics. Unlike the spherical symmetric case the free energy here can not be written as a function of the eigenvalues of the corresponding interaction matrix. The method in this paper relies on a dense sub-graph conditioning technique introduced by Banerjee . The proof of the approximation by the linear spectral statistics part is close to Banerjee and Ma. | |

High order finite element discretizations of Helmholtz problems

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

It has received much attention since it is widely employed in applications,

but still challenging to numerically simulate in the high-frequency regime.

In this seminar, we focus on acoustic waves for the sake of simplicity

and consider finite element discretizations. The main goal of the

presentation is to highlight the improved performance of high order

methods (as compared to linear finite elements) when the frequency is large.

We will very briefly cover the zero-frequency case, that corresponds to the well-studied Poisson equation. We take advantage of this classical setting to recall central concepts of the finite element theory such as quasi-optimality and interpolation error.

The second part of the seminar is devoted to the high-frequency case.

We show that without restrictive assumptions on the mesh size,

the finite element method is unstable, and quasi-optimality is lost.

We provide a detailed analysis, as well as numerical examples, which

show that higher order methods are less affected by this phenomena,

and thus more suited to discretize high-frequency problems.

Before drawing our main conclusions, we briefly discuss advanced topics,

such as the use of unfitted meshes in highly heterogeneous media

and mesh refinements around re-entrant corners.

For further information about the seminar, please visit this webpage.

]]>zum Thema Random Diophantine Equations

Tags: TAG Events DMI, TAG Events Forschung Mathematik, TAG Events Perlenkolloquium]]>

Zero-one laws for first order properties of random graphs]]>

Zero-one laws for first order properties of random graphs]]>

Direct/Inverse Hopfield model and Restricted Boltzmann Machines]]>

Mean-field methods fail to reconstruct the parameters of the model when the dataset is clusterized. This situation is found at low temperatures because of the emergence of multiple thermodynamic states. The paradigmatic Hopfield model is considered in a teacher-student scenario as a problem of unsupervised learning with Restricted Boltzmann Machines (RBM). For different choices of the priors on units and weights, the replica symmetric phase diagram of random RBM’s is analyzed and in particular the paramagnetic phase boundary is presented as directly related to the optimal size of the training set necessary for a good generalization. The connection between the direct and inverse problem is pointed out by showing that inference can be efficiently performed by suitably adapting both standard learning techniques and standard approaches to the direct problem. | |

]]>

Prelog Chow rings and stable rationality in semistable degenerations

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

On error estimates for optimal control problems with functions of bounded variation

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We consider a convex optimal control problem governed by a partial differential equation in one space dimension which is controlled by a right-hand-side living in the space of functions with bounded variation. These functions tend to favor optimal controls that are piecewise constant with often finitely many jump poins. We are interested in deriving finite element discretization error estimates for the controls when the state ist discretized with usual piecewise linear finite elements, and the controls is either variationally discrete or piecwise constant. Due to the structure of the objective function, usual techniques for estimating the control error cannot be applied. Instead, these have to be derived from (suboptimal) error estimates for the state, which can later be improved.

For further information about the seminar, please visit this webpage.

]]>Classification of simple singularities]]>

Birational models of terminal sextic double solids

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Automorphisms of Del Pezzo surfaces of degree 4]]>

Maximum Likelihood Estimation of Toric Fano Varieties

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Unlikely Intersections with Isogeny Orbits

Tags: TAG Events DMI, TAG Events Perlenkolloquium, TAG Events Forschung Mathematik]]>

Leonhard Euler's methods of celestial mechanics (Beginn: 15:15 Uhr)

Tags: TAG Events Forschung Mathematik, TAG Events Forschung Informatik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Details zum Anlass finden sich auf der Webseite des Departements Physik.

]]>Anisotropic vector fields: quantitative estimates and applications to the Vlasov-Poisson equation

Tags: TAG Events DMI, TAG Events Forschung Mathematik, TAG Events Perlenkolloquium]]>

The Cauchy problem for the quantum hydrodynamics system and the stability of weak solutions

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Oscillatory behaviour in coagulation-fragmentation models

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Numerical aspects of elliptic diffusion problems on random domains

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

For further information about the seminar, please visit this webpage.

]]>Algebraic torus actions

Tags: TAG Events DMI, TAG Events Perlenkolloquium, TAG Events Forschung Mathematik]]>

Lower bounds of non-vanishing sums (part 2) ]]>

Dual complexes of degenerations and Berkovich geometry

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

]]>

Aposteriori error analysis and adaptive schemes for the wave equation

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Notes: This is based on joint work with E.H. Georgoulis, C. Makridakis and J.M. Virtanen.

References:

[1] W. Bangerth and R. Rannacher, J. Comput. Acoust. 9(2):575–591, 2001.

[2] C. Bernardi and E. Süli, Math. Models Methods Appl. Sci. 15(2):199--225, 2005.

[3] E. H. Georgoulis, O. Lakkis, and C. Makridakis. IMA J. Numer. Anal., 33(4):1245–1264, 2013, arxiv.org/abs/1003.3641

[4] E. H. Georgoulis, O. Lakkis, C. Makridakis, and J. M. Virtanen. SIAM J. Numer. Anal., 54(1), 2016, arxiv.org/abs/1411.7572

For further information about the seminar, please visit this webpage.

]]>Complex projective structures on Riemann surfaces]]>

Invitation to random tensors]]>

Birational Kleinian groups and birational structures

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Multirate explicit stabilized integrators for stiff differential equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The efficiency of these methods deteriorates as the system becomes stiffer, even if stiffness is induced by only few degrees of freedom. In the framework of discretized parabolic PDEs, the number of function evaluations has to be chosen inversely proportional to the smallest element size in order to achieve stability, thus largely wasting computational resources on locally-refined meshes. We first tackle this issue by replacing the right hand side of the PDE with an averaged force, which is obtained by damping the high modes down using the dissipative effect of the equation itself and which is cheap to evaluate. Combining RKC methods with the averaged force we give rise to multirate RKC schemes, for which the number of expensive function evaluations is independent of the small elements' size.

The stability properties of our method are demonstrated on a model problem and numerical experiments confirm that the stability bottleneck caused by a few of fine mesh elements can be overcome without sacrificing accuracy.

For further information about the seminar, please visit this webpage.

]]>On the frequency of height values]]>

On bulk deviations for the local behavior of random interlacements]]>

In this talk we will discuss some recent large deviation asymptotics concerning the local behavior of random interlacements on Z^d, d≥3. In particular, we will describe the link with previous results concerning macroscopic holes left inside a large box, by the the adequately thickened connected component of the boundary of the box in the vacant sets of random interlacements. | |

29.10.2019 18:15 - 29.10.2019 19:00, Fachgruppe Mathematik und Informatik, DMI, Spiegelgasse 1, Seminarraum 00.003

Christian Graf, Non-life pricing for commercial lines at Allianz Suisse

Tags: TAG Events DMI]]>

**Next event:**

November 12, 2019

Orhan Yildirim, Data Scientist at IWB (Deutsch)

]]>

Graded Rings and Birational Geometry

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

The main purpose of the talk is to give a short introduction to the theory and describe some of its applications.

]]>Lower bounds of non-vanishing sums]]>

C^0-closedness of the set of symplectomorphisms, spherical symplectic nonsqueezing, and holomorphic maps

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in Hamilton's equation. A (smooth) diffeomorphism on a symplectic manifold is called a symplectomorphism iff it preserves the symplectic form. This happens iff the diffeomorphism solves a certain inhomogeneous quadratic first order system of PDE's. In classical mechanics symplectomorphisms play the role of canonical transformations.

A famous result by Eliashberg and Gromov states that the set of symplectomorphisms is $C^0$-closed in the set of all diffeomorphisms. This is remarkable, since in general, the $C^0$-limit of a sequence of solutions of a first order system of PDE's need not solve the system. A well-known proof of the Eliashberg-Gromov theorem is based on Gromov's symplectic nonsqueezing theorem for balls.

In my talk I will sketch this proof. Furthermore, I will present a symplectic nonsqueezing result for spheres that sharpens Gromov's theorem. The proof of this result is based on the existence of a holomorphic map from the (real) two-dimensional unit disk to a certain symplectic manifold, satisfying some Lagrangian boundary condition. Such a map solves the Cauchy-Riemann equation for a certain almost complex structure. ]]>

22.10.2019 18:15 - 22.10.2019 19:00, Fachgruppe Mathematik und Informatik, DMI, Spiegelgasse 1, Seminarraum 00.003

Laurent Huber, Quantitative Insurance Specialist at Schroders, Investment Management

Tags: TAG Events DMI]]>

**Next events:**

October 29, 2019

Christian Graf, Non-life pricing for commercial lines at Allianz Suisse (English)

-------

November 12, 2019

Orhan Yildirim, Data Scientist at IWB (Deutsch)

]]>

tba

Tags: TAG Events DMI, TAG Events Forschung Mathematik, TAG Events Perlenkolloquium]]>

Two Puzzles in Particle Physics]]>

ℚ-homology planes satisfying the Negativity Conjecture

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Blowing up and automorphisms on a blow-up]]>

On continuous automorphisms of Cremona groups

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Pricescapes - Competitive Pricing Landscapes]]>

Mathematical modelling of transmission dynamics of Opisthorchis viverrini

Tags: TAG Events DMI, TAG Events Forschung Mathematik, TAG Events Perlenkolloquium]]>

Latest news about forthcoming Work Programme; Experiences and tips from successful applicants

Tags: TAG Events Forschung Informatik, TAG Events Forschung Mathematik, TAG Events Perlenkolloquium, TAG Events DMI]]>

In addition, information is provided how to identify topics of interest and find partners to submit proposals from basic research to market application.

Horizon 2020 project winners from academia and industry will give insights into running projects.

For the 2020 Work Programme, the areas of Artificial Intelligence (AI), digitizing European Industry, HPC, Big Data, Cloud technologies, 5G, Next Generation Internet (NGI) and key technologies for digital transformation from photonics to software, advanced and smart connectivity are prioritized.

Please register online.

]]>04.09.2019 08:30 - 06.09.2019 16:00

Summer School of the PhD Program Data Analytics

Tags: TAG Events DMI]]>

More information on the summer school and a link for the enrollment can be found here.

]]>02.09.2019 08:00 - 12.09.2019 17:00, Alte Universität, Rheinsprung 9

One week of mini-courses and a week-long conference held from September 2 till 13, 2019

Tags: TAG Events DMI, TAG Events Forschung Mathematik, TAG Events Zahlentheorie]]>

Website: numbertheory.dmi.unibas.ch/trip2019

]]>

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Finite subgroups of tame polynomial automorphisms

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Symmetry of Ground States by Fourier Rearrangement

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Tensor B-Spline Numerical Method for PDEs: a High Performance Approach

Tags: TAG Events DMI, TAG Events Forschung Informatik]]>

ABSTRACT:

Solutions of Partial Differential Equations (PDEs) form the basis of many mathematical models

in physics and medicine. In this work, a novel Tensor B-spline methodology for numerical

solutions of linear second-order PDEs is proposed. The methodology applies the B-spline

signal processing framework and computational tensor algebra in order to construct high-

performance numerical solvers for PDEs. The method allows high-order approximations, is

mesh-free, matrix-free and computationally and memory efficient.

The first chapter introduces the main ideas of the Tensor B-spline method, depicts the main

contributions of the thesis and outlines the thesis structure.

The second chapter provides an introduction to PDEs, reviews the numerical methods for

solving PDEs, introduces splines and signal processing techniques with B-splines, and describes

tensors and the computational tensor algebra.

The third chapter describes the principles of the Tensor B-spline methodology. The main

aspects are 1) discretization of the PDE variational formulation via B-spline representation

of the solution, the coefficients, and the source term, 2) introduction to the tensor B-spline

kernels, 3) application of tensors and computational tensor algebra to the discretized variational

formulation of the PDE, 4) tensor-based analysis of the problem structure, 5) derivation of

the efficient computational techniques, and 6) efficient boundary processing and numerical

integration procedures.

The fourth chapter describes 1) different computational strategies of the Tensor B-spline

solver and an evaluation of their performance, 2) the application of the method to the forward

problem of the Optical Diffusion Tomography and an extensive comparison with the state-

of-the-art Finite Element Method on synthetic and real medical data, 3) high-performance

multicore CPU- and GPU-based implementations, and 4) the solution of large-scale problems

on hardware with limited memory resources.

Der Vortrag ist universitätsöffentlich.

]]>

Gaussian fluctuations in directed polymers]]>

Recent results on singular limits for Vlasov-Poisson

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The Vlasov-Poisson system is a kinetic equation that models collisionless plasma. A plasma has a characteristic scale called the Debye length, which is typically much shorter than the scale of observation. In this case the plasma is called ‘quasineutral’. This motivates studying the limit in which the ratio between the Debye length and the observation scale tends to zero. Under this scaling, the formal limit of the Vlasov-Poisson system is the Kinetic Isothermal Euler system.

The Vlasov-Poisson system itself can formally be derived as the limit of a system of ODEs describing the dynamics of a system of N interacting particles, as the number of particles approaches infinity. The rigorous justification of this mean field limit remains a fundamental open problem.

In this talk we present how the mean field and quasineutral limits can be combined to derive the Kinetic Isothermal Euler system from a regularised particle model.

Quantitative seismic imaging using reciprocity waveform inversion with arbitrary probing sources

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We investigate the deployment of new devices developed in the acoustic setting: the dual-sensors, which are able to capture both the pressure field and the vertical velocity of the waves. For solving the inverse problem, we define a new cost function, adapted to these two types of data and based upon the reciprocity. We first note that the stability of the problem can be shown to be Lipschitz, assuming piecewise linear parameters. In addition, reciprocity waveform inversion allows a separation between the observational and numerical acquisitions. In fact, the numerical sources do not have to coincide with the observational ones, offering new possibilities to create adapted computational acquisitions, consequently reducing the numerical cost. We illustrate our approach with three-dimensional medium reconstructions, where we start with minimal information on the target models. We also extend the methodology for elasticity.

Eventually, if time allows, we shall explore the model representation in numerical seismic inversion, where the adaptive eigenspace method appears as a promising approach to have a compromise between number of unknowns and resolution.

References

[1] G. Alessandrini, M. V. de Hoop, F. Faucher, R. Gaburro and E. Sincich, Inverse problem for the Helmholtz equation with Cauchy data: reconstruction with conditional well-posedness driven iterative regularization, ESAIM: M2AN (2019).

[2] E. Beretta, M. V. De Hoop, F. Faucher, and O. Scherzer, Inverse boundary value problem for the Helmholtz equation: quantitative conditional Lipschitz stability estimates. SIAM Journal on Mathematical Analysis, 48(6), pp.3962-3983 (2016).

[3] M. J. Grote, M. Kray, and U. Nahum, Adaptive eigenspace method for inverse scattering problems in the frequency domain. Inverse Problems, 33(2), 025006 (2017).

[4] H. Barucq, F. Faucher, and O. Scherzer, Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation. arXiv preprint arXiv:1903.08991 (2019).

For further information about the seminar, please visit this webpage.

AlumniComputerScience

Tags: TAG Events Alumni Informatik, TAG Events DMI, TAG Events Forschung Informatik]]>

The talk will present some of the successful projects at Apple where Computer Vision and Machine Learning was used. It will include a deeper dive into the technology behind the real time facial motion capture on iPhone.

Dr. Brian Amberg studied „Information Science Cognition“ in Freiburg i. Brsg. From 2005 until 2010, he was a PhD in the research group of Prof. Dr. Vetter. In 2012, Amberg co-founded at EPFL

The Kuzenetsov Lemma for Conservation Laws with Space and Time Dependent Flux]]>

Thresholds for Measuring Degree in fractional Sobolev Spaces

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The degree of a map between two spheres of same dimension can be estimated by Sobolev norm of said map (of the right class). In this talk I will discuss to what extend this is possible for the Hopf degree as well – and why the estimate we have is “analytically optimal” but probably not “topologically optimal”. Joint work with J. Van Schaftingen.

]]>21.05.2019 14:00, Seminarraum 05.002, Spiegelgasse 5, Departement Mathematik und Informatik

Tags: TAG Events DMI, TAG Events Studium Informatik]]>

]]>

Mathematical Models of Cancer Evolution

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Adaptive Iterative Linearization Galerkin Methods for Nonlinear Partial Differential Equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

For further information about the seminar, please visit this webpage. ]]>

A Brief Look at Genomic Testing in Terms of Algebraic Statistics]]>

Random walk on a simple exclusion process]]>

In this talk we will study the asymptotic behavior of a random walk that evolves on top of a simple symmetric exclusion process. This nice example of a random walk on a dynamical random environment presents its own challenges due to the slow mixing properties of the underlying medium. We will discuss a law of large numbers that has been proved recently for this random walk. Interestingly, we can only prove this law of large numbers for all but two exceptional densities of the exclusion process. The main technique that we have employed is a multi-scale renormalization that has been derived from works in percolation theory. | |

14.05.2019 18:15, Seminarraum 00.003, Spiegelgasse 1, Departement Mathematik und Informatik

Stephan Heule, SBB, Data Scientist

Tags: TAG Events DMI, TAG Events Studium Informatik]]>

Wie sieht Deine berufliche Zukunft aus? Diese Vortragsreihe zeigt Dir spannende Karrierewege.

Der Vortrag findet um 18.15 Uhr im Seminarraum 00.003 in der Spiegelgasse 1 (Departement Mathematik und Informatik) statt.

]]>

14.05.2019 12:15 - 14.05.2019 13:15, DMI, Spiegelgasse 5, Basel, Raum 05.002

Heuristic Search: Something Old and Something New

Tags: TAG Events DMI, TAG Events Forschung Informatik, TAG Events Alumni Informatik]]>

I begin this talk with a review of long-established results in heuristic search and the early history of bidirectional heuristic search. I then describe a recent (2016) breakthrough in bidirectional heuristic search (the MM algorithm), which challenges long-held assumptions and exposes exciting new research directions. Although the technical details in this talk are focused on heuristic search, the general lessons with which I conclude are relevant to researchers in all branches of A.I.

Speaker Bio:

Professor Emeritus Robert Holte of the Computing Science Department at the University of Alberta is a former editor-in-chief of the journal "Machine Learning" and co-founder and former director of the Alberta Innovates Center for Machine Learning (AICML, now known as Amii). In addition to machine learning, Professor Holte has made seminal contributions to the subfield of A.I. known as single-agent heuristic search, most notably his recent work on bidirectional heuristic search.

Professor Holte was elected a Fellow of the Association for the Advancement of Artificial Intelligence (AAAI) in 2011 and received a Lifetime Achievement Award for his work on heuristic search from the Symposium on Combinatorial Search in 2018. Another Lifetime Achievement Award will be announced at the end of May

]]>

Irrationality of quotient varieties

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

13.05.2019 17:00, Fachgruppe Mathematik, Kollegienhaus der Universität Basel, Petersplatz 1, Hörsaal 117003

Dr. Jitao David Zhang, Dr. Thomas Albrecht, Dr. Volker Herdtweck, F. Hoffmann-La Roche Ltd.

Tags: TAG Events DMI]]>

Principal Scientist, Computational Biology

Pharmaceutical Sciences, Pharma Research and Early Development

Roche Innovation Center Basel

**Dr. Thomas Albrecht**

Senior Scientist, Scientific Solution Engineering & Architecture

Pharma Research and Early Development Informatics

Roche Innovation Center Basel**Dr. Volker Herdtweck**

HR Campus Marketing Manager/University Relations

F. Hoffmann-La Roche Ltd.

Alle Vorträge sind auf Englisch und sind für Studierende der Mathematik, Computer Science und Computational Science geeignet.

Der anschliessende Apéro findet im Foyer der Aula vom Kollegienhaus statt.

]]>

The Local Density Approximation in Density Functional Theory

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Density Functional Theory (DFT) is the main method used in practical computations to approximate solutions of the many-body Schrödinger equation. The main idea is to express everything in terms of the one-particle density instead of the many-body wave function. The Local Density Approximation (LDA) is the simplest nonlinear functional used in this context and it has played a central role in the construction of better empirical approximations since the 60s.

In this talk I will first explain what DFT and LDA are. I will then give the first rigorous justification of the LDA. More precisely, I will show that the exact Levy-Lieb functional of DFT converges to the LDA in the limit of very flat densities. Joint work with Elliott H. Lieb (Princeton) and Robert Seiringer (IST Austria).

Fluid-squeezing singularities for the incompressible Euler equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

I will discuss a new result on the existence of a stationary solution with a fluid-squeezing singularity for the two-fluid incompressible Euler equations. The proof is based on a new set of estimates that permits us to analyze a fluid squeezed by a self-intersecting interface. We will exploit these estimates in the dynamical case and show a local existence result starting from a fluid-squeezing singularity.

]]>The geometry of random walk isomorphisms]]>

The classical random walk isomorphism theorems relate the local time of a random walk to the square of a Gaussian free field. I will present non-Gaussian versions of these theorems, relating hyperbolic and hemispherical sigma models (and their supersymmetric versions) to non-Markovian random walks interacting through their local time. Applications include a short proof of the Sabot-Tarres limiting formula for the vertex-reinforced jump process (VRJP) and a Mermin-Wagner theorem for hyperbolic sigma models and the VRJP. This is joint work with Tyler Helmuth and Andrew Swan. | |

Birational transformations of tetragonal conic bundles

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

In this talk, I will first discuss application of Sarkisov program to the rationality problem of algebraic varieties having conic bundle structures.

Then I concentrate on some special, so-called

The second part of the talk is based on the joint work in progress with V. Shokurov. ]]>

06.05.2019 18:00, Studienfach Mathematik, Spiegelgasse 1, Seminarraum 00.003

Stephanie Meier-Rohr, Softwareentwicklerin, CUBUS AG, Engineering Software

Tags: TAG Events DMI]]>

Im Anschluss daran sind alle herzlich zu einem kleinen Apéro eingeladen.

]]>

Some recent results on the approximation of conservation laws: revisiting the notion of conservation

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

However, this is not the end of the story. All these works use a one dimensional way of thinking: the main player is the normal flux across cell interfaces. In addition there are several excellent numerical methods that do not fit the form of the lax Wendroff theorem.

In that talk, I will introduce a more general setting and show that any reasonable scheme for conservation law can be put in that framework. In addition, I will show that an equivalent flux formulation, with a suitable definition of what is a flux, can be explicitly constructed (and computed), so that any reasonable scheme can be put in a finite volume form.

I will end the talk by showing some applications: how to systematically construct entropy stable scheme, or starting from the non conservative form of a system-say the Euler equations-, how to construct a suitable discretisation. And more.

This is a joint work with P. Bacigaluppi (now postdoc at ETH) and S. Tokareva (now at Los Alamos).

For further information about the seminar, please visit this webpage.

]]>

Plateau's Problem: The Geometric Measure Theory Approach]]>

On numerical schemes for hyperbolic conservation laws]]>

A distributional approach to fractional Sobolev spaces and fractional variation

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Somewhat surprisingly, the first appearance of the concept of a fractional derivative is found in a letter written to de l'Hôpital by Leibniz in 1695. Since then, Fractional Calculus has fascinated generations of mathematicians and several definitions of fractional derivatives have appeared. In more recent years, the fractional operator defined as the gradient of the Riesz potential has received particular attention, since it has revealed to be a useful tool for the study of fractional-order PDEs and fractional Sobolev spaces. In a joint work with G. E. Comi, combining the PDE approach developed by Spector and his collaborators with the distributional point of view adopted by Šilhavý, we introduced new notions of fractional variation and fractional Caccioppoli perimeter in analogy with the classical BV theory. Within this framework, we were able to partially extend De Giorgi’s Blow-up Theorem to sets of locally finite fractional Caccioppoli perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets. In this talk, after a quick overview on Fractional Calculus, I will introduce the main features of the fractional operators involved and then give an account on the main results on the fractional variation we were able to achieve so far.

]]>Real algebraic curves in real minimal del Pezzo surfaces

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Theory of Deep Learning 5: Information theoretic approach to deep learning theory: a test using statistical physics methods]]>

Relying on the heuristic replica method from statistical physics we present an estimator for entropies and mutual informations in models of deep model networks. Using this new tool, we test numerically the relation between generalisation and information. ]]>

Recent methods for solving the high-frequency Helmholtz equation on a regular mesh

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

For further information about the seminar, please visit this webpage. ]]>

Flexibility and Rigidity of Isometric Embeddings

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The problem of embedding abstract Riemannian manifolds isometrically (i.e. preserving the lengths) into Euclidean space stems from the conceptually fundamental question of whether abstract Riemannian manifolds and submanifolds of Euclidean space are the same. As it turns out, such embeddings have a drastically different behaviour at low regularity (i.e. C^1) than at high regularity (i.e. C^2): for example, it's possible to find C^1 isometric embeddings of the standard 2-sphere into arbitrarily small balls in R^3, and yet, in the C^2 category there is (up to translation and rotation) just one isometric embedding, namely the standard inclusion. Analoguous to the Onsager conjecture, one might ask if there is a regularity threshold in the Hölder scale which distinguishes these behaviours. In my talk I will give an overview of what is known concerning the latter question.

]]>Theory of Deep Learning 4: Training Neural Networks in the Lazy and Mean Field Regimes]]>

On Compensated Compactness and Hardy Spaces]]>

Theory of Deep Learning 3: Neural Tangent Kernel: Convergence and Generalization of Deep Neural Networks]]>

Multilevel Uncertainty Quantification with Sample-Adaptive Model Hierarchies

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

This is joint work with Gianluca Detommaso (Bath), Tim Dodwell (Exeter) and Jens Lang (Darmstadt).

For further information about the seminar, please visit this webpage. ]]>

On (Hilbert, Isabelle) and universal pairs]]>

The Dirichlet Problem for the Logarithmic Laplacian

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

I will report on some recent results - obtained in joint work with Huyuan Chen - on Dirichlet problems for the Logarithmic Laplacian Operator, which arises as formal derivative of fractional Laplacians at order s= 0. I will discuss the functional analytic framework for these problems and show how it allows to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of fractional Laplacians as the order tends to zero. Furthermore, I will discuss necessary and sufficient conditions on domains giving rise to weak and strong maximum principles for the logarithmic Laplacian. If time permits, I will also discuss regularity estimates for solutions to corresponding Poisson problems.

]]>Theory of Deep Learning 2: Over-parametrization in neural networks: an overview and a definition]]>

We will start by reviewing some of the recent literature on the geometry of the loss function, and how SGD navigates the landscape in the OP regime. Then we will see how to define OP by finding a sharp transition described by the models fitting abilities to its training set. Finally, we will discuss how this critical threshold is connected to the generalization properties of the model, and argue that life beyond this threshold is (more or less) as good as it gets. ]]>

Bogoliubov Theory in the Gross-Pitaevskii Limit]]>

Tags: TAG Events DMI, TAG Events Forschung Informatik, TAG Events Forschung Mathematik, TAG Events Studium Informatik]]>

Tags: TAG Events DMI, TAG Events Forschung Informatik, TAG Events Forschung Mathematik, TAG Events Studium Informatik]]>

Tags: TAG Events DMI, TAG Events Forschung Informatik, TAG Events Forschung Mathematik, TAG Events Studium Informatik]]>

Theory of Deep Learning 1: Introduction to the main questions]]>

In this first talk I will introduce the main theoretical questions about deep neural networks:

1. Representation - what can deep neural networks represent?

2. Optimization - why and under what circumstances can we successfully train neural networks?

3. Generalization - why do deep neural networks often generalize well, despite huge capacity?

As a preface I will review the basic models and algorithms (Neural Networks, (stochastic) gradient descent, ...) and some important concepts from machine learning (capacity, overfitting/underfitting, generalization, ...). ]]>

The essential skeletons of pairs and the geometric P=W conjecture

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Linebündeli]]>

]]>

Vektorbündeli]]>

Symmetries of foliation: transverse action

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

I will briefly recall a criterion for the finiteness of the transverse action in the case of algebraically integrable foliations (i.e. foliations whose leaves coincide with the fibres of a fibration). Then I will explain how the presence of certain transverse structures on the foliation allow to recover the same result; in this case, one can study the monodromy of such a structure (which is defined in an analogous way as that of a more familiar (G,X)-structure) and apply factorization results in order to reduce the problem to subvarieties of quotients of the product of unit discs, whose geometry is now quite well understood.

]]>Tags: TAG Events Forschung Mathematik, TAG Events Perlenkolloquium, TAG Events DMI, TAG Events Zahlentheorie]]>

Solution Spaces in Complex Systems Design

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

For further information about the seminar, please visit this webpage. ]]>

28.02.2019 14:15 - 28.02.2019 15:15, 00.003 Spiegelgasse 1

Towards effective André-Oort (after Kühne et al.)

Tags: TAG Events DMI, TAG Events Forschung Mathematik, TAG Events Zahlentheorie]]>

Since 2012 various special cases of the André-Oort conjecture has been proved effectively, most notably in the work of Lars Kühne. In my talk I will restrict to the case of the "Shimura variety" C^n and will try to explain on some simple examples how the effective approach of Kühne works.

No previous knowledge about André-Oort conjecture is required, I will give all the necessary background. ]]>

26.02.2019 12:15 - 26.02.2019 14:00, DMI, Spiegelgasse 5, Basel, Raum 05.002

From Data Integration to Knowledge Management

Tags: TAG Events DMI, TAG Events Forschung Informatik, TAG Events Alumni Informatik]]>

The time to develop new drugs can be dramatically reduced through intelligent data-integration and re-use. To reduce this time by a factor of 10, Roche Technical Development, an organisation of 2500 colleagues, has embraced a central information integration approach. In this talk I will present our vision and technology platform, give an update on where we are on our journey, and mention some of the barriers we encountered.

Our approach to balance real world needs with accuracy and speed has gained some good success over the course of the last four years for a complex information landscape with hundreds of information entities.

Speaker Bio:

Etzard leads the global Information / Knowledge Management effort in Pharma Technical Development for F. Hoffmann-La Roche in Basel, with a focus on processes and tools for effective knowledge utilisation. Etzard has worked at the interface of the Life- and Computer-Sciences for more than 20 years, in technical, managerial as well as strategic roles.

Before joining Roche, Etzard worked as CIO at the Jackson Lab (a US based genomics research institutes with 1800 employees), and was CTO for the Life Sciences at Hewlett Packard. Etzard has earned academic degrees in Biology, Bio-Informatics and Informatics, with a PhD in Computer Science from ETH Zurich on “A Scalable Architecture for Scientific Databases”.

]]>

A reduced SQP Method for Shape Optimisation of Flow Control Problems

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

For further information about the seminar, please visit this webpage. ]]>

Inequalities on Conductors]]>

Tags: TAG Events DMI]]>

**Jubiläumsveranstaltung und Gründung AlumniActuarialScience**

Am 6. Februar 1939 wurde Versicherungslehre an der Universität Basel als selbständiges Examensfach eingeführt. Anlässlich des 80jährigen Bestehens findet am Mittwoch, 6. Februar 2019, im Hörsaal 101 der Alten Universität eine Jubiläumsveranstaltung statt mit Gründung der Fachalumni Actuarial Science.

Alle Interessierten sind herzlich eingeladen. Der Besuch der Veranstaltung ist von der SAV als Weiterbildung anerkannt (1 Credit Point SAV).

**Anmeldung für AlumniEssen**

Für die Vorträge und den anschliessenden Apéro in der Alten Universität ist keine Anmeldung erforderlich.

Das AlumniEssen findet im Restaurant *Zum Isaak* statt und ist für Mitglieder *AlumniActuarialScience* und solche, die es gerne werden wollen. Bitte anmelden bis am 27. Januar 2019 an info@*clutter*AlumniActuarialScience.ch.

*AlumniActuarialScience ***werden**

Mitglied werden in der Fachalumni Actuarial Science können ehemalige Studierende Actuarial Science/Versicherungsmathematik, Dozierende und Emeriti, SAV-Hörer*innen und alle am Studiengang interessierten Privatpersonen und Institutionen.

Weitere Informationen: AlumniActuarialScience.

]]>Cost Partitioning: Raising Prices by Lowering Costs

Tags: TAG Events DMI, TAG Events Forschung Informatik, TAG Events Forschung Mathematik, TAG Events Perlenkolloquium]]>

Dr. Florian Pommerening (Forschungsgruppe Artificial Intelligence / Prof. Dr. Malte Helmert) hat im Rahmen des diesjährigen Dies Academicus den Fakultätspreis der Phil.-Nat. Fakultät für seine Dissertation erhalten.

Im Anschluss an den Vortrag wird ein Apéro offeriert.

]]>Gibbs measures of nonlinear Schrödinger equations and many-body quantum mechanics

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Many time-dependent nonlinear Schrödinger equations admit an invariant Gibbs measure, which is a probability measure on the space of distributions that is left invariant by the time evolution. Such measures have been extensively studied as tool to construct global solutions of time-dependent nonlinear Schrödinger equations with rough initial data. I review some recent progress on deriving these measures in dimensions 1,2,3 as high-temperature limits of many-body quantum mechanics. In one dimension, I also explain how time-dependent correlation functions of the nonlinear Schrödinger equation arise as limits of corresponding quantum many-body correlation functions.

]]>Metastability results for the Navier-Stokes equations and related models

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We study diffusion and mixing in the incompressible Navier-Stokes equations and related scalar models. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequency. In turn, mixing acts to enhance the dissipative forces, giving rise to what we refer to as enhanced dissipation: this can be understood by the identification of a time-scale faster than the purely diffusive one. This talk is based on two recently obtained results: (1) a general quantitative criterion that links mixing rates (in terms of decay of negative Sobolev norms) to enhanced dissipation time-scales, and (2) a precise identification of the enhanced dissipation time-scale for the Navier-Stokes equations linearized around the Poiseuille flow, along with metastability results and nonlinear transition stability thresholds.

]]>Nonlinear Spectral Decomposition

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

For further information about the seminar, please visit this webpage.

]]>Securing a World of Physically Capable Computers

Tags: TAG Events Alumni Informatik, TAG Events DMI, TAG Events Forschung Informatik, TAG Events Forschung Mathematik, TAG Events Perlenkolloquium]]>

**Abstract:**

Computer security is no longer about data; it's about life and property. This change makes an enormous difference, and will shake up our industry in many ways. First, data authentication and integrity will become more important than confidentiality. And second, our largely regulation-free Internet will become a thing of the past. Soon we will no longer have a choice between government regulation and no government regulation. Our choice is between smart government regulation and stupid government regulation. Given this future, it's vital that we look back at what we've learned from past attempts to secure these systems, and forward at what technologies, laws, regulations, economic incentives, and social norms we need to secure them in the future.

**Speaker Bio:**

Bruce Schneier is an internationally renowned security technologist, called a security guru by the Economist. He is the author of 14 books -- including the best-seller Click Here to Kill Everybody -- as well as hundreds of articles, essays, and academic papers. His influential newsletter Crypto-Gram and blog Schneier on Security are read by over

250,000 people. Schneier is a fellow at the Berkman Klein Center for Internet and Society at Harvard University; a Lecturer in Public Policy at the Harvard Kennedy School; a board member of the Electronic Frontier Foundation, AccessNow, and the Tor Project; and an advisory board member of EPIC and VerifiedVoting.org. He is also a special advisor to IBM Security and the Chief Technology Officer of IBM Resilient.

]]>

TBA]]>

Equations with complex analytic coefficients

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Geometry inpainting with preserving singularities and curvature

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

For further information about the seminar, please visit this webpage. ]]>

Gaussian Processes for Parametric Modeling of Time Series

Tags: TAG Events Forschung Informatik, TAG Events Studium Informatik]]>

Almost homogeneous curves and surfaces

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

In this talk, I will explain how to classify the pairs (X,G) where X is a curve or a surface and G is a smooth and connected algebraic group acting on X with a dense orbit.

For curves, I will mainly focus on the regular ones, defined over an arbitrary field. Over an algebraically closed field, the "natural" notion of non-singularity is "smoothness". However, over an arbitrary field, the weaker notion of "regularity" is more suitable. I will recall the difference between those two notions and show that there exist regular homogeneous curves which are not smooth.

For surfaces, I will restrict to the smooth ones, defined over an algebraically closed field. The situation is more complicated than for curves. Moreover, new phenomena and several difficulties appear in positive characteristic, and I will highlight them. ]]>

29.11.2018 14:15 - 29.11.2018 15:15

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Tamm's Theorem for log-analytic functions]]>

Finite Energy Weak Solutions of the Navier-Stokes-Korteweg equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In this talk I will present some results concerning the analysis of finite energy weak solutions of the Navier-Stokes-Korteweg equations, which model the dynamic of a viscous compressible fluid with diffuse interface. A general theory of global existence is still missing, however for some particular cases of physical interest I will present results regarding the global existence and the compactness of finite energy weak solutions. The talk is based on a series of joint works with Paolo Antonelli (GSSI - Gran Sasso Science Institute, L’Aquila).

]]>TBA]]>

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

16:15 | Vortrag von Thomas Steiner, Basel: |

17:15 | Apéro mit Präsentation eines neuen Bandes der Euler-Edition durch das Bernoulli-Euler-Zentrum, das Departement Mathematik und Informatik der Universität Basel und Birkhäuser Basel Leonhardi Euleri Opera Omnia, Bd. IVA/8: Herausgegeben von Andreas Kleinert und Thomas Steiner |

18:15 | Vortrag von Andreas Kleinert, Halle/Saale:Eulers hallesche Korrespondenten |

]]>

Probabilistic numerical methods, Bayesian inference and multiscale inverse problems

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

In the second part of the talk, we will discuss a Bayesian approach for inverse problems involving elliptic partial differential equations with multiple scales. Computing repeated forward problems in a multiscale context is computationnally too expensive and we propose a new strategy based on the use of "effective" forward models originating from homogenization theory. Convergence of the true posterior distribution for the parameters of interest towards the homogenized posterior is established via G-convergence for the Hellinger metric. A computational approach based on numerical homogenization and reduced basis methods is proposed for an efficient evaluation of the forward model in a Markov Chain Monte-Carlo procedure.

References:

A. Abdulle, G. Garegnani, Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration, Preprint (2018), submitted for publication.

A. Abdulle, A. Di Blasio, Numerical homogenization and model order reduction for multiscale inverse problems, to appear in SIAM MMS.

A. Abdulle, A. Di Blasio, A Bayesian numerical homogenization method for elliptic multiscale inverse problems, Preprint (2018), submitted for publication.

For further information about the seminar, please visit this webpage. ]]>

22.11.2018 14:15 - 22.11.2018 15:15

On bounded automorphisms of fields with operators

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Adaptive Eigenspace Inversion for Breast Tumour Detection ]]>

Propagation of regularity and mixing estimates for solutions of the incompressible continuity equation with Sobolev vector field

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Since the work by DiPerna and Lions (1989) the continuity and transport equation under mild regularity assumptions on the vector field have been extensively studied, becoming a florid research field. The applicability of this theory is very wide, especially in the study of partial differential equations and very recently also in the field of non-smooth geometry.

The aim of this talk is to give an overview of the quantitative side of the theory initiated by Crippa and De Lellis. We address the problem of mixing and propagation of regularity for solutions to the continuity equation drifted by Sobolev fields. The problem is well understood when the vector field enjoys a Sobolev regularity with integrability exponent p>1 and basically nothing is known (at the quantitative level) in the case p=1.

We present sharp regularity estimates for the case p>1 and new attempts to attack the challenging question in the case p=1. This is a join work with Quoc-Hung Nguyen.

]]>Local law and eigenvector delocalization for supercritical Erdos-Renyi graphs]]>

Joint work with Yukun He and Matteo Marcozzi.

]]>Hyperkähler manifolds and their Betti numbers

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Optimal transport in seismic imaging based on full waveform inversion

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

For further information about the seminar, please visit this webpage.

]]>MOONSTAR - Sentinel Lymph Node Fingerprinting]]>

Oriented first passage percolation on the hypercube]]>

Positivity, Graphs and Unknotting

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Wavenumber-Explicit Analysis for High-Frequency Maxwell Equations

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

This talk comprises joint work with Prof. Markus Melenk, TU Wien.

For further information about the seminar, please visit this webpage.

]]>08.11.2018 16:15 - 08.11.2018 18:00, DMI, Spiegelgasse 1, Basel, Seminarraum 00.003 im EG

Decentralized Collective Learning for Self-managed Sharing Economies

Tags: TAG Events DMI, TAG Events Forschung Informatik, TAG Events Alumni Informatik]]>

Abstract:

The Internet of Things equips citizens with phenomenal new means for online participation in sharing economies. When agents self-determine options from which they choose, for instance their resource consumption and production, while these choices have a collective system-wide impact, optimal decision-making turns into a combinatorial optimization problem known as NP-hard. In such challenging computational problems, centrally managed (deep) learning systems often require personal data with implications on privacy and citizens’ autonomy. This paper envisions an alternative unsupervised and decentralized collective learning approach that preserves privacy, autonomy and participation of multi-agent systems self-organized into a hierarchical tree structure.

Remote interactions orchestrate a highly efficient process for decentralized collective learning. This disruptive concept is realized by I-EPOS, the Iterative Economic Planning and Optimized Selections, accompanied by a paradigmatic software artifact. Strikingly, I-EPOS outperforms related algorithms that involve non-local brute-force operations or exchange full information. This paper contributes new experimental findings about the influence of network topology and planning on learning efficiency as well as findings on techno-socio-economic trade-offs and global optimality. Experimental evaluation with real-world data from energy and bike sharing pilots demonstrates the grand potential of collective learning to design ethically and socially responsible participatory sharing economies.

]]>

08.11.2018 14:15 - 08.11.2018 15:15

Solving polynomial-exponential equations.

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

$$e^z+e^{\root 9 \of {1-z^9}}=1$$

one approach uses theta functions on ${\bf C}^{28}$. ]]>

Primitive root problems]]>

Cusp Universality for Wigner-type Random Matrices]]>

06.11.2018 12:15 - 06.11.2018 13:30, DMI, Spiegelgasse 5, Basel, Seminarraum 05.002 im 5. OG

Modeling and Individualizing Learning in Computer-Based Environments

Tags: TAG Events DMI, TAG Events Forschung Informatik, TAG Events Alumni Informatik]]>

Abstract:

Learning technologies are becoming increasingly important in today's education. This includes game-based learning and simulations, which produce high volume output, and MOOCs (massive open online courses), which reach a broad and diverse audience at scale. The users of such systems often are of very different backgrounds, for example in terms of age, prior knowledge, and learning speed. Adaptation to the specific needs of the individual user is therefore essential. In this talk, I will present two of my contributions on modeling and predicting student learning in computer-based environments with the goal to enable individualization. The first contribution introduces a new model and algorithm for representing and predicting student knowledge. The new approach is efficient and has been demonstrated to outperform previous work regarding prediction accuracy. The second contribution introduces models, which are able to not only take into account the accuracy of the user, but also the inquiry strategies of the user, improving prediction of future learning. Furthermore, students can be clustered into groups with different strategies and targeted interventions can be designed based on these strategies. Finally, I will also describe lines of future research.

]]>Around a big mapping class group

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

]]>

Transparent boundary conditions for sound propagation in human lungs, modelled by fractal trees

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

The construction of such transparent conditions relies on the approximation of the Dirichlet-to-Neumann (DtN) operator, whose symbol is a meromorphic function that satisfies a certain non-linear functional equation. We present two approaches to approximate the DtN in the time domain, alternative to the low-order absorbing boundary conditions, which appear inefficient in this case.

The first approach stems from the use of the convolution quadrature (cf. [Lubich 1988], [Banjai, Lubich, Sayas 2016]), which consists in constructing an exact DtN for a semi-discretized in time problem. In this case the combination of the explicit leapfrog method for the volumic terms and the implicit trapezoid rule for the boundary terms leads to a second-order scheme stable under the classical CFL condition.

The second approach is motivated by the Engquist-Majda ABCs (cf. [Engquist, Majda 1977]), and consists in approximating the DtN by local operators, obtained from the truncation of the meromorphic series which represents the symbol of the DtN. We show how the respective error can be controlled and provide some complexity estimates.

This is a joint work with Patrick Joly (INRIA, France) and Adrien Semin (TU Darmstadt, Germany).

For further information about the seminar, please visit this webpage.

]]>01.11.2018 14:15 - 01.11.2018 15:15

]]>

Fields of definition and abelian varieties]]>

High-dimensional Gaussian fields with isotropic increments seen through spin glasses]]>

Mathematics in drug discovery: a practitioner’s view

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Jitao David Zhang was a Marie-Curie Fellow at European Bioinformatics Institute (2007-2008) and an International PhD Programm Fellow at German Cancer Research Center (2008-2011). He obtained the doctorate degree in computational biology and biostatistics in 2011 and in that year joined Pharma Research and Early Development (pRED), the research unit of F. Hoffmann-La Roche AG in Basel, Switzerland. He supports early drug-discovery projects, develops algorithms and software for data analysis and interpretation, and collaborates to devise new technologies and platforms. His research interests include biological data modelling and integration, biological network analysis, molecular phenotyping, and applied machine learning in disease understanding and preclinical drug discovery.

]]>Spline curves in Riemannian Shape Spaces

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

For further information about the seminar, please visit this webpage.

]]>Die AlumniComputerScience werden im Anschluss an das DMI-Kolloquium mit einem Apéro eröffnet. Sie sind herzlich dazu eingeladen.

Tags: TAG Events Alumni Informatik]]>

18.10.2018 17:15 - 18.10.2018 19:00, Hörsaal 001, Kollegienhaus, Petersplatz 1, Basel

Öffentliche Antrittsvorlesungen

Tags: TAG Events DMI, TAG Events Forschung Informatik, TAG Events Forschung Mathematik, TAG Events Alumni Informatik, TAG Events Perlenkolloquium]]>

Prof. Dr. Jiří Černý, Professor für Wahrscheinlichkeitstheorie*Random Walk Percolation and Level Sets of Random Fields*

Prof. Dr. Julia Vogt, Professorin für Adaptive Systems and Medical Data Science*Machine Learning for Clinical Data Analysis and Precision Medicine*

Im Anschluss an die Vorträge findet das Kickoff der Alumni Computer Science statt und es wird ein Apéro offeriert. Der Anlass ist öffentlich.

]]>Talks to present past and current work.

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

On variational methods in Fourier analysis

Tags: TAG Events Forschung Mathematik]]>

Johann Bernoulli im Prioriätsstreit zwischen Leibniz und Newton

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

04.10.2018 14:15 - 04.10.2018 15:15

On Goormaghtigh's equation

Tags: TAG Events Zahlentheorie, TAG Events Forschung Mathematik]]>

\begin{equation}\label{eq}

\frac{x^m-1}{x-1} = \frac{y^n-1}{y-1}, \; \; y>x>1, \; m > n > 2.

\end{equation}

There are two known solutions $(x, y,m, n)=(2, 5, 5, 3), (2, 90, 13, 3)$ and it is believed that these are the only solutions. It is not known if this equation has finitely or infinitely many solutions, and not even if that is the case if we fix one of the variables. It is known that there are finitely many solutions if we fix any two variables. Moreover, there are effective results in all cases, except when the two fixed variables are the exponents $m$ and $n$. If the fixed $m$ and $n$ additionally satisfy $\gcd(m-1, n-1)>1$, then there is an effective finiteness result. My co-authors and me showed that if $n \geq 3$ is a fixed integer, then there exists an effectively computable constant $c (n)$ such that $\max \{ x, y, m \} < c (n)$ for all $x, y$ and $m$ that satisfy Goormaghtigh's equation with $\gcd(m-1,n-1)>1$. In case $n \in \{ 3, 4, 5 \}$, we solved the equation completely, subject to this non-coprimality condition. ]]>

Talks to present past and current work.

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Immanuel van Santen will speak about „Embeddings and tame automorphisms in affine geometry",

Anne Lonjou will explain some link between "Cremona group and geometric group theory", and

Jérémy Blanc will present us "Birational geometry of surfaces and threefolds".

]]>28.09.2018 12:15 - 28.09.2018 14:00, Computer Science, Spiegelgasse 1, Raum 00.003

Medical Video Analytics

Tags: TAG Events Forschung Informatik, TAG Events DMI]]>

**Abstract:**In hospitals all around the world, medical multimedia information systems have gained high importance over the last few years. One of the reasons is that an increasing number of interventions is performed in a minimally invasive way. These endoscopic inspections and surgeries are conducted with a tiny camera – the endoscope – which enables clinicians to control the intervention via an external display. Apart from the viewing purpose, the video signal can also be recorded and used for post-procedural scenarios, such as communicating operation techniques (i.e., training and teaching), planning future interventions, and medical forensics and analytics. The problem, however, is the sheer amount of unstructured video data that is added to the multimedia archive on a daily basis. Without proper management and content analytics the videos cannot be used efficiently by clinicians. In this talk I will summarize our recent investigations in this challenging research field and conclude on the achieved results.

**Bio:**Dr. Klaus Schöffmann is an Associate Professor in the Distributed Multimedia Systems research group at the Institute of Information Technology (ITEC) at Klagenfurt University, Austria. He received his PhD in 2009 and his habilitation (venia docendi) in 2015, both in computer science and from Klagenfurt University. His research focuses on video analytics and interactive multimedia systems, particularly in the medical domain. He has co-authored more than 100 publications on various topics in multimedia, inclusive of more than 25 on different aspects of medical multimedia systems. He has co-organized several international conferences, workshops, and special sessions in the field of multimedia. Furthermore, he is co-founder of the Video Browser Showdown (VBS) – an international live evaluation competition of interactive video search. He is a member of the IEEE and the ACM, and a regular reviewer for international conferences and journals in the field of multimedia. Klaus Schöffmann teaches various courses in computer science, including mobile app development, video retrieval, distributed multimedia systems, and operating systems.

Tags: TAG Events Forschung Informatik]]>

27.09.2018 14:15 - 27.09.2018 15:15

Extreme values of the Riemann Zeta function, and log-correlated Gaussian fields ]]>

From Biology to Polynomials and Beyond: A Brief Look at Algebraic Statistics

Tags: TAG Events Forschung Mathematik]]>

Talks to present past and current work.

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Julia will speak about "A_k-singularities of plane curves of fixed bidegree".

Egor will answer to "What transformation groups in algebraic, differential and metric geometry have in common?".

Philipp will speak about "Algebraic Statistics: Gaussian Mixtures and Beyond".

]]>Tags: TAG Events Forschung Mathematik, TAG Events Perlenkolloquium]]>

Wasserstein metric based full waveform inversion

Tags: TAG Events Forschung Mathematik]]>

The Ginibre ensemble and Gaussian multiplicative chaos]]>

(Joint work in progress with Paul Bourgade and Guillaume Dubach).

]]>Der Vorkurs dient zur Vorbereitung des Studiums und erlaubt es, Mathematikkenntnisse aus der Schule aufzufrischen und allfällige Lücken zu erkennen. Er ist fakultativ und richtet sich vor allem an Studienanfängerinnen und Studienanfänger der Philosophisch-Naturwissenschaftlichen Fakultät. Weitere Details finden Sie auf unserer Webseite unter Studium.]]>

Ubiquity of phases in some percolation models with long-range correlations

Tags: TAG Events Forschung Mathematik]]>

This talk is based on joint works with A. Prévost (Köln) and P.-F. Rodriguez (Bures-sur-Yvette).

]]>13.07.2018 10:45, Fabrizio Barroero, Philipp Habegger, Alte Universität U101

Tags: TAG Events Forschung Mathematik, TAG Events Zahlentheorie, TAG Events DMI]]>

03.07.2018 18:30 - 03.07.2018 19:30, Congress Center Basel, Montreal Room

Tags: TAG Events DMI, TAG Events Forschung Informatik, TAG Events Forschung Mathematik]]>

Emerging real-world graph problems include: detecting and preventing disease in human populations; revealing community structure in large social networks; and improving the resilience of the electric power grid. Unlike traditional applications in computational science and engineering, solving these social problems at scale often raises new challenges because of the sparsity and lack of locality in the data, the need for research on scalable algorithms and development of frameworks for solving these real-world problems on high performance computers, and for improved models that capture the noise and bias inherent in the torrential data streams. In this talk, Bader will discuss the opportunities and challenges in massive data-intensive computing for applications in social sciences, physical sciences, and engineering.

David Bader is Professor and Chair of the School of Computational Science and Engineering at Georgia Institute of Technology, and is regarded as one of the world’s leading experts in data sciences. His interests are at the intersection of high performance computing (HPC) and real-world applications, including cybersecurity, massive-scale analytics, and computational genomics. Bader has co-authored over 200 articles in peer-reviewed journals and conferences, and is an associate editor for high-impact publications including IEEE Transactions on Computers, ACM Transactions on Parallel Computing, and ACM Journal of Experimental Algorithmics. He is a Fellow of the IEEE and AAAS, and has served on a number of advisory committees in scientific computing and cyber-infrastructure, including the White House's National Strategic Computing Initiative. Bader has served as a lead scientist in several DARPA programs and is a co-founder of the Graph500 list, a rating of "Big Data" computing platforms. He was recognized as a “Rock Star of HPC” by InsideHPC and as HPCwire's “People to Watch” in 2012 and 2014.

More details:

pasc18.pasc-conference.org/program/public-lecture/

]]>

02.07.2018 00:00 - 04.07.2018 00:00, Computer Science, Congress Center Basel

Tags: TAG Events DMI, TAG Events Forschung Informatik, TAG Events Forschung Mathematik]]>

The PASC18 theme is

The PASC18 technical program (pasc18.pasc-conference.org/program/schedule/) is organized around eight scientific domains: Chemistry and Materials, Climate and Weather, Computer Science and Applied Mathematics, Emerging Domains in HPC, Engineering, Life Sciences, Physics, and Solid Earth Dynamics. Over the course of three full days, the program includes, technical sessions with keynotes, panels, minisymposia, peer-reviewed papers and poster sessions. Full papers are published in the ACM Digital Library.

We look forward to advancing scientific computing with you at the Basel Congress Center on July 2-4!

]]>

Tags: TAG Events Forschung Mathematik, TAG Events Perlenkolloquium]]>

21.06.2018 10:00 - 21.06.2018 12:00, Computer Science, Spiegelgasse 1, Seminar Room 00.003 (ground floor)

Advances in the TAU Performance System and the Need for Dynamic Performance Introspection, In Situ Analysis, and Adaptation

Tags: TAG Events Forschung Informatik, TAG Events DMI]]>

**Abstract:** Since the beginning of ``high-performance'' parallel computing, observing

and analyzing performance for purposes of finding bottlenecks and

identifying opportunities for improvement has been at the heart of

delivering the performance potential of next-generation scalable systems.

Interestingly, it is the ever-changing parallel computing landscape that is

the main driver of requirements for parallel performance technology and the

improvements necessary beyond the current state-of-the-art. Indeed, the

development and application of our TAU Performance System over many years

largely follows an evolutionary path of addressing measurement and analysis

problems in new parallel machines and programming environments.

However, the outlook to future parallel systems with high degrees of

concurrency, heterogeneous components, dynamic runtime environments,

asynchronous execution, and power constraints suggests a new perspective

will be needed on the role of performance observation and analysis in

respect to tool technology integration and performance optimization

methods. The reliance on post-mortem analysis of application-level

("1st person") performance measurements is prohibitive for

exascale-class machines because of the performance data volume, the

primitive basis for performance data attribution, and the fundamental

problem of performance variation that will exist. Instead, it will be

important to provide introspection support across the exascale software

stack to understand how system ("3rd person") resources are used during

execution. Furthermore, the opportunity to couple a global performance

introspection capability (a "performance backplane") with online

performance decision analytics inspires the concept of an autonomic

performance system that can feed back policy-based decisions to guide

the computation to better states of execution.

The talk will explore these issues by giving a brief retrospective on

performance tool evolution, setting the stage for current research

projects where a new performance perspective is being pursued. It will

also speculate on what might be included in next-generation parallel

systems hardware, specifically to make the exascale machines more

performance-aware and dynamically-adaptive. **Speaker Bio: **Allen D. Malony is a Professor in the Department of Computer and

Information Science at the University of Oregon (UO) where he directs

parallel computing research projects, notably the TAU parallel

performance system project. He has extensive experience in performance

benchmarking and characterization of high-performance computing systems,

and has developed performance evaluation tools for a range of parallel

machines during the last 30 years. Malony is also interested in

computational and data science. He is the Director of the new Oregon

Advanced Computing Institute for Science and Society (OACISS) at UO.

Malony was awarded the NSF National Young Investigator award, was a

Fulbright Research Scholar to The Netherlands and Austria, and received

the Alexander von Humboldt Research Award for Senior U.S. Scientists by

the Alexander von Humboldt Foundation. Last year he was the

Fulbright-Tocqueville Distinguished Chair to France. Recently, he was

award the Fulbright for the Future by the Franco-Americaine Fulbright

Commission. Malony is the CEO of ParaTools, Inc., which he founded with

Dr. Sameer Shende in 2004. ParaTools SAS is a French company they

started in 2014, and is wholly-owned company by ParaTools, Inc.

ParaTools specializes in performance analysis and engineering, HPC

applications and optimization, and parallel software, hardware, and

tools.

How and why to pack in big dimensions

Tags: TAG Events Forschung Mathematik, TAG Events Forschung Informatik, TAG Events DMI, TAG Events Perlenkolloquium]]>

In this talk we will speak about packing problems in multidimensional spaces. We will give an overview of this interesting class of mathematical questions and recent progress in the area. Also we will explain the importance of good packing configurations in applications such as internet, telecommunication, and data storage.

The talk is part of the annual meeting 2018 of the Swiss Mathematical Society.

]]>Speakers: Nicolas Bergeron, Gal Binyamini, Valentin Blomer, Ana Caraiani, Laura DeMarco

Tags: TAG Events Forschung Mathematik]]>

On energy-stable discontinuous Galerkin approximations for scattering problems in complex elastic media with adaptive curvilinear meshes

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

The maximal number of lines going through the same point on del Pezzo surfaces of degree one]]>

Abelian quotients of the Cremona groups

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Localization of orthonormal functions in spectral clusters

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Localization of eigenfunctions via an effective potential

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Lu = -div(A∇u) + Vu

on a Lipschitz domain Ω and, more generally, on a manifold with or without boundary. The eigenfunctions of L are often localized, as a result of disorder of the potential V , the matrix of coefficients A, irregularities of the boundary, or all of the above. In earlier work, Filoche and Mayboroda introduced the function u solving Lu = 1, and showed numerically that it strongly reflects this localization. Here, we deepen the connection between the eigenfunctions and this landscape function u by proving that its reciprocal 1/u acts as an effective potential. The effective potential governs the exponential decay of the eigenfunctions of the system and delivers information on the distribution of eigenvalues near the bottom of the spectrum.

]]>Optimal transport in seismic imaging based on full waveform inversion

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

24.05.2018 15:00 - 24.05.2018 16:00, Computer Science, Spiegelgasse 1, Room 05.013, 5th floor

From Location to Location Pattern Privacy

Tags: TAG Events Forschung Informatik]]>

**Abstract:** Location privacy is getting increased attention due to mass use of Location Based Services (LBSs). Location privacy aims at protecting individuals' exact location from LBS providers. For this purpose, sharing coarse locations respecting users' location privacy profiles are commonly applied. The problem gets more complicated especially in urban areas where user movements are restricted on roads and the visited places are semantically annotated like bars, night clubs etc. This way location can act as an intrusive inference on people's personal preferences and habits. So, location privacy is not confined to privacy of location itself. Moreover, any sensitive pattern in users' trajectories can be resurfaced by data mining.

In this talk, especially for urban area mobility, I will give how location privacy profiles can be defined and maintained w.r.t. possible attack models. Then, I will introduce temporally annotated sequences as a means to define sensitive location patterns. Finally, a sensitive location pattern suppression mechanism running in online stream fashion of LBS access will be detailed. **Speaker Bio:** Osman Abul is currently an associate professor of computer science at TOBB University of Economics and Technology, Ankara, Turkey. He received his PhD degree in computer engineering from Middle East Technical University, Ankara, Turkey. He held visiting posts in University of Calgary, Norwegian University of Science and Technology, and Italian Institute of Information Science and Technology. His research interests include data mining, data privacy and bioinformatics.

On the N-body problem and the Vlasov-Poisson equation]]>

Automorphism groups of Danielewski surfaces (TALK CANCELLED)

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Machine Learning Driving Innovative Computing Concepts

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

The study of dynamical systems has a long history. Many people consider Henri Poincaré as the founder (at the end of the 19th century) of the modern theory of dynamical systems. Further important contributions at the beginning of the 20th are due to George David Birkhoff, Gaston Julia and Pierre Fatou.

In the talk we will discuss some aspects of the theory of dynamical systems with a view toward arithmetic. As an example we will show that Birkhoff's multiple recurrence Theorem can be used to provide an alternative proof of van der Waerden's Theorem on arithmetic progressions. We will also show that Julia sets are useful to understand the dynamic given by a rational function. ]]>

An introduction to Cayley-Bacharach theorems]]>

Complex plane curves, their intersection with round spheres, and knot concordance

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Noetherianity up to conjugation

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

An Adaptive Multiscale Approach for Electronic Structure Methods

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Two-level Domain Decomposition preconditioners for Maxwell equations

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

[1] I. G. Graham, E. A. Spence, and E. Vainikko. Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption. Mathematics of Computation, 86(307):2089–2127, 2017.

]]>Percolation on the Erdös-Renyi model]]>

Isomorphisms between complements of unicuspidal curves in the projective plane

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The Mordell-Weil Theorem and the rank of elliptic curves]]>

Rational simple connectedness for Fano varieties

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In the current joint project with Laurent Gruson and Nicolas Perrin, we study some examples of Fano varieties in low dimension via explicit birational methods. ]]>

Mesh Refinement for T-splines in any dimension

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

As an outlook to future work, we outline an approach for the handling of zero knot intervals and multiple lines in the interior of the domain, which are used in CAD applications for controlling the continuity of the spline functions, and we also sketch basic ideas for the local refinement of two-dimensional meshes that do not have tensor-product structure.

]]>SPDE's and regularity structures]]>

Some affine plane bundles over the punctured affine plane

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The famous Dolgachev-Weisfeiler conjecture predicts that such fibrations are in fact isomorphic to the trivial bundle. We will show that this holds true in some particular examples. For instance, we will recover a result of Drew Lewis which states that the $\mathbb{A}^2$-fibration induced by the second Vénéreau polynomial is trivial.

Our proof is inspired by a previous work of Kaliman and Zaidenberg and consists in first showing that the considered fibrations have a fiber bundle structure when restricted over the punctured affine plane.

This is a joint work in progress with Jérémy Blanc. ]]>

Optimal adaptivity in finite and boundary element methods

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

In this talk, we first consider an adaptive FEM with hierarchical splines of arbitrary degree for linear elliptic PDE systems of second order with Dirichlet boundary condition for arbitrary dimension d≥2. We assume that the problem geometry can be parametrized over the d-dimensional unit cube. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted-residual a posteriori error estimator. In [1], we proved linear convergence of the error estimator with optimal algebraic rate.

Next, we consider an adaptive BEM with hierarchical splines of arbitrary degree for weakly-singular integral equations of the first kind that arise from the solution of linear elliptic PDE systems of second order with constant coefficients and Dirichlet boundary condition. We assume that the boundary of the geometry is the union of surfaces that can be parametrized over the unit square. Again, we propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions, where adaptivity is driven by some weighted-residual a posteriori error estimator. In [2], we proved linear convergence of the error estimator with optimal algebraic rate. In contrast to prior works, which are restricted to the Laplace model problem, our analysis allows for arbitrary elliptic PDE operators of second order with constant coefficients.

Finally, for one-dimensional boundaries, we investigate an adaptive BEM with standard splines instead of hierarchical splines. We modify the corresponding algorithm so that it additionally uses knot multiplicity increase which results in local smoothness reduction of the ansatz space. In [3], we proved linear convergence of the employed weighted-residual error estimator with optimal algebraic rate.

[1] G. Gantner, D. Haberlik, and Dirk Praetorius, Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines. Math. Mod. Meth. in Appl. S., Vol. 27, 2017.

[2] G. Gantner, Optimal adaptivity for splines in finite and boundary element methods, PhD thesis, TU Wien, 2017.

[3] Michael Feischl, Gregor Gantner, Alexander Haberl, and Dirk Praetorius. Adaptive 2D IGA boundary element methods. Eng. Anal. Bound. Elem., Vol. 62, 2016.

]]>Tags: TAG Events Forschung Mathematik, TAG Events Perlenkolloquium]]>

Automorphisms of pointless surfaces

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We will also discuss some particular types of surfaces with interesting automorphism groups, including Severi-Brauer surfaces. ]]>

Shape Optimization by Finite Elements]]>

Non-rationality of conic-bundles over IP^3

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

27.02.2018 12:15 - 27.02.2018 14:00, Computer Science, Spiegelgasse 5, Seminarraum 05.002, 5. OG

The Pathfinding Engine of Dragon Age: Origins

Tags: TAG Events Forschung Informatik]]>

**Abstract:** What does it take to build a high-quality pathfinding engine? One might think that pathfinding in games is a simple case of using A*, which finds shortest paths. But, in practice, there are many other considerations for finding high-quality paths quickly in dynamic environments. This talk will give an overview of the techniques required to build the pathfinding engine of Dragon Age: Origins. Five iterations of improvements will be presented, starting from basic pathfinding approaches, and moving to a final system that creates high-quality, smooth paths. To conclude, we will discuss open challenges in pathfinding and what it would take to create a pathfinding system that finds human-quality paths.

**Speaker Bio:** Nathan Sturtevant is an Associate Professor in the Computer Science Department at the University of Denver. His scientific research focuses on search in Artificial Intelligence. This includes work on heuristic and combinatorial search for single and multiple agents, including bidirectional search, automated abstraction, heuristics, refinement search, search for game design, heuristic learning, inconsistent heuristics, cooperative search, large-scale and parallel search.

Particular applications include pathfinding and planning in memory-constrained real-time environments (e.g. commercial video games) as well as algorithms for building and using memory-based heuristics via large-scale search. Other work considers theoretical and practical issues in games with more than two players, including opponent modeling, learning, and imperfect information.

Cremona contractibility]]>

Birational geometry of pairs

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Sparse grids in optimal control]]>

Regularity of free boundaries in obstacle problems for integro-differential operators

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We present a brief overview of the regularity theory for free boundaries in different obstacle problems. We describe how a monotonicity formula of Almgren plays a central role in the study of the regularity of the free boundary in some of these problems. Finally, we explain new strategies which we have recently developed to deal with cases in which monotonicity formulas are not available.

]]>Quantitative estimates of propagation of chaos for stochastic systems with singular interactions

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. We have to develop for this new laws of large numbers at the exponential scale. But our result only requires very weak regularity on the interaction kernel in the negative Sobolev space W^{-1,∞}, thus including the Biot-Savart law and the point vortices dynamics for the 2d incompressible Navier-Stokes.

Tidal energy in Newtonian two-body motion

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In this work, which is based on an essential linear analysis by Christodoulou, we study the tidal energy for the motion of two gravitating incompressible fluid balls with free boundaries, obeying the Euler-Poisson equations. The orbital energy is defined as the mechanical energy of the center of mass of the two bodies. When the fluids are replaced by point masses, according to the classical analysis of Kepler and Newton, the conic curve describing the trajectories of the bodies is a hyperbola when the orbital energy is positive and an ellipse when the orbital energy is negative. If the point masses are initially very far, then the orbital energy, which is conserved in the case of point masses, is positive corresponding to hyperbolic motion. However, in the motion of fluid balls the orbital energy is no longer conserved, as part of the conserved energy is used in deforming the boundaries of the bodies. This energy is called the tidal energy. If the tidal energy becomes larger than the total energy during the evolution, the orbital energy must change its sign, signaling a qualitative change in the orbit of the bodies. We will show that under appropriate conditions on the initial configuration this change of sign occurs. Our analysis relies on an a-priori estimates which we establish up to the point of closest approach.

(This is a joint work with Sohrab Shahshahani from UMass Amherst.)

]]>McKay Correspondence]]>

Dirac operators with magnetic links

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We consider Dirac operators on the 3-sphere with singular magnetic fields which are supported on links, that is on one-dimensional manifolds which are diffeomorphic to finitely many copies of S^{1}. Each connected component carries a flux 2πα which exhibits a 2π-periodicity, just like Aharonov-Bohmsolenoids in the complex plane. We study the kernel of such operators through the spectral flow of loops corresponding to tuning some flux from 0 to 2π, that is the number of eigenvalues crossing 0 along the loop (counted algebraically). It turns out that the spectral flow is generically non-zero and depends on the shape of the curves and their linking number. Through the stereographic projection the result extends to R^{3}. And then by smearing out the magnetic fields we obtain new solutions (ψ,A) to the zero-mode equation on R^{3}:

σ·(-i∇+A)=0,

(ψ,A) ∈ H^{1}(R^{3})^{2} × \dot{H}^{1}(R^{3})^{3 }∩ L^{6}(R^{3})^{3},

where σ=(σ)_{j=1...3} denotes the family of the Pauli matrices, A is the magnetic potential associated to the magnetic field ∇×A, and σ⋅(-i∇+A) is the corresponding Dirac operator in R^{3}.

(Joint work with Fabian Portmann and Jan Philip Solovej)

]]>Signature morphisms of the Cremona group of the plane

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We consider the motion of a rigid body due to the pressure of a surrounded two-dimensional irrotational perfect incompressible fluid, the whole system being confined in abounded domain with an impermeable condition on a part of the external boundary. Thanks to an impulsive control strategy we prove that there exists an appropriate boundary condition on the remaining part of the external boundary (allowing some fluid going in and out the domain) such that the immersed rigid body is driven from some given initial position and velocity to some final position and velocity in a given positive time, without touching the external boundary. The controlled part of the external boundary is assumed to have a nonvoid interior and the final position is assumed to be in the same connected component of the set of possible positions as the initial position.

]]>Aspects of Adaptive Galerkin FEM for random PDEs

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Better bounding]]>

Fibre-like Fano manifolds: a bestiary

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Patterns formation for minimizers of a local/non-local interaction functional in general dimension

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We study a functional consisting of a perimeter term and a non-local term which are in competition, both in the discrete and continuous setting.In the discrete setting such functional was introduced by Giuliani, Lebowitz, Liebe and Seiringer. For both the continuous and discrete problem, we show that the global minimizers are exact periodic stripes. One striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one dimensional. Such behaviour for a smaller range of exponents in the discrete setting had been already shown,using different techniques. This is a joint work with E. Runa.

]]>A Mixed Finite-Element Method for Gas Transport on Pipeline Networks

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

The beauty of elliptic curves]]>

Characterisation of varieties by their automorphisms

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Let X be a quasi-affine irreducible variety such that Aut(X) \simeq Aut(A

(1) X is a Q-acyclic open subset of a smooth affine rational variety, and dim(X) is a most equal to n;

(2) X is toric and dim(X) is at least equal to n. After giving a brief history on some related results that concern the characterisation of geometric objects via their automorphisms, we give the key ideas of the proof of our main result. ]]>

The Porous Medium Equation with Large Initial Data on Negatively Curved Riemannian Manifolds

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We discuss existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan–Hadamard manifolds satisfying suitable lower bounds on Ricci curvature, with initial data that can grow at inﬁnity at a prescribed rate, that depends crucially on the curvature bounds. Furthermore, we give a precise estimate for the maximal existence time, and we show that in general solutions do not exist if the initial data grow at inﬁnity too fast. Such results have been recently obtained jointly with G. Grillo and M. Muratori.

]]>Introduction to Deep Reinforcement Learning for Non-Machine-Learners]]>

On birational transformations of P3 of low degree

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

SQP Methods for Shape Optimization Based on Weak Shape Hessians

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

The talk introduces shape optimization as a special sub-class of PDE constraint optimization problems. The main focus here will be on generating Newton-like methods for large scale applications. The key for this endeavor is the derivation of the shape Hessian, that is the second directional derivative of a cost functional with respect to geometry changes in a weak form based on material derivatives instead of classical local shape derivatives. To avoid human errors, a computer aided derivation system is also introduced.

The methodologies are tested on problem from fluid dynamics and geometric inverse problems.

]]>Matrix Factorizations and Chern classes for modules with (eventually) periodic resolutions]]>

Geometry of the fibres of some integrable systems]]>

Commensurating actions of birational groups

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Small G-varieties

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

An affine G-variety X is called

A striking consequence is the following result.

Theorem. Let n > 4. Then a smooth $\SL_n$-variety of dimension d < 2n-2 is an $\SL_n$-vector bundle over a smooth variety of dimension d-n. There are also interesting applications to actions of the affine group $\Aff_n$. This was the starting point of this joint work with with Andriy Regeta and Susanna Zimmermann. ]]>

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Optimizing strategies for releasing sterile mosquitoes]]>

Cremona group and hyperbolic spaces

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Del Pezzo fibrations in positive characteristic

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Some remarks on Poonen's conjecture

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We are interested in cycles of φ

An extension problem for Riemannian metrics

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The problem of extending a given metric on a finite three dimensional domain to one which is asymptotically flat and which satisfies the constraint equations of general relativity was proposed by Robert Bartnik. He proposed that the minimal mass of such an extension would be a measure of the quasi-local mass of the domain and this is called the Bartnik mass. In this talk we will describe this problem and recent results on it including a comparison of the Bartnik mass to other quasi-local mass notions.

]]>A fully non-linear flow with surgeries for 2-convex hyper-surfaces

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The lecture describes joint work with Simon Brendle on the motion of closed hyper surfaces in a Riemannian manifold, on which the sum of any two principal curvatures is positive at all points. We exhibit a fully non-linear, parabolic choice of deformation law that preserves this curvature condition and allows only neck-like singularities which can be overcome by finitely many surgeries. The lecture explains the analytical techniques involved and sharp geometric conclusions made possible by this flow.

]]>Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

A Family of 65-Nodal Sextics]]>

Pseudotrees and Not Getting Slapped Too Often]]>

Quantum Machine Learning (QML)

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Stochastic two-dimensional decomposition of high-dimensional solution spaces for robust design]]>

Finite Energy Weak Solutions of the Quantum Navier-Stokes equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In this talk I will present a result concerning the global existence of finite energy weak solutions of the quantum Navier-Stokes equations. The novelty of the result is that we are able to consider the vacuum in the definition of weak solutions. The main tools are a new formulation of the equations which allows us to get an additional a priori estimate to prove compactness and a non trivial choice of the approximation system consistent with the a priori estimates.

This is a joint work with Paolo Antonelli (GSSI)

]]>Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Rational and non-rational varieties

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

For example, non-singular plane cubic curves (elliptic curves) cannot be rationally parametrized. In dimension one and two, the problem of existence of such parametrization is completely solved.

In higher dimensions, the picture is less clear. In this talk, I will survey know results and open problems in this direction.

]]>Local time discretisation based on transmission problem for linear wave equations: construction and analysis

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Igusa quartic and and Wiman-Edge sextics

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Arnaud Beauville proved that all but four of these quartic threeffolds are irrational. Later Cheltsov and Shramov proved that the remaining threefolds in this pencil are rational. In this talk, I will give an alternative prove of both these results. To do this, I will describe Q-factorizations of the double cover of the four-dimensional projective space branched over the Igusa quartic, which is known as Coble fourfold. Using this, I will show that $\mathbb{S}_6$-symmetric quartic threefolds are birational to conic bundles over quintic del Pezzo surfaces whose degeneration curves are contained in the pencil studied by Wiman and Edge.

This is a joint work with Alexander Kuznetsov and Constantin Shramov (Moscow). ]]>

Mathematical modelling of transmission dynamics of Opisthorchis viverrini]]>

Nonlocal obstacle problems: regularity of the solutions and the free boundaries

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We will introduce some recent results in collaboration with L. Caffarelli and X. Ros-Otonon the optimal regularity of the solutions and the regularity of the free boundaries (near regular points) for nonlocal obstacle problems.The main novelty is that we obtain results for different operators than the fractional Laplacian. Indeed, we can consider infinitesimal generators of non rotationally invariant stable L ́evy processes.

]]>From noncommutative motivic measures to subgroups of the Cremona group

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

I will introduce the Grothendieck ring of triangulated categories, and show how, using Bondal-Larsen-Lunts motivic measure, a subgroup of such ring will define a subgroup of the group Bir(X) of birational self-maps of X. A main example is given by the filtration via the motivic dimension, which induces a filtration on Bir(X). As a consequence, in the case X=P

Blowup for NLS]]>

Small volume asymptotics for elliptic equations and their use in impedance tomography

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Geometry of the moduli of parabolic bundles on elliptic curves

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The former moduli space is itself related to the moduli space of rank 2 parabolic bundles over a 5-punctured P

The Arrow of Time

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

recap of properties of (relative) entropy, the following examples of irreversible behavior are discussed:

(1) The second law of thermodynamics is discussed within quantum statistical mechanics.

(2) It is indicated how Brownian motion of a quantum particle immersed in a heat bath can be derived from unitary quantum dynamics of the total system.

(3) Motion with friction is discussed in an example of Hamiltonian evolution equations for a particle moving through a dispersive wave medium.

(4) It is sketched that and why quantum dynamics is fundamentally irreversible. ]]>

Curves and moduli]]>

Blowup for Fractional NLS

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In this talk, we give sufficient criteria for blowup of solutions to nonlocal Schrödinger equations with focusing power-type nonlinearity. To give an outline of the arguments used in the proof, let us mainly focus on the mass-supercritical problem posed on the whole space R^{n} with prescribed radial initial datum of negative energy.

This is a joint work with Thomas Boulenger and Enno Lenzmann

]]>Liouville type equation on R^n]]>

IGA BEM for Maxwell eigenvalue problems

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

To live up to the promises of IGA, namely closing the gap bewteen design and analysis, we suggest an IGA boundary element method (BEM). We will review the state-of-the-art of all relevant building blocks. We will address the B-spline de Rham complex on a boundary manifold, the Galerkin discretization of the electric field integral equation, and present a convergence result. We will discuss a recent contour integral method [1] to solve the resulting non-linear eigenvalue problem. Aspects of integrating so-called ”fast methods” will also be presented, in particular Adaptive Cross Approximation [5] and Calderón preconditioning [4].

[1] W.-J. Beyn. An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl, 436(10):3839–3863, 2012.

[2] A. Buffa, G. Sangalli, and R. Vázquez. Isogeometric analysis in electromagnetics: B-splines approximation. Comput Method Appl M, 199:1143–1152, 2010.

[3] J. Corno, C. de Falco, H. De Gersem, and S. Schöps. Isogeometric simulation of Lorentz detuning in superconducting accelerator cavities. Comput Phys Commun, 201:1–7, February 2016.

[4] J. Li, D. Dault, B. Liu, Y. Tong, and B. Shanker. Subdivision based isogeometric analysis technique for electric field integral equations for simply connected structures. J Comput Phys, 319:145–162, 2016.

[5] B. Marussig, J. Zechner, G. Beer, and T.-P. Fries. Fast isogeometric boundary element method based on independent field approximation. Comput Method Appl M, 284:458–488, 2015.

The work of Stefan Kurz is supported by the ’Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt.

]]>Recent approaches to significant pattern mining in genetics]]>

Rademacher’s Theorem, Cheeger’s Conjecture and PDEs for measures

Tags: TAG Events Forschung Informatik, TAG Events DMI]]>

The classical Rademacher Theorem asserts that every Lipschitz function is differentiablea lmost everywhere with respect to Lebesgue measure. On the other hand, Preiss (’90) gave a surprising example of a nullset in the plane such that every Lipschitz function is differentiable at at least one point of this set. Thus, it is a natural question to ask whether there exists a singular measure such that all Lipschitz functions are differentiable with respect to this singular measure. It turns out that this question has an intricate connection to the geometric structure of normal one-currents. In this talk I will present a converse to Rademacher’s Theorem, which settles the question in the negative in all dimensions: if a positive measure μ has the property that all Lipschitz functions are μ-a.e. differentiable, then μ is absolutely continuous with respect to Lebesgue measure (in the plane, this question was already solved by Alberti, Csornyei and Preiss in ’05). In a geometric context, Cheeger conjectured in ’99 that in all Lipschitz differentiability spaces (which are essentially Lipschitz manifolds in which Rademacher’s Theorem holds) likewise there is a “functional converse” to Rademacher’s Theorem. As the second main result, I will present a recent solution to this conjecture.Technically, the proofs of both of these theorems are based on a recent structure result for the singular parts of PDE-constrained measures, its corollary on the structure of normalone-currents, and the powerful theory of Alberti representations.

This is a joint work with A. Marchese and G. De Philippis

]]>Towards hyperelliptic curve cryptography]]>

Betti numbers and pseudo effective cones in 2-Fano varieties

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Fractional div-curl quantities and applications to nonlocal geometric equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractional div-curl quantities. We demonstrate how these quantities appear naturally in nonlocal geometric equations, which can be used to obtain a regularity theory for fractional harmonic maps and critical systems with nonlocal antisymmetric potential.

This is a joint work with Armin Schikorra

]]>Optimal integration in reproducing kernel Hilbert spaces

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Uncertainty Quantification for PDEs with Anisotropic Random Diffusion]]>

Hyperbolicity of moduli spaces of abelian varieties

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Hilbert's third problem]]>

Partial Regularity for Holonomic Minimisers of Quasiconvex Functionals

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We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Several applications to variational problems in condensed matter physics with broken symmetries will also be discussed related to the manifold constraint condition.

]]>Orthogonal tensor decomposition from an algebraic perspective

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

(Joint work with Ada Boralevi, Emil Horobet, and Elina Robeva.) ]]>

Self-organized dynamics: from emergence of consensus to social hydrodynamics

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Rationale Kurven in P^3 – Flächen und Sekanten ]]>

A Morse index formula for the Lane-Emden problem

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We consider the semilinear Lane-Emden problem (E_{p}):

-Δu = |u|^{p-1}u in B

u = 0 on ∂B

where B is the unit ball of R^{N}, N≥3, centered at the origin and 1< p < p_{S}, p_{S}=(N+2)/(N−2). We compute the Morse index of any radial solutionup of (E_{p}), for p sufficiently close to p_{S}. The proof exploits the asymptotic behavior of u_{p} as p→p_{S} and the analysis of a limit eigenvalue problem.

This is a joint work with F. De Marchis and F. Pacella

]]>Integral points on polygons and modular forms]]>

The space of Kähler metrics on singular varieties

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

This is based on a joint work with Vincent Guedj. ]]>

Reproducing kernel methods

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Furthermore, we will present a deterministic a priori (often exponential) convergence analysis via sampling inequalities which can be employed to analyze a large class of regularized reconstruction schemes.

Such an analysis enables us to derive a priori couplings of various discretization and regularization parameters. Such parameters can range from iteration numbers in numerical linear algebra, numerical evaluation of input parameters to rounding errors.

An important issue is the choice of the reproducing kernel. We will discuss some implications of such choices and address the problem of approximating the solution of a parametric partial differential equation using problem adapted kernels.

This is partly based on joint work with M. Griebel and B. Zwicknagl (both Bonn University).

]]>On projectivity of the moduli space of stable surfaces in characteristic p > 5

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Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Variational aspects of Liouville equations and systems on compact surfaces

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A class of Liouville equations and systems on compact surfaces is considered: we focus on the Toda system which is motivated in mathematical physics by the study of models in non-abelian Chern-Simons theory and in geometry in the description of holomorphic curves in complex analysis. We discuss its variational aspects which yield existence results.

]]>Curves on a surface - How they self-intersect and why this number can be less than zero]]>

Numerical homogenization beyond periodicity and scale separation with applications to wave propagation in heterogeneous media

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A knot theorists approach to singularities and their deformations

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About irreversibility in rarefied gas dynamics

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Quasi-Optimal Schwarz Domain Decomposition Methods for Time Harmonic Waves

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Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

From condensed matter physics to probability theory

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The basic laws governing atoms and electrons are well understood, but it is impossible to make predictions about the behaviour of large systems of condensed matter physics. A popular approach is to introduce simple models and to use notions of statistical mechanics. I will review quantum spin systems and their stochastic representations in terms of random permutations and random loops. I will also describe the ‘universal’ behaviour that is common to loop models in dimensions 3 and more.

]]>Mathematical Modelling of Rabies Elimination in N’Djamena, Chad]]>

An Adaptive Wavelet Method for the Solution of Boundary Integral Equations in Three Dimensions

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Groundstate solutions for a nonlinear Choquard equation

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

I will discuss the existence of groundstate solutions for the Choquard equation in the whole space R^{N}. I will first consider the case of a homogeneous nonlinearity F(u) =|u|^{p}, then I will prove the existence of solutions under general hypotheses. In particular, the cases N=2 and N≥3 will have to be treated differently. The solutions are found through a variational mountain pass strategy.

Arithmetic via Hyperelliptic Curves]]>

Astronomica im Briefwechsel zwischen Leonhard Euler und Daniel Bernoulli

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Algorithmic patterns for hierarchical matrices on many-core processors

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

We are interested to use these many-core processors for the full H-matrix construction and application process. A motivation for this interest lies in the well-known claim that future standard processors will evolve towards many-core hardware, anyway. In order to be prepared for this development, we want to discuss many-core parallel formulations of classical H-matrix algorithms and adaptive cross approximations.

In the presentation, the use of H-matrices is motivated by the model application of kernel-based approximation for the solution of parametric PDEs, e.g. PDEs with stochastic coefficients. The main part of the talk will be dedicated to the challenges of H-matrix parallelizations on many-core hardware with the specific model hardware of GPUs. We propose a set of parallelization strategies which overcome most of these challenges. Benchmarks of our implementation are used to explain the effect of different parallel formulations of the algorithms.

]]>A generalization of the beer mats puzzle]]>

Expanding large global solutions of the equations of compressible fluid mechanics

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In a recent work Sideris constructed a finite-parameter family of compactly supported affine solutions to the free boundary isentropic compressible Euler equations satisfying the physical vacuum condition. The support of these solutions expands at a linear rate in time. We show that if the adiabatic exponent gamma belongs to the interval (1,5⁄3] then this moduli space of affine motions is non-linearly stable without any symmetry assumptions; small perturbations lead to global-in-time solutions that remain close to an affine motion, they are smooth in the interior of their support, and no shocks are formed in the process.Our strategy relies on two key ingredients: a new interpretation of the affine motions using an (almost) invariant action of GL(3) on the compressible Euler system and the use of Lagrangian coordinates. The former suggests a particular rescaling of time and a change of variables that elucidates a stabilisation mechanism, while the latter requires introducing new ideas with respect to the existing well-posedness theory for vacuum free boundary fluid equations.

(Joint work with Juhi Jang (USC))

]]>Functional analytic approach to dynamical systems

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In the last decade the functional analytic approach to the study of the statistical properties of dynamical systems has become very prominent. I will try to present, in their simpler form, some of the ideas involved and review some of the results.

]]>Algebraic aspects of hyperbolic volume

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The main part deals with similar features and questions about hyperbolic volume of (non-)arithmetic Coxeter orbifolds of higher dimensions. We discuss the volume problem for certain Coxeter pyramids. This family gives rise to interesting (non-)arithmetic reflection groups whose commensurability classification has recently been performed (joint work with R. Guglielmetti and M. Jacquemet).

This is a report about ongoing work. ]]>

Höhenschranken für gewisse algebraische Zahlen ]]>

Verifiable conditions for convergence rates of regularization methods for inverse problems

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Over the last years Variational Source Conditions (VSCs) have become a standard assumption for the analysis of these methods. Compared to spectral source conditions they have a number of advantages: They can be used for general nonquadratic penalty and data fidelity terms, lead to simpler proofs, are often not only sufficient, but even necessary for certain convergence rates, and they do not involve the derivative of the forward operator (and hence do not require restrictive assumptions such as a tangential cone condition). However, so far only few sufficient conditions for VSCs for specific inverse problems are known.

To overcome this drawback, we propose a general strategy for the verification of VSCs, which consists of two sufficient conditions: One of them describes the smoothness of the solution, and the other one the degree of ill-posedness of the operator. For a number of important linear inverse problems this leads to equivalent characterizations of VSCs in terms of Besov spaces and necessary and sufficient conditions for rates of convergence. We also discuss the application of our strategy to nonlinear parameter identification and inverse medium scattering problems where it provides sufficient conditions for VSCs in terms of standard function spaces.

]]>On the number of minimal models of a smooth threefold of general type

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Hilbert’s tenth problem: Diophantine equations and decidability]]>

Hierarchical Tensor Approximation

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We propose an alternative: the Galerkin trial space is constructed using a partition of unity. By multiplying local cut-off functions with polynomials, we can obtain discretizations of arbitrary order, and local grid refinement can be realized by reducing the supports of the cut-off functions. The main challenge lies in the construction of the corresponding system matrix, since even determining the sparsity pattern involves interactions between cut-off functions on different levels of the mesh hierarchy.

Our approach leads to a sparse system matrix, the basis functions are convenient tensor products of functions on lower-dimensional domains, and local regularity can be exploited by variable-order interpolation in order to obtain close to optimal complexity.

]]>Moduli space of cubic surfaces and their anti canonical divisors

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Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Fractional closed geodesics and applications]]>

Tropical compactifications, Mori Dream Spaces and Minkowski bases

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

This is a work in progress with Elisa Postinghel. ]]>

Die Tate-Lichtenbaum Paarung und Supersinguläre Elliptische Kurven]]>

Towards code modernization and porting on Intel many-core architecture: the story of Gadget code

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Conjugacy classes of n-tuples in semi-simple Jordan algebras

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Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

The complement problem in the plane]]>

Regularization of Rational Group Actions - part 2

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Dynamics of automorphism of a particular family of affine cubic surfaces]]>

Regularization of Rational Group Actions - part 1

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Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

]]>

On the Wright's complex

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Uniqueness of weak solutions to transport equation with two-dimensional nearly incompressible BV vector field

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Given a bounded, autonomous vector field b: R^{2} → R^{2}, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation

(1) ∂tu + b · ∇u = 0.

We prove that uniqueness of weak solutions holds under the assumptions that b is of class BV and it is nearly incompressible. Our proof is based on a splitting technique (introduced previously by Alberti, Bianchini and Crippa) that allows to reduce (1) to a family of 1-dimensional equations which can be solved explicitly, thus yielding uniqueness for the original problem.

In order to perform this program, we use Disintegration Theorem and known results on the structure of level sets of Lipschitz maps: this is done after a suitable localization of the problem, in which we exploit also Ambrosio’s superposition principle.

This is joint work with S. Bianchini and N. A. Gusev.

]]>Parabolic equations with random coefficients on moving hypersurfaces

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

This is a joint work with Charles M. Elliott (University of Warwick, UK), Ralf Kornhuber (Free University Berlin, Germany) and Thomas Ranner (University of Leeds, UK). ]]>

Jordan groups and birational automorphisms of algebraic varieties

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Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Discontinuous Galerkin Method for seismic imaging in frequency domain

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Quasi-invariant gaussian measures for the 4th order NLS under sharp regularity assumptions

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Blowup Dynamics in the Keller-Segel Model of Chemotaxis

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The Keller-Segel equations model chemotaxis of bio-organisms. In a reduced form, considered in this talk, they are related to Vlasov equation for self-gravitating systems and are used in social sciences in descriptions of crime patterns. It is relatively easy to show that in the critical dimension 2 and for the mass of initial conditions greater than 8π, the solutions break down in finite time. Understanding the mechanism of this breakdown turned out to be a subtle problem defying solution for a long time. Preliminary results indicate that the solutions “blowup”. This blowup is supposed to describe the chemotactic aggregation of the organisms and understanding its universal features would allow comparison of theoretical results with experimental observations. In this talk I discuss recent results on dynamics of solutions of the (reduced) Keller-Segel equations in the critical dimension 2, which include a formal derivation and partial rigorous results on the blowup dynamics of solutions.

(The talk is based on joint work with S. I. Dejak, D. Egli and P.M. Lushnikov.)

]]>Robust Coarse Space Construction for Domain Decomposition Methods

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The existence of morphisms from a Calabi-Yau manifold

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p-adic numbers and their interplay with complex geometry]]>

BMO, quasiconformal maps, and transport equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We will explain two results about (linear and nonlinear) transport equations, quasiconformal maps, and vector fields with unbounded divergence. Originally, these results are motivated by a difficult problem on Muckenhoupt weights and elliptic PDE. However, classical harmonic analysis tools allow to reformulate this problem in variational BMO terms, and then a theorem by H. M. Reimann brings naturally the connection to the transport theory.

]]>Stability and Regularity Properties of the Navier-Stokes Equation

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We will discuss some regularity properties of weak solutions to the three-dimensional Navier–Stokes equation. We will first recall the classical partial regularity theory, developed by Scheffer and later by Caffarelli–Kohn–Nirenberg. Then we will present some new results in both the small data and perturbative frameworks.

]]>Model-aware Newton-solver for electrical impedance tomography

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Isomonodromic deformations of the five punctured sphere arising from complex plane quintics

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Logarithmic flat connections and monodromy]]>

Eulerian and Lagrangian continuous solutions to a non-convex balance law

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We consider continuous solutions to the single balance law ∂_{t}u+∂_{x}(f(u)) = g, g bounded, f ∈ C^{2}. We discuss correspondences among the source terms in the Eulerian and Lagrangian settings, extending previous works relative to the flux f(u) = u^{2} when possible. Counterexamples point out a new behavior of solutions when f is non-convex, and when the set of inflection points of f is not negligible, stressing the difference among the Lagrangian/Eulerian formulations in this context.

This is a joint work with G. Alberti and S. Bianchini.

]]>Scalar conservation laws with discontinuous flux

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In order to obtain uniqueness for solutions of scalar conservation laws with discontinuous flux, Kruzhkov’s entropy conditions are not enough and additional dissipation conditions have to be imposed on the discontinuity set of the flux. Understanding these conditions requires to study the structure of solutions on the discontinuity set. I will show that under quite general assumptions on the flux, solutions admit traces on the discontinuity set of the flux. This allows to show that any pair of solutions satises a Kato type inequality with an explicit reminder term concentrated on the discontinuities of the flux. Applications to uniqueness is then discussed.

]]>Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Moduli spaces and enumerative geometry]]>

On a classical question about Chern numbers

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Adaptive evolution and concentrations in parabolic PDEs

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

PSL(2,C), the exponential and some new free groups

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Modular forms and their applications]]>

Torus equivariant K-stability

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

On the concentration of entropy dissipation for scalar conservation laws

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

After a brief overview about the classical well-posedness result for scalar conservation laws, we investigate the structure of bounded solutions. In particular we prove that the entropy dissipation measure is concentrated on a countably 1-rectifiable set. In order to prove this result we introduce the notion of Lagrangian representation of the solution.

This is a joint work with Stefano Bianchini.

]]>A^1-contractibility of Koras-Russell threefolds

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Automorphism groups of affine algebraic surfaces preserving an A1-fibration

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In particular, we derive an example of a surface with infinite discrete automorphism group. ]]>

On linear inviscid damping, boundary effects and blow-up

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The Euler equations of fluid dynamics are time-reversible equations and possess many conserved quantities, including the kinetic energy and entropy. Furthermore, as shown by Arnold, they even have the structure of an infinite-dimensional Hamiltonian system. Despite these facts, in experiments one observes a damping phenomenon for small velocity perturbations to monotone shear flows, where the perturbations decay with algebraic rates. In this talk, I discuss the underlying phase-mixing mechanism of linear inviscid damping, its mathematical challenges and how to establish decay with optimal rates for a general class of monotone shear flows. Here, a particular focus will be on the setting of a channel with impermeable walls, where boundary effects asymptotically result in the formation of singularities.

]]>Automorphism groups of positive entropy of normal projective varieties

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“Breaking Bad with Nonlocality“ & “Private Index Lookup – fetching data without revealing what is retrieved”

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Finite Element Methods in piecewise smooth domains]]>

A new approach to the classification of singular Fano-Mori 3-folds.

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Additive group actions on algebraic varieties

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In this talk we present some recent results about additive group actions on non-necessarily affine algebraic varieties that generalize the usual description of additive group actions on affine varieties via locally nilpotent derivations. In particular, we provide a characterization of additive group actions on a wide class of algebraic varieties in terms of a certain type of integrable vector fields. ]]>

Möbius invariant versions of the Willmore flow and related evolution equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

While the Willmore energy is invariant under Möbius transformations, its negative L^{2}-gradient flow is not - simply because the L^{2}-scalar product used in its definition does not have this invariance. In this talk we present Möbius invariant versions of the Willmore flow picking up ideas of Ruben Jakob and Oded Schramm. We will discuss its uses and limitations and prove well-posedness of the Cauchy problem and attractivity of local minimizers.

Noncommutative invariant algebras and weighted homographic actions

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In the particular case of homography groups, we will give in this talk an overview of some results about this topic involving transvectants, modular forms and Rankin-Cohen brackets. ]]>

Calculating the energy of molecules. A computational introduction to chemistry and wavelet boundary element methods.]]>

Networks of curves in the plane moving by curvature

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I consider the motion by curvature of a network of curves in the Euclidean plane and I discuss existence, uniqueness, and asymptotic behavior of the evolution. In particular, I focus on two model cases: a regular embedded network composed by three curves with fixed endpoints (triod) and a regular embedded network composed by two curves, one of which is closed (spoon). After talking about the state of art of the problem, I will present some new and possibly ”incoming” results obtained with Carlo Mantegazza and Matteo Novaga.

]]>Cohomological action of automorphisms of compact Kaehler threefolds

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Groups of automorphisms of some affine varieties

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Intersection product and first Chern class]]>

On Moser type inequalities in the whole space

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The Trudinger-Moser inequality is a substitute for the well known Sobolev embedding theorem when the limiting case is considered. We discuss Moser type inequalities in the whole space which involve complete and reduced Sobolev norm.Then we investigate the optimal growth rate of the exponential type function both in the first order case and in the higher order case.

]]>Model order reduction with adaptive finite element POD and application to uncertainty quantification

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Standard techniques assume that all snapshots use one and the same spatial mesh. I present a generalization for unsteady adaptive finite elements, where the mesh can change from time step to time step and, in the case of stochastic sampling, from realization to realization. I will answer the following questions: How can the coding effort for creating such a reduced-order model be minimized? How can the union of all snapshot meshes be avoided? What is the main difference between static and adaptive snapshots in the error analysis of Galerkin reduced-order models?

As a numerical test case I consider a two-dimensional viscous Burgers equation with smooth initial data multiplied by a normally distributed random variable. The results illustrate the convergence properties with respect to the number of POD basis functions and indicate possible savings of computation time.

]]>Quadratic maps with a periodic critical point of period 2

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Holomorphic self-maps with relatively compact image]]>

The finiteness problem for minimal surfaces of bounded index in a 3-manifold

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Given a closed, Riemannian 3-manifold (N, g) without symmetries (more precisely: generic) and a non-negative integer p, can we say something about the number of minimal surfaces it contains whose Morse index is bounded by p? More realistically, can we prove that such number is necessarily finite? This is the classical ”generic finiteness” problem, which has a rich history and exhibits interesting subtleties even in its basic counterpart concerning closed geodesics on surfaces. We settle such question when g is a bumpy metric of positive scalar curvature by proving that either finiteness holds or N does contain a copy of RP^{3} in its prime decomposition and we discuss the obstructions to any further generalisation of such result. When g is assumed to be strongly bumpy (meaning that all closed, immersed minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by White) then the finiteness conclusion is true for any compact 3-manifold without boundary.

Dimension-adaptive sparse grid simulations of multiscale viscoelastic flows

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Singular Moser-Trudinger inequalities and their extremals]]>

Nonlinear Bounds in Hölder Spaces for the Monge-Ampère Equation

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It is well known that for many second-order PDEs the solution v gains two derivatives with respect to the right-hand side g in Hölder spaces. Often, however, it is useful to have a quantitative understanding of regularity. In ’89, Caffarelli proved interior a priori estimates for fully nonlinear, uniformly elliptic equations. Specifically, he showed that ‖v‖_{C2,α(B_{1/2})}≤C(‖v‖_{L∞(B_{1})}+‖g‖_{Cα(B_{1})}) and C∼1/α as α→0. The natural question to ask is then: Can one extend such quantitative estimates to other equations? An equation that appears frequently in analysis, geometry, and applications is the Monge-Ampére equation det(D^{2}u) = f. The Monge-Ampère equation enjoys the same qualitative regularity gains as its linear counterpart the Poisson equation in the appropriate setting, and so we ask whether or not the quantitative picture is also the same. This is not the case. In this talk, we will first review Caffarelli’s interior a priori estimates. Then, we will move to the Monge-Ampère equation and see a different picture.

(Joint work with Alessio Figalli and Connor Mooney)

]]>Self-similar blowup in nonlinear wave equations

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Blowup via self-similar solutions is ubiquitous in nonlinear wave equations, in particular in the energy-supercritical case. There are many examples where a self-similar solution is actually conjectured to describe the generic blowup of the equation. I will report on recent progress in understanding the stability of self-similar solutions in semilinear wave equations.

]]>On electrorheological fluids

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We discuss the existence theory for different models describing the motion of electrorheological fluids.

]]>Intrinsic Finite Element Methods

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Heights and elliptic curves]]>

Regularity issues for some nonlocal and nonlinear elliptic equations

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In this talk, I will review some regularity results for weak solutions of nonlocal variants of the p-Laplace equation. The model case is given by the Euler-Lagrange equation of an Aronszajn–Gagliardo–Slobodeckij seminorm. In particular, I will present a higher differentiability result for solutions, recently obtained in collaboration with Erik Lindgren (KTH). I will also discuss some connections of these equations to an Optimal Transport problem with congestion effects.

]]>Separating invariants and local cohomology

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

(Joint with Jack Jeffries) ]]>

Lipschitz extensions of maps between metric spaces

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A pair of metric spaces (X;Y) is said to have the Lipschitz extension property if any Lipschitz map from a subset of X into Y can be extended to a globally defined Lipschitz map to the whole space X. In this talk I will first recall some classical extension results for spaces with a linear structure, and I will present recent results for the case when the target space Y is the Heisenberg group.

]]>Controllability method for time-harmonic wave equations

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Universal Groebner bases and Cartwright-Sturmfels ideals

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These ideas will then be applied to the study of universal Groebner bases of ideals of minors, which admit such a grading. The main tools and concepts are Z^{m} gradings, multigraded Hilbert series, universal Groebner bases, initial and generic initial ideals. I will introduce and study two families of multigraded ideals with the property that any two ideals in the same family that have the same Hilbert series also have the same generic initial ideal. I will show that ideals of minors belong to one of these families, and derive some result about their universal Groebner bases.

Combination of discretization and modelling error estimates]]>

Local eigenvalue statistics for random regular graphs

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I discuss results on local eigenvalue statistics for random regular graphs. Under mild growth assumptions on the degree, we prove that the local semicircle law holds at the optimal scale, and that the bulk eigenvalue statistics coincide with those of the GOE from random matrix theory.

(Joint work with R. Bauerschmidt, J. Huang and H.-T. Yau.)

]]>Invariance of Plurigenera for foliations on surfaces

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Purcell swimmer and sub-Riemannian geometry]]>

Effective bounds on positive characteristic singular surfaces

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Modular Forms in Number Theory and Combinatorics]]>

Crystallographic Applications of Uniform Distribution Theory

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Real algebraic surfaces with many handles in (CP^1)^3

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Multiplicative and modular diophantine problems

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Area formula for Hyperbolic Triangles and Lexell problem]]>

Invariant manifolds for the porous medium equation

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We investigate the speed of convergence and higher-order asymptotics of solutions to the porous medium equation. Applying a nonlinear change of variables, we rewrite the equation as a diffusion on a fixed domain with quadratic nonlinearity.The degeneracy is cured by viewing the dynamics on a hypocycloidic manifold. It is in this framework that we can prove a differentiable dependency of solutions on the initial data, and thus, dynamical systems methods are applicable. Our main result is the construction of invariant manifolds in the phase space of solutions which are tangent at the origin to the eigenspaces of the linearized equation. We show how these invariant manifolds can be used to extract information on higher-order long-time asymptotic expansions.

]]>Multilevel Monte Carlo methods for multiscale problems

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In this talk, I will give an introduction to multilevel Monte Carlo methods in the case when the variance of the detail does not converge uniformly. The idea is illustrated by the calculation of the expectation for an elliptic problem with a random multiscale coefficient and then extended to approximations of statistical solutions to the Navier-Stokes equations.

]]>Some properties of the group GL_2(C[X_1, . . . , X_m]) and some applications to the polynomial automorphism group Aut(A^n) of the affine space A^n

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Multiplication polynomials and relative Manin-Mumford

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

0→**G**_{a} →G→E→0

where Ga is the additive group. The famous Manin-Mumford conjecture, proved by Hindry for such G, states that the intersection of a curve in G with its set of torsion points is finite unless the curve is contained in an algebraic subgroup. We will present a relative version of this theorem for families of additive extensions. Afterwards we will discuss some consequences of this result for classical problems such as Pell’s equation in polynomials and elementary integration.

]]>Calculating Gauss norms in polynomial continued fractions]]>

Barycenter technique and two remaining questions in Geometric Analysis

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In this talk, we will present our recent solutions of the remaining cases of the boundary Yamabe problem and the Riemann mapping problem asked by Escobarin 1992. Rather than discussing our arguments of proofs, we will focus more on explaining the barycenter technique of Bahri-Coron which we employ. We hope by doing this to allow an easier understanding for the audience, since it seems to us, that even among experts, the barycenter technique is not known like the minimizing technique of Aubin-Schoen. Moreover, we hope also the audience to see how naturally the barycenter technique fits into conformally invariant variational problems verifying the structure of quantization and strong interaction phenomena.

(Joint work with M. Mayer of University of Giessen)

]]>On the fibres of Mori fibre spaces

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Persistent Homology]]>

Adams inequality on the Hyperbolic space

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In this talk we will discuss the classical Adams inequality and its versions in the hyperbolic space. We will also discuss the hyperbolic versions of Adachi-Tanaka type inequalities and the exact growth.

]]>Conforming Approximations of Bingham Type Fluid Flows

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In the general context of a convex lower semi-continuous functional on a Hilbert space, we prove the convergence of time implicit space conforming approximations, without viscosity and for non-smooth data. Then we introduce a general class of total variation functionals, for which we can apply the regularization method. We consider the time implicit regularized, linearized or not, algorithms, and prove their convergence for general total variation functionals.

]]>A symplectic version of the Chevalley restriction theorem for polar representations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Flows of singular vector fields and applications to fluid and kinetic equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Distinguishing Knots]]>

On compactness of solution of the Quantum Navier-Stokes equations

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In this talk we focus on a new compactness result about weak solutions of the quantum Navier-Stokes equations. The novelty of the result is that we are able to consider the vacuum in the definition of weak solutions. The main tool is a new formulation of the equations which allows us to get an additional a priori estimate to prove compactness. Some remarks concerning the choice of the approximation system to get global existence will be made.

(Joint work with Paolo Antonelli)

]]>Family of exotic affine spheres

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On Bressan's Mixing Conjecture and Related Results]]>

Monte Carlo Methods and Mean Field Approximation for Stochastic Particle Systems

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In 2012, my Master thesis developed different versions of Multilevel Monte Carlo (MLMC) for particle systems, both with respect to time steps and number of particles and proposed using particle antithetic estimators for MLMC. In that thesis, I showed moderate savings of MLMC compared to Monte Carlo. In this talk, I recall and expand on these results, emphasizing the importance of antithetic estimators in stochastic particle systems. I will finally conclude by proposing the use of our recent Multi-index Monte Carlo method to obtain improved convergence rates.

]]>Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

High-Order Accurate Energy-Stable Finite Difference Methods]]>

On the geometry of matrix multiplication

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Schwarz Methods as Preconditioners for Non-linear Problems

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On Cremona Representations

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In this talk I will recall some properties of Cr

Local Exponential Runge-Kutta Methods]]>

Compactness and stability issues for Einstein-Lichnerowicz constraints system

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I will survey recent results on the Einstein-Lichnerowicz constraints system which appears in general relativity when trying to formulate the Cauchy problem for the Einstein equation coupled with a scalar field. I will discuss existence, uniqueness,compactness and stability for this system. This is a joint work with Bruno Premoselli.

]]>What is the (fractional) Laplacian?

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

No previous expertise on the subject is needed.

]]>Numerical modeling of light interaction with matter on the nanoscale using DGTD methods

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This is a joint work with Claire Scheid and Jonathan Viquerat.

[1] Fezoui, L., S. Lanteri, S. Lohrengel, and S. Piperno. Convergenceand stability of a discontinuous Galerkin time-domain method for the3D heterogeneous Maxwell equations on unstructured meshes. ESAIM:Math. Model. Numer. Anal., Vol. 39, No. 6, 1149-1176, 2005.

]]>Birational equivalence of divisors and T-varieties

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Efficient Explicit Time Integration for the Simulation of Acoustic and Electromagnetic Waves

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Geschichte der Mathematik früher und heute am Beispiel einer Antrittsrede von Johann I Bernoulli (1667-1748)]]>

Geometry in the Euler equations of hydrodynamics: h-principle and convex integration

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The following dichotomy between rigidity and flexibility is now well known in geometry: while uniqueness holds for smooth solutions to the isometric embedding problem, the set of solutions becomes unimaginably large if one allows rough ones. What is surprising is that this dichotomy holds for problems coming from mathematical physics,and in particular the Euler equations of fluid dynamics. In this (mainly expository) talk I will explain the h-principle and the method of convex integration. Convex geometry is the heart of the matter and profuse figures will attempt to illustrate the difficulties and how to tame them.

]]>Compactness Properties for Singular Liouville Equations and Systems

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I will give a brief overview of the main results concerning topological methods for singular Liouville equations on compact surfaces, and I will show how to extend some of them to special elliptic systems. My analysis will focus on sharp forms of the Moser-Trudinger inequality and on mass-quantization results for the SU(3) Toda System.

]]>Analytical and Numerical Homogenization of Maxwell's Equations

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This is a joint work with Sonia Fliss and Patrick Ciarlet.

]]>Efficient Explicit Time Integration for the Simulation of Acoustic and Electromagnetic Waves]]>

Optimal trace ideals properties of the Fourier restriction operator and applications

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We study the trace ideals properties of the Fourier restriction operator to hypersurfaces. Equivalently, we generalize the theorems of Stein-Tomas and Strichartz to systems of orthonormal functions, with an optimal dependence on the number of such functions. As an application, we deduce new Strichartz inequalities describing the dispersive behaviour of the free evolution of quantum systems with an infinite number of particles. This is a joint work with Rupert Frank.

]]>Some recent results on the Jacobian equation

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Given two functions f and g,we want to find a map φ such that

g(φ(x)) det∇φ(x)=f(x) x∈Ω,

φ(x)=x x∈∂Ω.

Local case. We first consider the (local) existence, uniqueness and optimal regularity for the problem

g_{i}(φ(x)) det∇φ(x)=f_{i}(x) for every 1≤i≤n

where g_{i}·f_{i}>0.

*Global case*. A necessary condition is then

∫_{Ω} f =∫_{Ω} g. (1)

(i) We discuss the case where g·f>0 and give three different ideas for the existence problem with optimal regularity.

(ii) We then briefly comment on the case where g>0 but f is allowed to change sign.

*A problem without the condition (1)*. We consider a more general problem of the form

det∇φ(x)=f(x,φ(x),∇φ(x)) x∈Ω,

φ(x)=x x∈∂Ω.

where no constraint of the type (1) is needed.

]]>

Recent progresses in nonlinear potential theory

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Passages through resonances and capture into resonance in dynamics of charged particles

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

A fully discrete approximation of the one-dimensional stochastic wave equation

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This is a joint work with Lluís Quer-Sardanyons, Universitat Autònoma de Barcelona.

]]>Special reductive groups over an arbitrary field

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Topology of real algebraic curves in the projective plane]]>

A counter-example concerning regularity properties for systems of conservation laws

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In 1973 Schaeffer established a result that applies to scalar conservation laws with convex fluxes and can be loosely speaking formulated as follows: for a generic smooth initial datum, the admissible solution is smooth outside a locally finite number of curves in the (t,x) plane. Here the term ”generic” should be interpreted in a suitable technical sense, related to the Baire Category Theorem. My talk will aim at discussing a recent explicit counter-example that shows that Schaeffer’s Theorem does not extend to systems of conservation laws. The talk will be based on joint works with Laura Caravenna.

]]>Adaptive inversion methods for the Helmholtz equation]]>

Results and conjectures about some isoperimetric problems with density

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The standard isoperimetric inequality states that among all sets with a given fixed volume(or area in dimension 2) the ball has the smallest perimeter. That is, written here for simplicity in dimension 2, the following infimum is attained by the ball

2πR= inf{∫_{∂Ω} 1 dσ(x) : Ω⊂**R**^{2} and ∫_{Ω} 1 dx=πR^{2}}.

The isoperimetric problem with density is a generalization of this question: given two positive functions f,g:**R**^{2}→**R**^{2} one studies the existence of minimizers of

I(C) = inf{∫_{∂Ω} g(x) dσ(x) : Ω⊂**R**^{2} and ∫_{Ω} f(x) dx=C}.

I will mainly talk about the situation when f(x) =|x|q and g(x) =|x|p.This is a reach problem with strong variations in difficulties depending on the values of p and q. Some cases are still an open problem. One case has an interesting application related to the Moser-Trudinger imbedding. I will also mention the situation when f=g=eψ is strictly positive and radial, which leads to the log-convex density conjecture.

]]>Affine n-Space is Characterized by its Automorphisms

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We also show that every torus appears as Aut(X) for a suitable affine variety X, but that Aut(X) cannot be isomorphic to a semisimple group. In fact, if Aut(X) is finitedimensional and if X \not\simeq

Concerning the structure of Aut(

Finally, we prove that every non-trivial homomorphism φ: SAut(

Numerical treatment of boundary integral equations and stochastic PDE]]>

A generalization of Gromov’s almost flat manifold theorem

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Almost flat manifolds are the solutions of bounded size perturbations of the equation Sec=0 (Sec is the sectional curvature). In a celebrated theorem, Gromov proved that the presence of an almost flat metric implies a precise topological description of the underlying manifold.

Integral pinching theorems express curvature assumptions in terms of certain L^{p}-norms and try to deduce topological conclusions. But typically one needs to require p >n2, where n is the dimension of the manifold, to prove such rigidity theorems.

During this talk we will explain how, under lower sectional curvature bounds, to imposeanL1-pinching condition on the curvature is surprisingly rigid, leading indeed to the same conclusion as in Gromov’s theorem under more relaxed curvature conditions.

This is a joint work with B. Wilking.

]]>Explicit counterexamples to the Laurent Cancellation Problem

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Segmentation and Classification of Inner Structures of the Spinal Cord on MRI Images with Continuous Graph Cuts and Principal Component Analysis]]>

One Shot Inverse Scattering and Source Splitting

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Besides standard regularization methods, which are often iterative, a completely different methodology - so-called qualitative reconstruction methods - has attracted a lot of interest recently. These algorithms recover specific qualitative properties of scattering objects or anomalies inside a medium in a reliable and fast way. They avoid the simulation of forward models and need no a priori information on physical or topological properties of the unknown objects or inhomogeneities to be reconstructed. One of the drawbacks of currently available qualitative reconstruction methods is the large amount of data required by most of these algorithms. It is usually assumed that measurement data of waves scattered by the unknown objects corresponding to infinitely many primary waves are given - at least theoretically.

We consider the inverse source problem for the Helmholtz equation as a means to provide a qualitative inversion algorithm for inverse scattering problems for acoustic or electromagnetic waves with a single excitation only. Probing an ensemble of obstacles by just one primary wave at a fixed frequency and measuring the far field of the corresponding scattered wave, the inverse scattering problem that we are interested in consists in reconstructing the support of the scatterers. To this end we rewrite the scattering problem as a source problem and apply two recently developed algorithms - the inverse Radon approximation and the convex scattering support - to recover information on the support of the corresponding source. The first method builds upon a windowed Fourier transform of the far field data followed by a filtered backprojection, and although this procedure yields a rather blurry reconstruction, it can be applied to identify the number and the positions of well separated source components. This information is then utilized to split the far field into individual far field patterns radiated by each of the well separated source components using a Galerkin scheme. Finally we compute the convex scattering supports associated to the individual source components as a reconstruction of the individual scatterers. We discuss this algorithm and present numerical results.

]]>Classical Invariant Theory and Category O

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Mapping class group and delta-hyperbolicity of the curve graph]]>

Global solutions and asymptotics of Teichmüller harmonic map flow

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Teichmüller harmonic map flow is a gradient flow of the Dirichlet energy which is designed to evolve parametrized surfaces towards critical points of the area. In this talk we will discuss the construction and some new results for this flow and show in particular that for non-positively curved targets the flow changes or decomposes arbitrary closed initial surfaces into minimal immersions (possibly with branch points) through globally defined smooth solutions.

]]>Lifting real algebraic curves to real algebraic knots

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In this talk I will explain how to associate a hyperplane arrangement to a nodal planar rational real algebraic curve. This arrangement describes the space of nonsingular liftings and allows us to calculate the homology (and thus, in particular, the number of liftings up to rigid isotopy). We will show that, up to degree 5, this hyperplane arrangement is a rigid isotopy invariant (of planar curves) and can provide real algebraic analogues of the classical Reidemeister moves. Obstructions in the case of higher degrees will be discussed. The talk should be accessible to nonspecialists. ]]>

The Dirichlet divisor problem and beyond]]>

Gibbs measures for the periodic derivative NLS equation

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The DNLS equation is an integrable PDE, in the sense that there are infinitely many Hamiltonians associated to it. The aim of the talk is to present the construction of infinitely many functional measures associated to these integrals of motion of the equation, each measure being supported on Sobolev spaces of increasing regularity. These are natural candidates to be the invariant measures associated to the DNLS eq. Invariant measures are a crucial tool in the theory of integrable PDEs, useful e.g. to prove long time properties of regular solutions. The introductory general aspects will be reviewed and the new results on DNLS, obtained in collaboration with R. Luc (ICMAT,Madrid) and D. Valeri (MSC, Beijing), will be presented.

]]>From linear to nonlinear Calderòn-Zygmund Theory

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Effective Approximation and Diophantine Applications]]>

Stationary Kirchhoff systems in closed manifolds

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BEM++ - Efficient solution of boundary integral equation problems

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Group actions, entire curves and rational points

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This is discussed in connection with algebraic groups and prinicipal bundles. More precisely, let X be a projective manifold, G an algebraic group and E a G-principal bundle on X, all defined over some number field K. Then E admits a Zariski

dense set of L-rational points for some finite extension field iff X does. This corresponds to the homotopy lifting property whose complex analytic analogue allows to lift entire curves. ]]>

"Trails in a kinetic land"]]>

Multilevel Monte Carlo methods for multiscale problems

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Geometric Lang-Vojta Conjecture in P^2

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In the (split) function field case the conjecture predicts (weak) algebraic hyperbolicity for log-general type varieties.When the completion of the variety is the projective plane the conjecture is known both if the divisor at infinity consits of four lines in general position (Brownawell-Masser and,independently, Voloch) and for a conic and two lines with five singular points (Corvaja and Zannier). With different methods Chen and Pacienza-Rousseau proved that the conjecture holds in the hyperbolic case, i.e. the complement of a very generic curve of degree at least 5.

In the talk, after an introduction to this fascinating subject, we will show how to prove the conjecture in general for the complement of a very generic curve of degree at least four.The proof relies on a deformation argument applied to a conic and two lines and on the theory of logarithmic stable maps as defined by Abramovich-Chen (and independently by Gross and Siebert) which extends usual stable maps to the logarithmic category (in the sense of Kato and Illusie). ]]>

On flexible Polyhedrons]]>

A very informal introduction to finite elements]]>

Performance Engineering and Sparse Matrices: Introduction, applications and supercomputing

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Residual periodicity in arithmetic dynamics

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Area-minimizing graphs in the Heisenberg group

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We consider the area functional for graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that,under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity cannot be improved even under the bounded slope condition. This is based on a joint work with A. Pinamonti, F. SerraCassano and G. Treu.

]]>Frobenius property for integral currents and decomposition of normal currents

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In this joint work with Giovanni Alberti, we prove a Frobenius property for inte-gral currents: namely, if R=[∑,ξ,θ] k-dimensional integral current with a simple tangent vector field ξ∈C^{1}(**R**^{d};Λ_{k}(**R**^{d})), then ξ is involtive at almost every point in ∑. This result is related to the following decomposition problem formulated by F.Morgan: given a k-dimensional normal current T, do there exist a measure space L and a family of rectifiable currents {R_{λ}}_{λ∈L} such that T = ∫_{L} R_{λ} dλ and the mass decomposes consistently as M(T) = ∫_{L} M(R_{λ}) dλ? The aforementioned Frobenius property allows us to provide a counterexample to the existence of such a decomposition with a family of integral currents.

Mathematical Models of Mosquito Population Dynamics and Malaria

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Mathematical models can help to determine more efficient combinations of existing and new interventions in reducing malaria transmission and delaying the spread of resistance. We present difference equation models of mosquito population dynamics and malaria in mosquitoes; and ordinary differential equation models of mosquito movement and population dynamics. We analyse these models to provide threshold conditions for the survival of mosquitoes and show the existence of invariant positive states; and run numerical simulations to provide quantitative comparisons of interventions that target mosquitoes with varying levels of resistance.

]]>Generalisations of the tame automorphisms over a domain of positive characteristic

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Given a domain R of characteristic p >0, there exist two subgroups of the group GAn(R) of polynomial automorphisms which are natural generalisations of the linear group GL_{n}(R).

1) The subgroup of additive automorphisms, i.e. automorphisms with n components satisfying f(x+y) =f(x)+f(y) where x and y are set of n variables.

2) The subgroup of automorphisms with a Jacobian matrix in GL_{n}(R).

The subgroup generated by the translations and automorphisms of the type 1 (resp. 2) is called geometrically affine (resp. differentially affine). This group, together with the triangular automorphisms, generates the subgroup of geometrically tame (resp. differentially tame) automorphisms. We study these groups in dimension 2. We prove that they are different and endowed with a nice structure of amalgamated product.

]]>Artificial compressibility approximation of suitable weak solutions of incompressible Navier-Stokes equations

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In this talk I will discuss the problem of the approximation of suitable weak solutions of Navier-Stokes equations in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. It is well-known that suitable weak solutions enjoy the partial regularity theorem proved in the famous paper of Caffarelli-Kohn-Nirenberg, hence they are more regular than a Leray weak solutions. However, since the uniqueness of weak solutions of Navier-Stokes is unknown we don’t know if different approximation methods lead to a suitable weak solution. I will present a recent result obtained with L. C. Berselli (University of Pisa) where we proved that weak solutions obtained by some artificial compressibility approximation are suitable. The novelty of the result is that the Navier-Stokes equations are considered in a bounded domain with Navier boundary conditions.

]]>Automorphisms of some affine surfaces]]>

Double centralizers of unipotent elements in simple algebraic groups.

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The structure of complex unimodular maps

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We describe the structure of maps u:(0,1)^{n} → S^{1} having a given Sobolev regularity. Such maps are described by their singularities and phases. This is the analog of the Weierstrass factorization theorem for holomorphic functions; the singularities of the Sobolev maps play the role of the zeroes of holomorphic maps. We will present implications of this result to functional analytic questions related to manifold valued maps. If the time permits it, we will discuss the question of the control of the phases, and present some applications to some model PDEs and nonlocal problems.

Ten good reasons for using kernel reconstructions in adaptive finite volume particle methods

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On Cremona contractibility of plane curves

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This is a joint work in progress with Ciro Ciliberto (Univ. Roma "Tor Vergata"). ]]>

Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation

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A classical question in differential geometry concerns which smooth functions f can arise as Gauss curvature of a conformal metric on a 2-dim Riemannian manifold M. This amounts to solve a PDE which is the Euler-Lagrange equation of an energy functional. In this talk we will discuss about compactness issues and bubbling phenomena for this equation on surfaces of genus greater than 1 (joint work with Borer and Struwe) and on the torus.

]]>Bi-orderability of mapping tori]]>

Oscillations versus dispersion in plasma physics

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Local solutions to a free boundary problem for the Willmore functional

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space

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Minimal energy problems with hypersingular Riesz potentials

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Holomorphically Equivalent Algebraic Embeddings

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Theorem. If f, g : X → **C**^{m} are algebraic embeddings and 2 dimX+ 1≤m, then there exists a holomorphic automorphism φ of **C**^{m} such that φ◦f=g.

In fact, the proof is based on an idea of Kaliman, with which he proved that two algebraic embeddings of **C** into **C**^{3} are holomorphically equivalent. In the course of this talk, we discuss this idea. Moreover, we provide examples of algebraic embeddings into **C**^{m} that are holomorphically non-equivalent.

On the Hardy-Schrödinger operator with a boundary singularity

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We investigate the Hardy-Schrödinger operator L_{γ}=-Δ-γ/|x|2 on domains Ω⊂**R**^{n}, whose boundary contain the singularity 0. The situation is quite different from the well-studied case when 0 is in the interior of Ω. For one, if 0∈Ω, then L is positive if and only if γ<(n-2)^{2}/4, while if 0∈∂Ω the operator L could be positive for larger value of γ, potentially reaching the maximal constant n^{2}/4 on convex domains.

We prove optimal regularity and a Hopf-type Lemma for variational solutions of corresponding linear Dirichlet boundary value problems of the form Lγ=a(x)u, but also for non-linear equations including Lγ=(|u|^{β-2}u)/(|x|^{s}), where γ < n^{2}/4, s∈[0,2) and β:=2(n-s)/(n-2) is the critical Hardy-Sobolev exponent. We also provide a Harnack inequality and a complete description of the profile of all positive solutions–variational or not– of the corresponding linear equation on the punctured domain. The value γ=(n-1)^{2}/4 turned out to be another critical threshold for the operator Lγ, and our analysis yields a corresponding notion of “Hardy singular boundary-mass” mγ(Ω) of a domain Ω having 0∈Ω, which could be defined whenever (n^{2}-1)/4 < γ < n^{2}/4.

As a byproduct, we give a complete answer to problems of existence of extremals for Hardy-Sobolev inequalities of the form

C( ∫_{Ω }(u^{β})/(|x|^{s}) dx )^{2/β }≤∫_{Ω} |∇u|^{2} dx - γ∫_{Ω} (u^{2})/(|x|^{s})dx

whenever γ<n^{2}/4, and in particular, for those of Caffarelli-Kohn-Nirenberg. These resultsextend previous contributions by the authors in the case γ=0, and by Chern-Lin for the case γ<(n-2)^{2}/4. Namely, if 0≤γ≤(n^{2}-1)/4, then the negativity of the mean curvature of ∂Ω at 0 is sucient for the existence of extremals. This is however not sufficient for (n^{2}-1)/4≤γ≤(n^{2})/4, which then requires the positivity of the Hardy singular boundary-massof the domain under consideration.

Joint work with Nassif Ghoussoub.

]]>On the Cremona Group]]>

Some examples of numerical resolution of PDEs from the physics with FreeFem++

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Piezoelectric problems

Thermal problems with thermal resistances

Elasticity problems

Problems of fluid mechanics like incompressible Navier-Stokes

Problem of melting and/or solidification of the ice. (Boussinesq with specific heat)

Good Reduction for Endomorphisms of the Projective Line in Terms of the Branch Locus

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Dissipative Hölder solutions to the incompressible Euler equations

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We consider the Cauchy problem for the incompressible Euler equations on the three-dimensional torus. According to a conjecture due to Onsager, which is well known in turbulence theory, while all the solutions which are uniformly α-Hölder continuous in space for any α>1/3 must conserve the total kinetic energy, for any α<1/3 there can be uniformly α-Hölder solutions which are strictly dissipative. While the first part of the conjecture is well established since a long time, the second part is still open in its full generality. In the result that we present we show that, for any α<1/5, there exist C^{α} vector fields being the initial data of infinitely many C^{α} solutions of the Euler equations which dissipate the total kinetic energy.

On automorphism groups]]>

Parametrized automorphisms of manifolds with the density property

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Properties of Coulombic eigenfunctions of atoms and molecules

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The eigenfunctions of the Schrödinger operator for (non-relativistic) atoms and molecules (in the Born-Oppenheimer/clamped nuclei approximation) are solutions of an elliptic partial differential equation with singular (total) potential (i.e., zero-order term). In this talk we give an overview over our results about the structure/regularity of the eigenfunctions at the singularities of the potential. These, in particular, improve on the well-known ’Kato Cusp Condition’. If time permits, we also discuss the implications for the electron density.

This is joint work with S. Fournais (Aarhus, Denmark), and M. and T. Hoffmann-Ostenhof (Vienna, Austria).

]]>Fourth order energy-preserving locally implicit discretization for linear wave equations

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Conjugacy classes of special automorphisms of the affine spaces

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Polynomial propagation of moments and global existence for a Vlasov-Poisson system with a point charge

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We consider the Cauchy problem associated to the Vlasov-Poisson system and we extend the well-posedness theory of Lions and Perthame to the case of initial data which include a Dirac mass. Moreover we provide polynomially growing in time estimates for the moments of the solution. This is a joint work with L. Desvillettes and E. Miot.

]]>On the closure theorems of Steiner and Poncelet]]>

Holomorphic Automorphisms of the Koras-Russell Cubic

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Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law

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Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a small set, showing in particular the validity of Young’s law at almost every point (joint with Francesco Maggi).

]]>The density property for complex manifolds - a strong form of holomorphic flexibility

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On the Cancellation Problem for algebraic tori

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Towards a theory of ind-groups? Presentation of a few ind-groups

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Is it possible to endow such groups with an infinite-dimensional algebraic group (for short: ind-group) structure? Can one endow such groups with a topology making them topological groups?

Finally, can one hope to develop a theory of ind-groups generalizing in some sense the well established theory of algebraic groups?

I will try to present this subject by explaining some of the involved notions. I will consider different examples, among which the *n*-th Cremona group.

This latter group is nothing else than the ground field fixing automorphism group of the field of rational functions in *n* indeterminates.

Rayleigh-Bénard convection at finite Prandtl number:bounds on the Nusselt number

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We consider Rayleigh-Bénard convection at finite Prandtl number as modelled by the Boussinesq equation. We are interested in the scaling of the average upward heat transport, the Nusselt number Nu, in terms of the Rayleigh number Ra, and the Prandtl number Pr.

Physically motivated heuristics suggest the scaling Nu∼Ra^{1⁄3} and Nu∼Ra^{1/2} depending on Pr, in different regimes.

In this talk I present a rigorous upper bound for Nu reproducing both physical scalings in some parameter regimes up to logarithms. This is obtained by a (logarithmically failing) maximal regularity estimate inL1and inL1for the nonstationary Stokes equation with forcing term given by the buoyancy term and the nonlinear term, respectively. This is a joint work with Felix Otto and Antoine Choffrut.

]]>Relative Manin-Mumford in additive extensions]]>

Moduli spaces of (G,h)-constellations

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Given a reductive group G acting on an affine scheme X, a Hilbert function h, and a stability condition θ, we explain how to construct the moduli space M of θ-stable (G,h)-constellations on X, which is a common generalization of the invariant Hilbert scheme after Alexeev and Brion and of the moduli space of θ-stable G-constellations for finite groups introduced by Craw and Ishii. The main tools for this construction are the geometric invariant theory and the invariant Quot schemes. Moreover, the moduli space M is naturally equipped with a morphism μ: M → X//G which turns to be a “nice” desingularization of the quotient X//G in many situations.

]]>Preservation of Volumes in Nonholonomic Mechanics

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The study of nonholonomic mechanical systems is challenging because the equations of motion are not Hamil- tonian. The dynamics of the system can however be described in terms of a bracket of functions that fails to satisfy the Jacobi identity. One now speaks of an almost Poisson bracket.

The failure of the Jacobi identity leads to phenomena that are not shared by usual Hamiltonian systems. Open questions in nonholonomic mechanics that have received attention in recent years include determining general conditions for measure preservation, existence of asymptotic equilibria, relationship between symmetries and con- servation laws, reduction, and integrability.

In the first part of this talk I will present a basic introduction to nonholonomic mechanics. I will then present my recent work with Y. Fedorov and J. C. Marrero in which we study the problem of measure preservation for nonholonomic systems possessing symmetries in a systematic manner. Our method allows us to identify specific parameter values for which there exists a preserved measure for concrete mechanical examples.

]]>Numerical analysis of boundary element methods for impedance transmission conditions

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With impedance transmission conditions (ITCs) we can propose boundary element formulations on the sheet mid-line (or mid-surface) only. In the beginning of the talk we give a motivation of meaningful impedance transmission conditions, which relate jumps and mean values of the Dirichlet and Neumann traces on the mid-line. This relation may

involve surface differential operators, as for boundary conditions of Wentzell's type, and depend on frequency, conductivity, sheet thickness and sheet geometry e.g. curvature). These parameters may take small or large values and may lead to singularly perturbed

boundary integral equations.

We will introduce related boundary element methods in two and three dimensions and analyse well-posedness and discretisation error depending on the model parameters. Numerical experiments confirm the convergence order of the discretisation error of the proposed BEM and that the discretisation error behaves for smooth enough sheets equivalent to the exact solution when varying the model parameters. The results obtained for the eddy current model, for which a Poisson equation has to be solved outside the mid-line, can be transfered to the Helmholtz equation and to transmission conditions

arising from other models.

Prime-order birational diffeomorphisms of the sphere

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Singular Liouville systems and non-abelian Chern Simons vortices

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We discuss a class of singular Liouville systems in the plane and their role in the construction of non-abelian Chern-Simons vortices

]]>Jordan properties of groups of birational automorphisms

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Analysis of explicit multirate Runge-Kutta schemes for conservation laws

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On Almgren's stratification principle

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Universality in graphene and Weyl semimetals

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Sparse grid quadrature and interpolation methods for solving the Schroedinger equation

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The Demazure roots of a spherical embedding

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Joint work with Alexander Perepechko. ]]>

Lagrange and the Early Calculus of Variations

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The Hartree equation for infinite quantum systems

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A Fermi gas occupying the whole euclidian space is an example of a translation-invariant quantum system with an infinite number of particles. We study its stability properties under the time-dependent nonlinear Hartree equation. If this system is slightly perturbed at the initial time, we show in particular that it returns to the translation-invariant state for large times. This is an instance of nonlinear dispersion for infinite quantum systems, which was recently studied by Frank, Lewin, Lieb and Seiringer in the linear case. This a joint work with Mathieu Lewin (CNRS/Cergy). I will also mention some recent work on Strichartz estimates for systems of orthonormal functions, joint with Rupert Frank (Caltech).

]]>Hyperbolic manifolds of small volume

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On orbits of the automorphism group on an affine toric variety

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In the second part we introduce the total coordinates on

Estimates for the topological degree and related problems

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In this talk, I first discuss estimates for the topological degree of maps from sphere into itself. Second, I present characterizations of Sobolev spaces based on the pointwise convergence or the Gamma-convergence of a sequence of nonlocal, nonconvex functionals related to these estimates. If time permits, I will also discuss the connection between these functionals with various filters in the denoising problem. The talk is based on joint works with Jean Bourgain and Haim Brezis.

]]>New results on the isoperimetric problem with density

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Dual variational methods and nonvanishing for the nonlinear Helmholtz equation

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Automorphisms of the Lie algebra of vector fields on affine n-space

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As an immediate consequence, we get the following result due to Kulikov. If every injective endomorphism of the Lie algebra VF(

Kinetic Models in Extended Phase Space

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Integro-Differential harmonic maps into manifolds

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I will present results and ideas for the proof for regularity theory for critical points of non-local, degenerate integro-differential energies into manifolds which are related to p-harmonic maps.

]]>Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Following [1], in this talk we show how to establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE’s associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into **R**^{∞}.

When specialized to the setting of Euclidean or infinite dimensional (e.g.Gaussian) spaces, large parts of previously known results are recovered at once.Moreover, the class of RCD(K,∞) metric measure spaces, recently introduced by Ambrosio, Gigli and Savar ́e, object of extensive recent research, fits into our framework. Therefore we provide, for the first time, well-posedness results forODE’s under low regularity assumptions on the velocity and in a non smooth context.

**References**:

[1] L. Ambrosio and D. Trevisan. Well posedness of Lagrangian flows and continuity equations in metric measure spaces. ArXiv e-prints, February 2014.

Preperiodic points over global fields

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We consider dynamical systems on the projective line given by rational functions. Let $\phi$ be an endomorphism of the projective line defined over a global field K. We prove a bound for the cardinality of the set of K-rational preperiodic points for $\phi$ in terms of the number of places of bad reduction. The result is completely new in the function fields case and it is an improvement of the number fields case. An important tool is an S-unit equation theorem in 2 variables.

]]>Perturbation methods and low rank approximations for the Darcy equation with lognormal permeability

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We consider perturbation methods based on Taylor expansion of the solution of the PDE around the nominal permeability value. Successive higher order corrections to the statistical moments such as pointwise mean and covariance of the solution can be obtained recursively from the computation of high order correlation functions which, on their turn, solve high dimensional problems. To overcome the curse of dimensionality in computing and storing such high order correlations, we adopt a low-rank format, namely the so called tensor-train (TT) format.

We show that, on the one hand, the Taylor series does not converge globally, so that it only makes sense to compute corrections up to a maximum critical order, beyon which the accuracy of the solution deteriorates insetad of improving. On the other hand, we show on some numerical test cases, the effectiveness of the proposed approach in case of a moderately small variance of the log-normal permeability field.

]]>Closed ind-subgroups with the same Lie algebras

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Unique Continuation for Fractional Schrödinger Equations with Rough Potentials

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This talk is focused on unique continuation principles for fractional Schrödinger equations with scaling-critical and rough potentials. The results are deduced via so-called Carleman estimates. In particular, these methods can be transferred to “variable coefficient” versions of fractional Schrödinger equations.

]]>The tame automorphism group of affine 3-space is not closed

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From asymptotic analysis for simulation to generalized Ventcel's boundary value problem

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I will explain how some mathematical questions about the loss of coercivity arise from the computation of the profiles appearing in the expansion. Ventcel boundary conditions are second order differential conditions that appears when looking for a transparent boundary condition for an exterior boundary value problem in planar linear elasticity. The goal is to bound the infinite domain by a large “box” to make numerical approximations possible. Like Robin boundary conditions, they lead to wellposed variational problems under a sign condition of a coefficient. Nevertheless situations where this condition is violated appeared in several works. The wellposedness of such problems was still open. I will present, in the generic case, existence and uniqueness result of the solution for the Ventcel boundary value problem without the sign condition. Then, I will consider perforated geometries and give conditions to remove the genericity restriction.

]]>Sufficient conditions for Willmore-immersions in R^3 to be minimal surfaces

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We provide two sharp sufficient conditions for immersed Willmore surfaces in **R**^{3}, definedon bounded C^{4}-subdomains of **R**^{2}, to be already minimal surfaces, i.e. to have vanishing mean curvatures on their entire domains. Our precise results read as follows:

**Theorem 1.** For some bounded C^{4}-domain Ω⊂**R**^{2} let X∈C^{4}(Ω,**R**^{3}) denote some immersed Willmore surface with Gauss map N and mean curvature H. Furthermore, assume that there exist constants c,d∈R and some fixed vector V∈S^{2} such that χ := cX+dV satisfies at least one of the following two conditions:

a) There is some “normal domain” G⊂Ω such that there hold H=0 on ∂G and H≥0 (or H≤0) in G∩O, where O⊂**R**^{2} is some open neighbourhood of ∂G, and

inf_{∂G}<χ,N> ≥ 0 as well as sup_{∂G}<χ,N> > 0;

b) H=0 on ∂Ω and

<χ,N> > 0 in Ω\A as well a sup_{∂Ω}<χ,N> > 0

for some finite set A⊂Ω.

Then H≡0 is satisfied in \bar{Ω}, i.e.X is a minimal surface on \bar{Ω}.

These results turn out to be particularly suitable for applications to Willmore graphs. We can therefore show that Willmore graphs on bounded C4 domains \bar{Ω} with vanishing mean curvatures on the boundary ∂Ω must already be minimal graphs. Our methods also prove that any closed Willmore surface in R3 which can be represented as a smooth graph over S2 has to be of constant, non-zero mean curvature and therefore a round sphere. Finally we demonstrate that our results are sharp by means of an examination of some certain part of the Clifford-Torus in R3.

]]>The absolute continuous spectrum of skew products of compact Lie groups

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T

have purely absolutely continuous spectrum in the subspace associated to

The unlikelihood of integrability in elementary terms

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Incompressible Extensions in Sobolev Spaces

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Given a bounded domain and boundary data, does there exist a vector-valued map on this domain which is incompressible, that is, a map whose Jacobian determinant is one (almost) everywhere? In a regular setting, this question has been essentially positively answered in a famous paper by Dacorogna and Moser. I will present an analogous result in Sobolev spaces of low regularity, which was recently achieved by a convex integration method jointly with K. Koumatos (Oxford) and F. Rindler (Warwick). I will also comment on several generalisations and applications.

]]>Unverzweigte Kohomologie

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- Zeigen, das eine Varietät nicht rational ist, d. h., ihr Funktionenkörper nicht rein transzendant ist.
- Wenn der Grundkörper ein Zahlkörper ist, zeigen, daß das Lokal-Global Prinzip für rationale Punkte nicht gilt (Brauer-Maninsche Hindernis).
- Die Picardgruppe, d.h. die Klassengruppe von Divisoren, untersuchen (Tatesche Vermutung).

Unverzweigte Kohomologiegruppen sind Verallgemeinerungen der Brauergruppe. Ich werde erklären, wie man sie zu ähnlichen Zwecken anwenden kann.

]]>Critical Mean Field Equations on multiply connected domains

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The uniqueness of solutions of the (Liouville) mean field-type equation on a simply connected domain and in the sub critical regime λ∈(0,8π) was first proved by T. Suzuki (1992). This result has been later improved by S.Y.A. Chang, C.C. Chen and C.S. Lin (2003) [CCL] to cover the critical value λ∈(0,8π]. The case where the domain is not simply connected has been a long-standing open problem which we have finally solved in a recent paper in collaboration with C.S. Lin. Our proof is based on a new generalization of a P.D.E. version of the Alexandrov-Bol's isoperimetric inequality on multiply connected domains. Another delicate problem is to understand the existence/non-existence of solutions for this equation on multiply connected domains at the critical parameterλ=8π. Criticality here means that the variational functional whose critical points are solutions of the equation is not anymore coercive for λ=8π, which implies in particular in this situation that existence/non existence of solutions depend on the geometry of the domain. I will discuss our generalization of a result in [CCL] which yield necessary and sufficient conditions for the existence of solutions for the mean field equation at the critical parameter λ=8π.

]]>On the robustification of optimum experimental design problems

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In this talk, we consider robust formulations for optimum experimental design that work under significant uncertainty. Our focus is on problem settings in which the model is described by differential equations of some type that are solved numerically. Our approach is based on a semi-infinite programming formulation in which we exploit additional problem structure, together with sparse grids, to ensure tractability. The talk includes numerical experiments to illustrate and compare the effectiveness of the approaches.

]]>Boundary regularity for elliptic integro-differential equations

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We study the boundary regularity of solutions to elliptic integro-differential equations. First we prove that, for the fractional Laplacian (-Δ)^{s} with s∈(0,1), solutions u satisfy that u/d^{s} is Hölder continuous up to the boundary, where d(x) is the distance to the boundary of the domain Ω. We will show that, in this fractional context, the quantity u/d^{s}|_{∂Ω} plays the role that the normal derivative plays in second order equations. Finally, we also present new boundary regularity results for fully nonlinear integro-differential equations.

Sharpness of Picard's theorem in space

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In the late 1960's and early 1970's, results of Reshetnyak and Martio-Rickman-Väisälä showed that mappings of bounded distortion, also called as quasiregular mappings, can be viewed as a counterpart for holomorphic mappings in quasiconformal geometry. One of the natural goals from the very beginning in this theory was obtain Picard-type results. In 1980, Rickman showed that a non-constant quasiregular mapping from the Euclidean n-space to the n-sphere omits only finitely many points, where the number depends only on the dimension and distortion. The sharpness of Rickman's theorem was not as simple issue as in the classical Picard theorem. In 1984, Rickman showed by a surprising and elaborate construction that given any finite set in the 3-sphere there exists a quasiregular from the Euclidean 3-space into the 3-sphere omitting exactly that set.

In this talk, I will discuss joint work with David Drasin on the sharpness of Rickman's Picard theorem in all dimensions. Especially, I will discuss the role of bilipschitz geometry in the proof which leads to a stronger stament on the metric properties of the map and is a crucial ingredient in dimensions n > 3. ]]>

A minimization approach to hyperbolic Cauchy problems

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On the chain rule for the divergence operator in $\mathbb{R}^2$

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Suppose that b:**R**^{d}**→R**^{d} is a vector field, β:**R****→R**^{ } is a smoth function and u:**R**^{2}**→R** is a scalar field. If both u and b are smooth then the following formula holds: div(β(u))b = β(u) - (β'(u) - uβ'(u)) div (b) + β'(u) div(ub). Generalizations of this formula when u∈L^{∞} and b belongs to Sobolev space or has bounded variation were studied by R. Di Perna, P.-L. Lions, L. Ambrosio, C. De Lellis, J. Maly and other authors. I will present a new result in this direction for d=2, which was obtained recently in collaboration with S. Bianchini. In particular our result holds when b is steady nearly incompressible BV vector field.

Inside-Outside Duality and Computation of Interior Eigenvalues from Far-Field Data

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Similar connections also exist for inverse exterior scattering problems - for instance, if zero is an eigenvalue of the far-field operator at a fixed wave number, then the squared wave number is an interior eigenvalue. Despite it is in general wrong that interior eigenvalues correspond to zero being an eigenvalue of the far field operator, one can prove a pretty direct characterization of interior eigenvalues via the behavior of the phases of the eigenvalues of the far-field operator.

In this talk, we present this characterization and sketch its proof for Dirichlet, Neumann, and Robin boundary conditions. Then we extend this theory to impenetrable scattering objects and show via a couple of numerical examples that one can indeed use this characterization to compute interior eigenvalues of unknown scattering objects from the spectrum of their far-field operators.

Our motivation to study this so-called inside-outside duality comes from a paper by Eckmann and Pillet (1995). This is joint work with Andreas Kirsch (KIT) and Stefan Peters (University of Bremen).

]]>Lie subalgebras of Vector Fields and Jacobian Conjecture

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(a) The Jacobian Conjecture holds in dimension 2;

(b) All Lie subalgebras

(c) All Lie subalgebras

Lifting automorphisms of adjoint representations

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We also showed that one can choose a biholomorphic lift Ψ such that Ψ(gv)=σ(g)Ψ(v), g ∈ G, v ∈ V, where σ is an automorphism of G. This leaves open the following questions: Can one lift holomorphic automorphisms of Z? Which automorphisms lift if V is not large? We answer the first question in the affirmative and also answer the second question. ]]>

Efficient multigrid calculation of the far field map of the Helmholtz equation

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Finiteness for Plücker varieties

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I will sketch a proof that any "bounded" Plücker variety is defined set-theoretically by equations of bounded degree, and in fact by finitely many equations up to symmetry. So far, this statement was unknown even for the first secant variety of the Grassmannian. The talk is based on joint work with Rob Eggermont, and inspired by Snowden's work on Delta-varieties. ]]>

Existence of physical solutions to the semigeostrophic equations

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The semigeostrophic equations are a set of equations which model large-scale atmospheric/ocean flows.

The system admits a dual version, obtained from the original equations through a change of variable. Existence for the dual problem has been proven in 1998 by Benamou and Brenier, but the existence of a solution of the original system remained open due to the low regularity of the change of variable.

In the talk we prove the existence of distributional solutions of the original equations, both in R^{3} and in a two-dimensional periodic setting. The proof is based on recent regularity and stability estimates for Alexandrov solutions of the Monge-Ampère equation, established by De Philippis and Figalli.

Lie Algebra generated by LNDs on surfaces {xy=p(z)}

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Plane Cremona transformations of fixed degree

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Rigidity of equality cases in symmetrization inequalities

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In this talk we first introduce a measure-theoretic notion of connectedness, inspired by Federer's notion of indecomposable current, that is then exploited to prove several characterization results of rigidity of equality cases, both in the case of the classical Steiner's inequality, as well as in the case of Ehrhard's inequality for Gaussian perimeter. This is a joint work with Filippo Cagnetti (U. Sussex), Maria Colombo (SNS Pisa), and Guido De Philippis (U. Bonn). ]]>

Rough solutions to parabolic equations

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]]>

The Hessian operator in Full Waveform Inversion: quantitative imaging of complex subsurface structures

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Essential dimension of fibered categories

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Quasimorphisms and defect spaces

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The fractional Yamabe problem

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In the last years, a substantial amount of work has been devoted to understand elliptic, parabolic and hyperbolic problems with non local diffusion. In this talk, I will introduce a new class of conformally covariant operators of fractional order generalizing the scalar and Paneitz curvature. I will describe the associated Yamabe problem, in the regular and singular settings. I will give some existence results and discuss open problems.

]]>Measure-valued solutions of transport equations - where discrete meets continuous

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We consider a one-dimensional transport (balance) equation with velocity which has non-Lipschitz zeroes. This leads to non-uniqueness and concentration of characterics and dynamics with both discrete and continuous components. To deal with these effects, we use measure-valued solutions and the so-called measure-transmission conditions. A metric in the space of Radon measures allowing to define unique and stable solutions is introduced. The equation under consideration was proposed as a structured population model of cell differentiation.

]]>A new approach to solve the inverse scattering problem for the wave equation

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1 - the Time-Reversed Absorbing Condition method: It combines time reversal techniques and absorbing boundary conditions to reconstruct and regularize the signal in a truncated domain that encloses the obstacle. This enables us to reduce the size of the computational domain where we solve the inverse problem, now from virtual internal measurements.

2 - the Adaptive Inversion method: It is an inversion method which looks for the value of the unknown wave propagation speed in a basis composed by eigenvectors of an elliptic operator. Then, it uses an iterative process to adapt the mesh and the basis and improve the reconstruction.

We present several numerical examples in two dimensions to illustrate the efficiency of the combination of both methods. In particular, our strategy allows (a) to reduce the computational cost, (b) to stabilize the inverse problem and (c) to improve the precision of the results.

]]>Extensions of principal G_a-bundles over the punctured plane

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the punctured affine plane

A Geometric Uncertainty Principle with an Application to Pleijel's estimate

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It is obvious that there is no tiling of the Euclidean plane with unit disks (any three disks have a gap in the middle): we prove a quantitative version of this statement. This simple insight has applications in spectral geometry: it tells us something about the topological structure of the vibration profile of a (possibly oddly-shaped) drum and allows us to recover an improved version of Pleijel's estimate (which was also recently done by Bourgain).

]]>Willmore spheres in Riemannian manifolds

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Given an immersion f of the 2-sphere in a Riemannian manifold (M,g) we study quadratic curvature functionals of the type: \int_{f(S^2)} H^2, \int_f(S^2) A^2, \int_{f(S^2)} )|Aº|^2, where H is the mean curvature, A is the second fundamental form, and Aº is the tracefree second fundamental form. Minimizers, and more generally critical points of such functionals can be seen respectively as GENERALIZED minimal, totally geodesic and totally umbilical immersions. In the seminar I will review some results (obtained in collaboration with Kuwert, Rivière and Shygulla) regarding the existence and the regularity of minimizers of such functionals. An interesting observation regarding the results obtained with Rivière is that the theory of Willmore surfaces can be usesfull to complete the theory of minimal surfaces (in particular in relation to the existence of canonical smooth representatives in homotopy classes, a classical program started by Sacks and Uhlenbeck).

]]>On the structure of null sets in Euclidean spaces

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Examples of rigid surfaces with infinitely transitive cylinders

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Rademacher theorem with respect to singular measures.

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Rademacher theorem states that every Lipschitz function on the euclidean space is differentiable almost everywhere with respect to the Lebesgue measure. In this talk I will explain how this statement should be modified when the Lebesgue measure is replaced by an arbitrary singular measure, and in particular I will show that the differentiability properties of Lipschitz functions with respect to such a measure are exactly described by the decompositions of the measure in terms of (measures on) rectifiable curves. This result is directly related to recent work by many authors, including myself, David Bate, Marianna Csornyei, Peter Jones, Andrea Marchese, and David Preiss.

]]>Adaptive Approximations of Parametric PDE-Constrained Control Problems

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Yet another level of challenge are control problems constrained by evolution PDEs involving stochastic or countably many infinite parametric coefficients: for each instance of the parameters, this requires the solution of the complete control problem.

Our method of attack is based on the following new theoretical paradigm. It is first shown for control problems constrained by evolution PDEs, formulated in full weak space-time form, that state, costate and control are analytic as functions depending on these parameters. Moreover, we establish that these functions allow expansions in terms of sparse tensorized generalized polynomial chaos (gpc) bases. Their sparsity is quantified in terms of p-summability of the coefficient sequences for some 0 < p <= 1. Resulting a-priori estimates establish the existence of an index set, allowing for concurrent approximations of state, co-state and control for which the gpc approximations attain rates of best N-term approximation. These findings serve as the analytical foundation for the development of corresponding sparse realizations in terms of deterministic adaptive Galerkin approximations of state, co-state and control on the entire, possibly infinite-dimensional parameter space.

The results were obtained with Max Gunzburger (Florida State University) and with Christoph Schwab (ETH Zuerich).

]]>The Field of Definition of Point-Sets in P^1

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Nonlinear bounds states on manifolds

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We will discuss the results of several joint ongoing projects (with subsets of collaborators Pierre Albin, Hans Christianson, Colin Guillarmou, Jason Metcalfe, Laurent Thomann and Michael Taylor), which explore the existence, stability and dynamics of nonlinear bound states and quasimodes on manifolds of both positive and negative curvature with various symmetry properties.

]]>New explicit examples of conformally constrained Willmore surfaces

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KAM for quasi-linear KdV

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Geometry and Invariants of the Affine Group

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Enstrophy dissipation in 2D incompressible fluids

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I will present some (old) results on the transport and dissipation of enstrophy in 2D incompressible flows. Enstrophy is half the space integral of vorticity squared, and it is a relevant quantity in 2D turbulence. I consider initial data with vorticity in L^{2} and its logarithmic refinements and study exact transport of enstrophy by the velocity field. I also consider data in the larger Besov space $B^{0}_{2,\infty}$ and study the existence of well-defined enstrophy defects, measuring the rate of enstrophy dissipation.

This is joint work with Milton Lopes Fihlo and Helena Nussenzveig Lopes.

]]>Stochastic Shape Optimization by Analogues from Operations Research

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At this point, there is a perfect match with two-stage stochastic programming: after having taken a non-anticipative decision in the first stage, and having observed the random data, a well-defined second-stage problem remains and is solved to optimality. Suitable objective functions complete the formal descriptions of the models, for instance, costs in the stochastic-programming setting and compliance or tracking functionals in shape optimization.

Stochastic programming now offers a wide collection of models to address shape optimization under uncertainty. This starts with risk neutral models, is continued by mean-risk optimization involving different risk measures, and will finally lead to analogues in shape optimization of decision problems with stochastic-order (or dominance) constraints.

In the talk we will present these models, discuss solution methods, and report some computational tests.

]]>Transitivity of automorphism groups of Gizatullin surfaces

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**Conjecture (Gizatullin):** The big orbit of a Gizatullin surface *V* coincides with its smooth locus, i. e. O = *V*_{reg} .

We show that the action of the automorphism group of a smooth Gizatullin surface with a distinguished and rigid extended divisor is not transitive in general. Thus such surfaces represent counterexamples to Gizatullin’s conjecture. For such surfaces we give an explicit orbit decomposition of the natural action of the automorphism group. Moreover, the automorphism group of such smooth Gizatullin surfaces can be represented as an amalgamated product of two automorphism subgroups.

]]>Mean field and Gross-Pitaevskii limits of quantum dynamics

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Topology preserving diffusions: the example of divergence-free vector fields

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The Lüroth problem and the Cremona group

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After a brief historical survey, I will recall the counter-examples found in the 70's; then I will describe a quite simple (and new) counter-example, and its application to the study of finite simple groups of birational automorphisms of

How many parameters are needed to define the general equation of degree n?

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Uniqueness of particle trajectories for incompressible fluids.

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We show some regularity results for some classes of 2D incompressible fluids, needed to show uniqueness of particle trajectories.

]]>Collocation methods for nonlinear Riemann-Hilbert problems on doubly connected domains

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The nonlinear boundary conditions for the desired holomorphic solutions lead to nonlinear singular integral equations on the boundary which belong to the class of quasiruled Fredholm maps defined on quasicylindrical domains in appropriate separable Banach spaces.

The closed boundary data give a priori estimates for the modulus of solutions which in turn implies a priori estimates in the Sobolev spaces considered here. For this class of problems, the Shnirelman--Efendiev degree of mappings can be defined which allows to investigate the existence of solutions if the boundary conditions satisfy some topological assumptions.

The lifting of the boundary value problem via holomorphic transformation onto the universal covering of the unit disc allows to construct a homotopic deformation of the lifted nonlinear singular integral equations to a uniquely solvable case which implies that the degree of mapping is 1 and existence of (in fact at least two) solutions follows.

If the nonlinear integral equations on the boundary are appoximated by trigonometric point collocation then the theory also implies that approximate solutions exist and converge asymptotically.

]]>Maximum-Likelihood duality for determinantal varieties

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(

with

In recent work by Hauenstein, Rodriguez, and Sturmfels the maximum-likelihood degree of determinantal varieties was studied. Extensive computations using numerical algebraic geometry led to the conjecture that the maximum-likelihood degree of the variety of rank-r matrices whose entries add up to 1 equals that of the variety of corank-(r-1) matrices whose entries add up to 1. I will present a proof of that conjecture, and variations of it for symmetric and skew-symmetric matrices. Joint work with Jose Rodriguez. ]]>

On the optimization of current carrying multicables

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Simulation results and experimental studies show that the positioning of the single cables has important influence on the maximum temperatures. In order to find an optimal cable design, i.e. to arrange the single cables with fixed cross section and current such that the maximum temperature is minimized, a shape optimization problem is formulated. We derive an adjoint system and the shape gradient using the formal Lagrange approach. The effect of the discontinuity of some coefficients on the shape gradient is shown. By application of different (nonlinear) optimizers combined with the finite element solver COMSOL Multiphysics, a solution is obtained numerically. In this talk, we present the modeling of the problem, the derivation of the shape gradient and numerical results.

This is joint work with Helmut Harbrecht and Thomas Apel.

]]>Real differential forms and currents in p-adic geometry

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In this talk, I will begin with a presentation of that of Berkovich. It roughly consists of 'adding plenty of points' to the usual

Then I will try to illustrate the following slogan: 'to see the good analog of a complex object in the *p*-adic world, one often has to work with Berkovich spaces', through three examples: spectral theory; dynamical systems; and the theory of real (p,q)-forms and related notions (integrals, boundary integrals, curvature forms of metrized line bundles) that as been recently developped in the framework of Berkovich spaces by Chambert-Loir and myself, and that I will try to describe in some detail.

Traffic flow modeling by conservation laws

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Several phenomena in traffic flow can be modeled through the use of conservation laws. We present two PDE-ODE coupled models that are used in different traffic situations. First, we consider a model that applies to moving bottlenecks and then we consider a model that applies in control problems for highway ramp metering. We provide a rigorous analytical framework for the Cauchy and Riemann problems and we show some numerical simulations.

]]>Infinite transitivity on universal torsors

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Cyclic sieving for generalised non-crossing partitions associated to complex reflection groups

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The role of anisotropy in the three-dimensional Navier-Stokes equations

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In this talk I will present some recent results in the study of the Cauchy problem for the three-dimensional Navier-Stokes equations. In particular using the fact that the two-dimensional equation is well-posed, I will try to explain the role of "spectral anisotropy" in the resolution of the equations.

]]>Blow up for critical wave equations on curved backgrounds.

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Following the work of Krieger, Schlag, and Tataru, we construct a family of blow-up solutions with finite energy norm to the equation

∂_{t}^{2}u - Δg u = |u|^{4} u.

This family has a continuous rate of blow up, but In contrast to the case where g is the Minkowski metric, the argument used to produce these solutions can only obtain blow up rates that are bounded above.

This is joint work with S. Shashahani.

]]>Regularity theory of degenerate elliptic equations in nondivergence form with applications to homogenization.

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I will present a regularity result for degenerate elliptic equations in nondivergence form. In joint work with Charlie Smart, we extend the regularity theory of Caffarelli to equations with possibly unbounded ellipticity - provided that the ellipticity satisfies an averaging condition. As an application we obtain a stochastic homogenization result for such equations which is equivalent to an invariance principle for random diffusions in random environments. The degenerate equations homogenize to uniformly elliptic equations, and we give an estimate of the ellipticity in terms of the averaging condition.

]]>On automorphisms of the affine Cremona group as an abstract group and as an ind-group

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In a next step we only consider those automorphisms of $\mathcal{G}_n$ that also respect its additional algebraic structure as an ind-group. It turns out that these are exactly the inner automorphisms of $\mathcal{G}_n$. To prove this I will follow an idea recently presented by Belov-Kanel and Yu, which uses tame approximation.

These are results from my Master's thesis under the supervision of Hanspeter Kraft.

]]>Two 45 minutes talks: "An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators" and "The group of automorphisms of the algebra of one-sided inverses of a polynomial algebra"

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In 1968, Dixmier posed six problems for the algebra of polynomial differential operators, i.e. the Weyl algebra. In 1975, Joseph solved the third and sixth problems and, in 2005, I solved the fifth problem and gave a positive solution to the fourth problem but only for homogeneous differential operators. The remaining three problems are still open. The first problem/conjecture of Dixmier (which is equivalent to the Jacobian Conjecture as was shown in 2005-07 by Tsuchimito, Belov and Kontsevich) claims that the Weyl algebra 'behaves' as a finite field extension. In more detail, the first problem/conjecture of Dixmier asks: is it true that an algebra endomorphism of the Weyl algebra is an automorphism? In 2010, I proved that this question has an affirmative answer for the algebra of polynomial integro-differential operators. In my talk, I will explain the main ideas, the structure of the proof and recent progress on the first problem/conjecture of Dixmier.

**The group of automorphisms of the algebra ****of one-sided inverses of a polynomial ****algebra**

The algebra *S*_{n} of one-sided inverses of a polynomial algebra *P _{n}* in

The group of automorphisms

When is the blow-up of points or curves in the projective space a weak Fano threefold?

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I will describe the theorem, give the proof, and describe the generalisation to dimension 3, by considering blow-ups of points and curves in the projective space. We can get similar descriptions, the conditions of generality are now in terms of multisecant lines, conic or twisted cubics.

Joint work with Stéphane Lamy.

]]>Vortex sheets for 2D incompressible ideal fluids in domains with boundary

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Following the work of Krieger, Schlag, and Tataru, we construct a family of blow-up solutions with finite energy norm to the equation

∂_{t}^{2}u - Δg u = |u|^{4} u.

This family has a continuous rate of blow up, but In contrast to the case where g is the Minkowski metric, the argument used to produce these solutions can only obtain blow up rates that are bounded above.

This is joint work with S. Shashahani.

]]>When the Garding inequality is ineffective: examples and remedies

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The Garding inequality states that positive pseudo-differential symbols are associated with semi-positive operators. It can be used in particular to show time-exponential growth of solutions to initial value problems for elliptic equations. I will give examples in which Garding fails to give appropriate bounds, and a way to overcome this difficulty. Examples include high-frequency asymptotics of systems based on Maxwell's equations, and compressible Euler systems with a Van der Waals pressure law. In these cases, appropriate bounds are derived via a description of the parametrix of a pseudo-differential system.

]]>On Uncertainties in Low-Frequency Electromagnetics

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Joint work with Sebastian Schöps and Thomas Weiland.

]]>Invariant Hilbert schemes and resolutions of quotient singularities

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In this talk, we will see that *H* is a smooth variety when the dimension of *V* is small, but that *H* is generally singular. When *H* is smooth, the Hilbert-Chow morphism *H* -> *X*//*G* is a canonical resolution of the singularities of the categorical quotient *X*//*G* (=Spec(*k*[*X*]^{G})). Then it is natural to ask what are the good geometric properties of this resolution (for instance if it is crepant).

To finish, we will mention some analogue results in the symplectic setting, that is to say by letting *p*=*q* and replacing *X* by the zero fiber of the moment map. The quotients that we get by doing this are isomorphic to the closures of nilpotent orbits, and the Hilbert-Chow morphism is a resolution of their singularities (sometimes a symplectic one).

Regularity theory for area-minimizing currents

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A well-known theorem of Almgren shows that area-minimizing integral *k*-dimensional currents in a Riemannian manifold of arbitrary dimension* N* are regular up to a set of closed dimension of Hausdorff dimension at most *N-2*. In a joint work with Emanuele Spadaro we give a much shorter proof of this statement in the euclidean setting, following the general program of Almgren but introducing new ideas at the various steps. In this talk I will explain some if these ideas. A generalization of our proof to the Riemannian case is work in progress.

Exponential Krylov subspace time integration for electromagnetic modeling: some recent advances

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Hyperbolic volume, estimates, and simplices

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Painlevé equations, and model theory

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Spectral methods in the study of semi-linear equations in Hilbert spaces.

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I will present the results on the existence of solutions to semi-linear equation

Lx+N(x)=0,

where L is a linear and N a nonlinear operator defined on Hilbert space. I concentrate on the case when 0 is in an essential spectrum of L. The two main methods which I use are: topological degree in infinite-dimensional spaces and the spectral theory for linear operators in Hilbert spaces. This results are part of my Ph.D. project.

]]>Ricci curvature lower bounds and branching geodesics in metric spaces

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Analytic tools for the study of spaces with Ricci curvature bounded from below

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- heat flow
- stability properties
- differentials and gradients in a non-smooth world
- Laplacian comparison estimates

If time permits, some consequences in terms of the geometry of these spaces will also be given.

A scattering theory construction of dynamical black holes

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Regularity issues for local minimizers of the Mumford-Shah functional in dimension two

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In this talk I shall focus on the higher integrability property enjoyed by the approximate gradients of local minimizers of the 2d Mumford-Shah energy. Related regularity issues shall be also discussed.

This is joint work with C. De Lellis (Universitaet Zuerich).

]]>Preperiodic points for rational maps

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Non-stadard solutions of the compressible Euler system

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The deceivingly simple–looking compressible Euler equations of gas dynamics have a long history of important contributions over more than two centuries. If we allow for discontinuous solutions, uniqueness and stability are lost. In order to restore such properties further restrictions on weak solutions have been proposed in the form of entropy inequalities. In this talk, we will discuss some counterexamples to the well–posedness theory of entropysolutions to the multi–dimensional compressible Euler equations. First, we show failure of uniqueness on a ﬁnite–time interval for entropy solutions starting from any continuously diﬀerentiable initial density and suitably constructed initial linear momenta. In other words, we prove that there exist wild initial data allowing for inﬁnitely many distinct entropy weak solutionsnof the compressible Euler system. Finally, we present a new upshot: a classical Riemann datum is a wild initial datum in 2 space–dimensions. All our methods are inspired by a new analysis of the incompressible Euler equations recently carried out by De Lellis and Székelyhidi and based on a revisited “h-principle”.

]]>Finite element approximation of large bending isometries

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Algebraic and geometric presentations of algebraic groups over local fields

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Mathematical billiards and flows on surfaces

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Spectral stability estimates for the Laplace operator with either Dirichlet or Neumann boundary conditions

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The talk will focus on the eigenvalue problem for the Laplace operator defined in an open and bounded domain, with homogenous conditions of either Dirichlet or Neumann type assigned at the boundary. Under fairly weak regularity assumptions on the domain, the problem admits a diverging sequence of nonnegative eigenvalues. I will discuss some new quantitative estimates controlling how each of the eigenvalues change when the domain is perturbed. These estimates apply to Lipschitz and to so-called Reifenberg-flat domains. The proof is based on an abstract lemma which applies to both the Neumann and the Dirichlet problem and which could be applied to other classes of domains.

The talk will be based on joint works with A. Lemenant and E. Milakis.

]]>A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics

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Adjoint methods for gradient-based optimization of oil recovery

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Well-behaved separating algebras

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Classification of solutions of the radial energy-critical wave equation

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About the mathematical derivation of Ohm's law

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A posteriori error analysis for optimal control problems governed by partial differential equations

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Endomorphisms of Varieties (joint work with Rafael Andrist)

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Valuative analysis of the dynamics of superattracting germs in C^2

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I shall present a joint work with William Gignac, on the growth of attracting rates for iterates of a superattracting germ in C

If time allows, I will present other applications and examples. ]]>

Antisymmetry and regularity

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Starting from the example of harmonic maps, we will find a class of PDE problems which enjoy an additional, at first glimpse hidden property: Antisymmetry! This feature enables us to deduce regularity assertions which heavily rely on Wente's theorem. For this latter, various approaches will be discussed. The presentation will be completed by a version of Wente's result for arbitrary dimension.

]]>Applications of Time Reversed Absorbing Condition

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On the Connectedness of Forcing Schemes

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Repeated modifications of orthogonal polynomials by linear divisors

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Monomial orders: from computational algebra to linguistic typology

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In this lecture I will first give an overview of this principle in both commutative and computational algebra. I will then use monomial orders to explain the analogies between two distinct, yet similar, theories in phonology: Optimality Theory and Harmonic Grammars. In particular I will show how to use a mathematical object, called Gröbner fan, to build models of linguistic typology.

]]>The vanishing viscosity problem for the Navier-Stokes equations in bounded domains

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In this talk I will discuss the vanishing viscosity problem for the Navier-Stokes equations in a bounded domain. It is well-known that when Dirichlet conditions are imposed on the boundary the inviscid limit is currently an open and difficult problem. On the other hand when other type of boundary conditions are considered the situation became simpler. In this talk a particular type of Navier boundary conditions involving only the vorticity of the velocity field are considered. In particular, I will discuss recent results obtained in collaboration with Luigi Berselli (University of Pisa) concerning the inviscid limit in energy norm of the Leray weak solutions and the inviscid limit in higher norms of local smooth solutions of the Navier-Stokes equations.

]]>The T-graph of the Hilbert scheme of points in the plane

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to an action of the torus on the Hilbert scheme of points in the plane.

The T-graph is the graph whose vertices are the fixed points under the

torus action (corresponding to monomial ideals), where two such vertices are connected by an edge if there exists a one-dimensional torus orbit whose closure contains the corresponding fixed points. I will describe some conditions for the existence of an edge between two given vertices.

This is joint work with Diane Maclaga.

]]>Numerics for Inverse Problems in Biomedical Imaging

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The resulting parameter estimation schemes have the underlying partial differential equations as side-constraints, and the solution of these optimization problems often requires solving the partial differential equation thousands or hundred of thousands of times. The development of efficient schemes is therefore of great interest for the practical use of such imaging modalities in clinical settings. In this talk, the formulation and efficient solution strategies for such inverse problems will be discussed, and we will demonstrate its efficacy using examples from our work on Optical Tomography, a novel way of imaging tumors in humans and animals. The talk will conclude with an outlook to even more complex problems that attempt to automatically optimize experimental setups to obtain better images.

]]>The Moser-Trudinger equation on a disk: blow-up behavior and non-existence

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We study the Moser-Trudinger equation Δu = λu Exp(u2), λ>0 on a 2-dimensional disk, arising from the Moser-Trudinger sharp embedding of H^{1}_{0}(Disk) into the Orlicz space of functions u with Exp(u2) integrable. We answer some long standing open questions:

a) The weak limit of a blowing-up sequence of solutions to the Moser-Trudinger equation on a disk is 0.

b) The Dirichlet energy of a blowing-up sequence of solutions on a disk converges to 4π.

c) For L large enough, the Moser-Trudinger equation on a disk admits no solution with Dirichlet energy larger than L.

This work is joint project with Andrea Malchiodi (SISSA - Trieste).

]]>The finite subgroups in the plane Cremona group over field of complex numbers

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nonsingular del Pezzo surfaces and conic bundles.

The description by equations of nonsingular Del Pezzo surfaces in weighted projective spaces and description of their automorphism groups is a very old completed problem. The description of the ''minimal'' finite subgroups of automorphism groups of nonsingular del Pezzo surfaces was done by I.V. Dolgachev and V.A. Iskovskikh in 2007. I constructed a method that allows us to describe by equations in weighted projective spaces the nonsingular conic bundles with an action of ''minimal'' subgroup of automorphisms . I applied the method in case, when the ''minimal'' subgroup of automorphisms is nonsolvable.

]]>The representation type of a projective variety

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ACM sheaves also provide a criterium to determine the complexity of the underlying variety. More concretely, this complexity can be studied in terms of the dimension and number of families of indecomposable ACM sheaves that it supports, namely its \emph{representation type}. Along this line, a variety that admits only a finite number of indecomposable ACM sheaves (up to twist and isomorphism) is called of

On the other extreme of complexity, we would find the varieties of

Microscopic derivation of the Ginzburg-Landau equation

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We give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.

]]>Euler characteristic and Moebius inversion for finite categories

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This talk is based on joint work with Tom Leinster (Glasgow). ]]>

On the Gauss problem with Riesz potential

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In **R**^{n} , n≥2, we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel |x−y|^{α−n}, where 1<α<n, for the Gauss variational problem, which is considered for finitely many compact, mutually disjoint, boundaryless (n−1)–dimensional Lipschitz manifolds Γ_{l}, l ∈ L, each Γ_{l} being charged with Borel measures with the sign α_{l} = ±1 prescribed.

For Newton potentials, i.e. the special case α = 2, this problem goes back to C. F. Gauss who used it as the model for electrostatic fields. Nowadays it is of interest for the storage of charges as produced by solar electricity modules. The more general Riesz kernels have also applications for finding numerical integration formulae on manifolds.

We show that the Gauss variational problem over an affine cone of Borel measures can alternatively be formulated as a minimum problem over an affine cone of surface distributions belonging to the Sobolev–Slobodetski space H^{−ε/2}(Γ), where ε := α−1 and Γ :=∪_{ l∈L} Γ_{l}. This allows the application of simple layer boundary integral operators on Γ. A corresponding numerical method is based on the Galerkin–Bubnov discretization with piecewise constant boundary elements. For n = 3 and α = 2, multipole approximation and in the case 1 < α < 3 = n wavelet matrix compression is applied to sparsify the system matrix. For a subclass of these problems, a dual formulation results in linear boundary integral equations although the original problem is nonlinear. Numerical results are presented to illustrate the approach.

Fast methods for computing pseudospectral quantities

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This talk is based on joint work with Nicola Guglielmi, Christian Lubich, and Bart Vandereycken.

]]>Nonscattering solutions and blowup at infinity for the critical wave equation

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I present some recent results, obtained in collaboration with Joachim Krieger, on novel types of solutions to the critical wave equation in 3 spatial dimensions. These solutions either blow up at infinity or vanish at a prescribed rate. The existence of such exotic dynamics violates a strong version of the soliton resolution conjecture.

Francois Bouchut:

TBA

]]>A generalized Newton's formula and Macdonald symmetric functions

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Non-conforming high order methods for seismic wave propagations in heterogeneous media

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Here, we consider a Discontinuous Galerkin (DGSE) and a Mortar (MSE) spectral element methods coupled with the leap-frog time integration scheme to simulate seismic wave propagations in two and three dimensional heterogeneous media. The main advantage with respect to conforming discretizations, as SE method, is that DGSE and MSE discretizations can accommodate discontinuities, not only in the parameters, but also in the wavefield, while preserving the energy. The domain of interest Ω is assumed to be union of polygonal subdomain Ω_{i}. We allow this subdomain decomposition to be geometrically non-conforming. Inside each subdomain Ω_{i}, a conforming high order finite element space associated to a partition *T*_{hi}(Ω_{i}) is introduced. We consider different polynomial approximation degrees within different subdomains. To handle non-conforming meshes and non-uniform polynomial degrees across ∂Ω_{i} , a DG or a Mortar discretization is considered.

Applications of the DGSE and MSE methods to simulate realistic seismic wave propagation problems are presented.

Joint work with: P.F. Antonietti, A. Quarteroni and F. Rapetti.

]]>Global existence and collisions for some configurations of nearly parallel vortex filaments

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

A system of simplified equations has been derived by Klein, Majda and Damodaran to describe the dynamics of nearly parallel vortex filaments in incompressible 3D fluids. This system combines a 1D Schrödinger-type structure together with the 2D point vortex system. Global existence for small perturbations of exact parallel filaments has been established by Kenig, Ponce and Vega in the case of two filaments and for particular configurations of three filaments. In this talk I will present large time existence results for particular configurations of four filaments and for other particular configurations of N filaments for any N larger than 2. I will also discuss some situations of finite time filament collapse. This is joint work with Valeria Banica.

]]>Integral closure and affine varieties with a torus action

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In this talk, we provide some examples whenever

Consider the group

We classify the

The problem of normalization for complexity zero case is well known (monomial or toric case). For the complexity one, the normalization of

Assume that

Ground states for the nonlinear Schrödinger-equation with interface

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We are interested in ground states for the nonlinear Schrödinger-equation (NLS) with an interface between two purely periodic media. This means that the coefficients in the NLS model two different periodic media in each halfspace. The resulting problem no longer has a periodic structure. Using variational methods we give conditions on the coefficients such that ground states are created/prevented by the interface.

]]>Choptuik's critical spacetime exists

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About twenty years ago, Choptuik studied numerically the gravitational collapse (Einstein field equations) of a massless scalar field in spherical symmetry, and found strong evidence for a universal, self-similar solution at the threshold of black hole formation. We give a rigorous, computer assisted proof of the existence of Choptuik's spacetime, and show that it is real analytic. This is joint work with E. Trubowitz.

]]>On h-vectors of matroids

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Dimensional reduction of the optimal transport problem with convex norm costs

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We consider the optimal transportation problem with cost functions given by generic convex norms in **R**^{d} and absolutely continuous first marginals. We show the existence of a partition of **R**^{d} into k-dimensional sets, k=0,...,d, such that every optimal transport plan can be characterized, via disintegration of measures, as a family of optimal transport plans each moving a conditional probability of the first marginal inside one of these k-dimensional sets, along the directions of an extremal k-dimensional cone of the convex norm. Moreover, the conditional probabilities of the first marginal on these sets are absolutely continuous with respect to the k-dimensional Hausdorff measure on the k-dimensional sets on which they are concentrated, thus settling the longstanding Sudakov's problem of the existence of locally affine decompositions of **R**^{d} that reduce norm cost transportation problem to families of lower dimensional ones. Finally, due to the minimality of our partition with respect to this "dimensional reduction" property, applications to secondary cost functions obtained first minimizing with respect to a convex norm and then with respect to a finer one (e.g., a strictly convex one) will be shown. These results were obtained in collaboration with Stefano Bianchini (SISSA, Trieste).

The QDMRG algorithm and recent advances in tensor approximation

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Joint work with Th. Rohwedder and S. Holtz

]]>An introduction to (partial) regularity theory for elliptic problems

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In this seminar we will give a survey on some aspects of the classical regularity theory for W^{1,p}-solutions to elliptic problems (convex variational integral or elliptic systems), restricting ourselves to simple model cases and explaining the challenges behind proving such results. For scalar valued solutions full regularity (continuous or even better) can be established under very mild assumptions, which is nowadays known as the De Giorgi-Nash-Moser theory. In the vectorial case instead, the various component functions and their partial derivative can interact in such a way that the system or variational integral under consideration allows discontinuous or even unbounded solutions, and in fact various counterexamples to full regularity have been constructed. As a consequence, only partial regularity can be expected, in the sense that the solution (or its gradient) is locally continuous outside of a negligible set (the singular set). We will give some heuristics on the general

approach to partial regularity results and then we briefly discuss how in some particular situations (small space dimensions, special structure conditions) an upper bound on the Hausdorff dimension of the singular set can be obtained.

T.B.A.

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

How primary decomposition of monoid congruences and binomial ideals is wrong

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Ideally one would want to carry out algebraic computations, such as primary decomposition of binomial ideals, entirely in this combinatorial language. We will present such a calculus, enabling one to compute by looking at pictures of monoids.

]]>All-electron Density Functional Theory Calculations for atoms and molecules using Multiwavelets and Massively Parallel Architectures

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Minimal representatives of even liaison classes

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In this talk we introduce a theoretical definition for 'minimal' representatives in any even linkage class. We show that these ideals

exist under reasonable assumptions on the linkage class, and, in general, if they exist they are essentially unique.

We then show that these ideals minimize homological invariants (e.g. Betti numbers, multiplicity, etc.) and they enjoy the best homological and local properties among all the ideals in their even linkage class. This justifies why they are, in some sense, the `best' possible ideals in the even linkage class.

We provide several classes of ideals that are the minimal representatives of their even linkage classes (including determinantal ideals) and, if time permits, show an easy application to produce more evidence towards the Buchsbaum-Eisenbud-Horrocks Conjecture. ]]>

Reduced Basis Surrogate Models for Parameter Optimization of Evolution Problems

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Von Euler, Schwarz, Ritz und Galerkin zum modernen Wissenschaftlichen Rechnen

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Generalized Inversion Statistics

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A Heterogeneous Multiscale Method for highly-oscillatory Hamiltonian systems with solution-dependent frequencies

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We will explain the Heterogeneous Multiscale Method (HMM)[1], which is believed to provide a numerical method for all kind of multiscale systems to overcome the difficulties of numerical integration generated by highly-oscillatory components. We will formulate the HMM for highly-oscillatory Hamiltonian systems with solution-dependent frequencies more precisely for the double spring pendulum with very stiff springs. Finally we will discuss the drawbacks of this method in case of solution-dependent frequencies.

[1] E, W.; Engquist, B.: The heterogeneous multi-scale method, *Comm. Math. Sci.*, **1**, 87--133, 2003.

On the annihilators of rational functions in the Lie algebra of derivations of k[x, y]

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Let us consider the Lie algebra W

of subalgebras in W

Let us consider natural action of the Lie algebra W_{2}(K) on the field of rational functions K(x,y). Recall that every derivation D

of W_{2}(K) of the ring K[x,y] can be uniquely extended to a derivation of the field K(x,y). It is natural to consider for a

fixed rational function u in K(x,y)\K the set A_{W_{2}}(u) of all derivations D of W_{2} such that D(u)=0. This set is called the annihilator

of u in W_{2}(K). It is a Lie subalgebra of W_{2}(K) and at the same time a K[x,y]-submodule of the K[x,y]-module W_{2}(K).

We show that it is a free K[x,y]-module of rank 1 and describe centralizers of elements and the maximal abelian subalgebras of the

Lie algebra A_{W_{2}}(u).

Time Domain Decomposition based on Multirate Multistep Timestepping Techniques for the Particle-In-Cell Method

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Holomorphic automorphisms of Danielewski surfaces

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In case of a Danielewski surface the so-called overshear group is dense in the group of holomorphic automorphisms. We describe the group structure of the overshear group with the help of Nevanlinna theory. ]]>

Hierarchical Matrix Approximation with Blockwise Constraints and h-Independent Convergence for Elliptic Problems

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

In this talk we present a new technique to locally preserve constraints inside the hierarchical matrix approximation. Numerical experiments indicate that imposing these local constraints leads to constant number of iteration steps when solving elliptic partial differential equations of second order while without preserving these constraints the number of iteration steps grow as *h* → 0. We will further discuss this approach from the theoretical point of view and will sketch why our approximate hierarchical *LU* decomposition leads to a spectral equivalent approximation.

This is joint work with Mario Bebendorf and Michael Bratsch from the University of Bonn.

]]>Hilbert depth and Stanley decomposition

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Stanley decompositions of multigraded modules M over polynomials rings have been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition, called the

We test our new notion on the syzygy modules of the residue class field of K[X

Related ideals are the powers of the irrelevant maximal ideal. For them the (standard graded) Hilbert depth can be computed

precisely. ]]>

A-hypergeometric functions

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

Explicit continuous and discontinuous Galerkin methods for 3D seismic modeling with the second order wave equation

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

T.B.A.

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Probabilistic UQ for PDEs with Random Data: A Case Study

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We compare two approaches: Gaussian process emulators and stochastic collocation combined with geostatistical techniques for determining the parameters of the input random field's probability law. The second approach involves the numerical solution of the PDE with random data as a parametrized deterministic system. The calculation of the statistics of the travel time from the solution of the stochastic model is formulated for each of the methods being studied and the results compared.

]]>Irreducible characters of spin wreath products

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Stochastic partial differential equations: examples and recent results

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

T.B.A.

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Multi-level Monte Carlo Finite Element method for parabolic stochastic partial differential equations

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T.B.A.

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Infinite-dimensional non-positively curved symmetric spaces of finite rank

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We will look at some infinite dimensional symmetric spaces of non-positive curvature which have a remarkable property : they have finite rank. There exists a positive integer p such that any isometrically embedded Euclidean space has dimension at most p.

The talk will be focused on the properties of these spaces and some group actions which come from (non-unitary) infinite-dimensional representations.

]]>Vertex operator realizations of Jack symmetric functions

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Contents:

1. partitions and symmetric functions

2. Some history and questions about symmetric functions

3. Vertex operator realization of rectangular Jack functions

4. A special case of Stanley's conjecture and the realization of general Jack functions

5. Open questions ]]>

Singularities of moduli spaces of principal bundles on curves

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Riemannian walk through the symplectic garden

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Finite separating sets and quasi-affine quotients

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Koszul algebras and their syzygies

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The main idea I will discuss is that the syzygies of Koszul algebras have some properties in common with the syzygies of algebras defined by monomials

of degree two. ]]>

Factorial algebraic group actions and categorical quotients

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We begin with a survey of known results concerning categorical quotients.

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we characterize existence of categorical quotient in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of a finitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorial quotients for actions on, e.g. complete, varieties with finitely generated Cox ring via lifting to the universal torsor. ]]>

Die Geometrie des Wortproblems

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Galois groups for inseparable field extensions

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T.B.A.

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Eine Metrik auf Rechtecken und Torusknoten

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Cohomology of line bundles on the cotangent bundle of a complete homogeneous space

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On the problem of connectedness for the Hilbert schemes of space curves

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

It is an open question whether H

I will discuss the state of affairs about this question, and briefly describe work in progress (with the help of Macaulay 2) showing that curves of type (a,a+4) on a smooth quadric surface are in the connected component of extremal curves; this problem was raised in Hartshorne's papers "On the connectedness of the Hilbert scheme of curves in P

On some geometric properties of orbital varieties

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Torsion in the symmetric algebra and images of rational maps

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Their approach was first put on firm mathematical bases by David Cox and several collaborators. The key point in this approach is to control the torsion in the symmetric algebra. This is performed using a construction of Herzog, Simis and Vasconcelos, that gives information on the equations of Rees algebras.

]]>Upgrades and Downgrades of p-divisors

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On the so-called Fermat–Pell Equation x^2−dy^2 =±1

Tags: TAG Events Forschung Mathematik, TAG Events DMI, TAG Events Perlenkolloquium]]>

11512 −92·1202 = 1, 17663190492 −61·2261539802 = 1

and

2275282 −103·224192 = 1,

hence they have not been found by a brute force search!

After a short introduction to this long history we explain the connection with Diophantine approximation and continued fractions, next we say a few words on more recent developments of the subject.

LVMB manifolds and triangulations of spheres

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The talk will mainly deal with a generalization due to Bosio of the LVM manifolds, emphazing the combinatorial aspect of the LVM manifolds. These new manifolds are known as LVMB manifolds. In particular, our aim will be to show the very strong connection between LVMB manifolds, toric varieties and triangulations of spheres.

]]>Shelling, constructing, counting

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Separating invariants for the basic $G_a$-actions

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Parametrizations of Ideals in K[x,y]

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Gorenstein projections and diptych varieties

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tba

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Holomorphic foliations on CP2 and Geometric Invariant Theory

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Birational constructions of automorphisms of affine 3-folds

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Projective Varieties of Low Degree

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Normal singularities with torus actions

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Additive group actions in positive chracteristic

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Quantum invariants of graphs and their geometry

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Algebraic approximations of diffeomorphisms of surfaces

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On rigidity of low degree del Pezzo fibrations

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Using polynomial invariants to compute geometric predicates

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A tropical proof of the Brill-Noether theorem

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The volume conjecture and Chern Simons theory

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Quivers and saturation for tensor products multiplicities for classical groups

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Isomorphism classes of Gorenstein local rings via Macaulay's inverse system

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Beispiel eines Sl_2-Hilbertschemas

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Versal actions with a twist

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A birational characterization of affine varieties with trivial ML invariant

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Picard groups of homogeneous varieties

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Algebraic actions on the affine plane

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h-Principle and fluid dynamics

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Limits of metabelian groups

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Rational points on cubic surfaces

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Was ist ein Faserbündel?

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Some examples coming from the study of the Danielewski hypersurfaces

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Essential dimension of algebraic tori

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Kepler, Newton und numerische Mathematik

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Das Leben von Kurt Heegner

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Aspects of class field theory for global function fields

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Normal subgroup generated by a plane polynomial automorphism

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Gorestein Liaison and determinantal schemes

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Some aspects of A^1-bundles

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On the Bhatwadekar-Dutta-Berson-Vénéreau Polynomials

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Des polygones réguliers aux chorégraphies et aux Hip-Hops

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An open problem on the coordinate ring of a variety with group action

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Spherical orbit closures in simple projective spaces and their normalization

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A sufficient criterion for the asymptotic stability of depths of ideal transformed Rees-modules

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Transcendence of automatic numbers(after Adamczewski and Bugeaud)

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Kann man Mathematik hören? Codes, Arithmetik und Topologie

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Symmetrien und Extremaleigenschaften in negativer Krümmung

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Von Amoeben zu Algen: tropische und arktische Einblicke in die Topologie von Laurent Polynomen

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Die Spirale des Theodorus, numerische Mathematik und spezielle Funktionen

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