A key tool to study the plane Cremona group is its action on a hyperbolic space. Saddly\, in higher rank such an action is not availabl e. Recently in geometric group theory\, actions on CAT(0) cube complexes t urned out to be a powerfull tool to study a large class of groups. \; \; \;

\nIn this talk\, based on a common work with Christia n 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 obta in.

DTEND;TZID=Europe/Zurich:20200225T120000 END:VEVENT BEGIN:VEVENT UID:news987@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191206T022316 DTSTART;TZID=Europe/Zurich:20191209T150000 SUMMARY:Seminar Algebra and Geometry: Michel van Garrel (Warwick) DESCRIPTION:In this joint work with Christian Böhning and Hans-Christian von Bothmer we apply Voisin's criterion of existence of a decomposition o f the diagonal to semistable degenerations. In doing so\, we obtain partia l results towards proving that very general cubic threefolds are stably ir rational. X-ALT-DESC:In this joint work with Christian \;Böhning \;and Hans- Christian von Bothmer we apply Voisin's criterion of existence of a decomp osition of the diagonal to semistable degenerations. In doing so\, we obta in partial results towards proving that very general cubic threefolds are stably irrational. DTEND;TZID=Europe/Zurich:20191209T163000 END:VEVENT BEGIN:VEVENT UID:news922@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191127T114131 DTSTART;TZID=Europe/Zurich:20191203T103000 SUMMARY:Seminar Algebra and Geometry: Erik Paemurru (University of Loughbor ough) DESCRIPTION:It is known that quasismooth 3-fold Fano hypersurfaces with ind ex 1 in weighted projective spaces over ℂ are birationally rigid (not bi rational to any other Fano 3-folds\, conic bundles or del Pezzo fibrations ). But very little is known when they carry non-orbifold singularities. I consider sextic double solids\, one of the simplest such 3-folds\, which h ave an isolated cA_n singularity. I have shown that n is at most 8\, and t hat rigidity fails for n > 3. In this talk\, I will illustrate this phenom enon by giving some examples. X-ALT-DESC:It is known that quasismooth 3-fold Fano hypersurfaces with inde x 1 in weighted projective spaces over ℂ are birationally rigid (not bir ational to any other Fano 3-folds\, conic bundles or del Pezzo fibrations) . But very little is known when they carry non-orbifold singularities. I c onsider sextic double solids\, one of the simplest such 3-folds\, which ha ve an isolated cA_n singularity. I have shown that n is at most 8\, and th at rigidity fails for n >\; 3. In this talk\, I will illustrate this phe nomenon by giving some examples. DTEND;TZID=Europe/Zurich:20191203T120000 END:VEVENT BEGIN:VEVENT UID:news924@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191112T123824 DTSTART;TZID=Europe/Zurich:20191126T103000 SUMMARY:Seminar Algebra and Geometry: Carlos Amendola (University of Munich ) DESCRIPTION:We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all 2-dimensional Gorenstein toric Fano varieties. We show tha t the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two singular points of type A1 and provide explicit expressions that allow to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then ex plore the reasons for the ML degree drop using A-discriminants and interse ction theory. Based on joint work with Dimitra Kosta and Kaie Kubjas. X-ALT-DESC:We study the maximum likelihood estimation problem for several c lasses of toric Fano models. We start by exploring the maximum likelihood degree for all 2-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two singular points of type A1 and provide explicit expressions that allow to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then exp lore the reasons for the ML degree drop using A-discriminants and intersec tion theory. Based \;on joint work with Dimitra Kosta and Kaie Kubjas. DTEND;TZID=Europe/Zurich:20191126T120000 END:VEVENT BEGIN:VEVENT UID:news927@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191104T113358 DTSTART;TZID=Europe/Zurich:20191112T103000 SUMMARY:Seminar Algebra and Geometry: Enrica Mazzon (Imperial College) DESCRIPTION:To a degeneration of varieties\, we can associate the dual inte rsection complex\, a topological space that encodes the combinatoric of th e central fiber and reflects the geometry of the generic fiber. The points of the dual complex can be identified to valuations on the function field of the variety\, hence the dual complex can be embedded in the Berkovich space of the variety.In this talk I will explain how this interpretation g ives an insight in the study of the dual complexes. I will focus on some d egenerations of hyper-Kähler varieties and show that we are able to deter mine the homeomorphism type of their dual complex using techniques of Berk ovich geometry. The results are in accordance with the predictions of mirr or symmetry\, and the recent work about the rational homology of dual comp lexes of degenerations of hyper-Kähler varieties\, due to Kollár\, Laza\ , Saccà and Voisin.This is joint work with Morgan Brown. X-ALT-DESC:To a degeneration of varieties\, we can associate the dual inter section complex\, a topological space that encodes the combinatoric of the central fiber and reflects the geometry of the generic fiber. The points of the dual complex can be identified to valuations on the function field of the variety\, hence the dual complex can be embedded in the Berkovich s pace of the variety.In this talk I will explain how this interpretation gi ves an insight in the study of the dual complexes. I will focus on some de generations of hyper-Kähler varieties and show that we are able to determ ine the homeomorphism type of their dual complex using techniques of Berko vich geometry. The results are in accordance with the predictions of mirro r symmetry\, and the recent work about the rational homology of dual compl exes of degenerations of hyper-Kähler varieties\, due to Kollár\, Laza\, Saccà and Voisin.This is joint work with Morgan Brown.DTEND;TZID=Europe/Zurich:20191112T120000 END:VEVENT BEGIN:VEVENT UID:news925@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191020T194626 DTSTART;TZID=Europe/Zurich:20191105T103000 SUMMARY:Seminar Algebra and Geometry: Shengyuan Zhao (University of Rennes) DESCRIPTION:Let Y be a smooth complex projective surface. Let U be a connec ted Euclidean open set of Y. Let G be a subgroup of Bir(Y) which acts by h olomorphic diffeomorphisms on U (i.e. preserves U and without indeterminac y points in U)\, in a free\, properly discontinuous and cocompact way\, so that the quotient X=U/G is a compact complex surface. Such a birational t ransformation group G\, or more precisely such a quadruple (Y\,U\,G\,X)\, will be called a birational Kleinian group. Once we have a birational Klei nian group\, the quotient surface is equipped with a birational structure\ , i.e. an atlas of local charts with rational changes of coordinates. I wi ll present some basic properties and subtleties of birational structures\, compared to the classical geometric structures. Then I will begin by stud ying birational structures on a special type of non-algebraic surfaces\, t he Inoue surfaces\, to reveal some of the general strategy. Using classifi cation of solvable and abelian groups of the Cremona group\, and by relati ng the foliations on Inoue surfaces with some birational dynamical systems via Ahlfors-Nevanlinna currents\, I will show that the Inoue surfaces hav e one unique birational structure. Then I will move on to the general stud y of birational Kleinian groups with the additional hypothesis that the qu otient surface is projective. I will explain how to use powerful results f rom Cremona groups\, holomorphic foliations and non-abelian Hodge theory t o get an almost complete classification of such birational Kleinian groups . X-ALT-DESC:

In this talk\, I will first discuss application of \; Sarkisov program to the rational ity problem of algebraic varieties having conic bundle structures. \;< br />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. DTEND;TZID=Europe/Zurich:20190507T120000 END:VEVENT BEGIN:VEVENT UID:news845@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190327T160809 DTSTART;TZID=Europe/Zurich:20190416T103000 SUMMARY:Seminar Algebra and Geometry: Matilde Manzaroli (École Polytechniq ue and Université de Nantes) DESCRIPTION:The study of the topology of real algebraic varieties dates bac k to the work of Harnack\, Klein and Hilbert in the 19th century\; in part icular\, the isotopy type classification of real algebraic curves with a f ixed degree in the real projective plane is a classical subject that has u ndergone considerable evolution. On the other hand\, apart from studies co ncerning Hirzebruch surfaces and at most degree 3 surfaces in the real pro jective space\, not much is known for more general ambient surfaces. In pa rticular\, this is because varieties constructed using the patchworking me thod are hypersurfaces of toric varieties. However\, there are many other real algebraic surfaces. Among these are the real rational surfaces\, and more particularly the real minimal surfaces. In this talk\, we present som e results about the classification of topological types realized by real a lgebraic curves in real minimal del Pezzo surfaces of degree. X-ALT-DESC:The study of the topology of real algebraic varieties dates back to the work of Harnack\, Klein and Hilbert in the 19th century\; in parti cular\, the isotopy type classification of real algebraic curves with a fi xed degree in the real projective plane is a classical subject that has un dergone considerable evolution. On the other hand\, apart from studies con cerning Hirzebruch surfaces and at most degree 3 surfaces in the real proj ective space\, not much is known for more general ambient surfaces. In par ticular\, this is because varieties constructed using the patchworking met hod are hypersurfaces of toric varieties. However\, there are many other r eal algebraic surfaces. Among these are the real rational surfaces\, and m ore particularly the real minimal surfaces. In this talk\, we present some results about the classification of topological types realized by real al gebraic curves in real minimal del Pezzo surfaces of degree. DTEND;TZID=Europe/Zurich:20190416T120000 END:VEVENT BEGIN:VEVENT UID:news637@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190314T120420 DTSTART;TZID=Europe/Zurich:20190319T103000 SUMMARY:Seminar Algebra and Geometry: Mirko Mauri (Imperial College London) DESCRIPTION:The geometric P=W conjecture is a conjectural description of th e asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In particular\, it is expected that the dual boundary complex of the compactification of character varieties is a sphere. In a joint wo rk with Enrica Mazzon and Matthew Stevenson\, we manage to compute the fi rst non-trivial examples of dual complexes in the compact case. This requ ires to develop a new theory of essential skeletons over a trivially-value d field. As a byproduct\, inspired by these constructions\, we show that certain character varieties appear in degenerations of compact hyper-Kä hler manifolds. In this talk we will explain how these new non-archime dean techniques can shed new light into classical algebraic geometry prob lems. X-ALT-DESC:The geometric P=W conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge&n bsp\;theory. In particular\, it is expected \;that the dual boundary&n bsp\;complex of the compactification of character varieties \;is a sph ere. In a joint work with Enrica Mazzon and Matthew Stevenson\, we manage to compute \;the first non-trivial \;examples of dual complexes in the compact case. This requires to develop a new theory of essential skel etons over a trivially-valued field. \;As a byproduct\, inspired by th ese constructions\, we show \;that certain character varieties appear in \;degenerations of compact hyper-Kähler manifolds. \; \;In this talk we will explain how these new \;non-archimedean techniques& nbsp\;can shed new light into classical algebraic geometry problems. \ ; DTEND;TZID=Europe/Zurich:20190319T120000 END:VEVENT BEGIN:VEVENT UID:news821@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190222T141344 DTSTART;TZID=Europe/Zurich:20190305T103000 SUMMARY:Seminar Algebra and Geometry: Federico Lo Bianco (Université de Ma rseille) DESCRIPTION:We consider a holomorphic (singular) foliation F on a projectiv e manifold X and a group G of birational transformations of X which preser ve F (i.e. it permutes the set of leaves). We say that the transverse acti on of G is finite if some finite index subgroup of G fixes each leaf of F. \\r\\nI will briefly recall a criterion for the finiteness of the transve rse action in the case of algebraically integrable foliations (i.e. foliat ions whose leaves coincide with the fibres of a fibration). Then I will ex plain how the presence of certain transverse structures on the foliation a llow to recover the same result\; in this case\, one can study the monodro my 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 t o reduce the problem to subvarieties of quotients of the product of unit d iscs\, whose geometry is now quite well understood. X-ALT-DESC:We consider a holomorphic (singular) foliation F on a projective manifold X and a group G of birational transformations of X which preserv e F (i.e. it permutes the set of leaves). We say that the transverse actio n of G is finite if some finite index subgroup of G fixes each leaf of F. \nI will briefly recall a criterion for the finiteness of the transverse a ction 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 red uce the problem to subvarieties of quotients of the product of unit discs\ , whose geometry is now quite well understood. DTEND;TZID=Europe/Zurich:20190305T120000 END:VEVENT BEGIN:VEVENT UID:news410@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181204T082628 DTSTART;TZID=Europe/Zurich:20181211T103000 SUMMARY:Seminar Algebra and Geometry: Luca Studer (University of Bern) DESCRIPTION:In the talk we discuss how Oka theory helps to solve systems of equations with complex analytic entries. A classical example is the fact that for every pair of complex analytic functions a\, b: C^n -> C with no common zero there are complex analytic functions x\, y: C^n -> C satisfyin g the Bézout identity ax+by=1. A more recent example is Leiterer's work\, where the solvability of xax^{-1}=b for complex analytic matrix-valued ma ps a\, b: C^n -> Mat(n x n\, C) is investigated. Both examples are brought into the context of the speakers research. X-ALT-DESC:In the talk we discuss how Oka theory helps to solve systems of equations with complex analytic entries. A classical example is the fact t hat for every pair of complex analytic functions a\, b: C^n ->\; C with no common zero there are complex analytic functions x\, y: C^n ->\; C sa tisfying the Bézout identity ax+by=1. A more recent example is Leiterer's work\, where the solvability of xax^{-1}=b for complex analytic matrix-va lued maps a\, b: C^n ->\; Mat(n x n\, C) is investigated. Both examples are brought into the context of the speakers research. DTEND;TZID=Europe/Zurich:20181211T120000 END:VEVENT BEGIN:VEVENT UID:news348@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181204T083213 DTSTART;TZID=Europe/Zurich:20181204T103000 SUMMARY:Seminar Algebra and Geometry: Bruno Laurent (Grenoble) DESCRIPTION:The varieties which are homogeneous under the action of an alge braic group are very symmetric objects. More generally\, we get a much wid er class of objects\, having a very rich geometry\, by allowing the variet ies to have not a unique orbit\, but a dense orbit. Such varieties are sai d to be almost homogeneous\; this includes the case of toric varities\, wh en the group is an algebraic torus.In this talk\, I will explain how to cl assify the pairs (X\,G) where X is a curve or a surface and G is a smo oth and connected algebraic group acting on X with a dense orbit.For cur ves\, I will mainly focus on the regular ones\, defined over an arbitrary field. Over an algebraically closed field\, the "natural" notion of non-si ngularity is "smoothness". However\, over an arbitrary field\, the weaker notion of "regularity" is more suitable. I will recall the difference betw een 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 complica ted than for curves. Moreover\, new phenomena and several difficulties app ear in positive characteristic\, and I will highlight them. X-ALT-DESC:The varieties which are homogeneous under the action of an algeb raic group are very symmetric objects. More generally\, we get a much wide r class of objects\, having a very rich geometry\, by allowing the varieti es to have not a unique orbit\, but a dense orbit. Such varieties are said to be almost homogeneous\; this includes the case of toric varities\, whe n the group is an algebraic torus.

In this talk\, I will explai n how to classify the pairs (X\,G) where \;X \;is a curve or a sur face and \;G \;is a smooth and connected algebraic group acting on \;X \;with a dense orbit.

For curves\, I will mainly f ocus on the regular ones\, defined over an arbitrary field. Over an algebr aically closed field\, the "\;natural"\; notion of non-singularity is "\;smoothness"\;. However\, over an arbitrary field\, the weak er notion of "\;regularity"\; is more suitable. I will recall the difference between those two notions and show that there exist regular hom ogeneous curves which are not smooth.

For surfaces\, I will res trict to the smooth ones\, defined over an algebraically closed field. The situation is more complicated than for curves. Moreover\, new phenomena a nd several difficulties appear in positive characteristic\, and I will hig hlight them. DTEND;TZID=Europe/Zurich:20181204T120000 END:VEVENT BEGIN:VEVENT UID:news341@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181204T082628 DTSTART;TZID=Europe/Zurich:20181120T103000 SUMMARY:Seminar Algebra and Geometry: Nikon Kurnosov (University of Georgia \, Athens\, US) DESCRIPTION:Hyperkähler manifolds are higher-dimensional generalizations o f K3 surfaces. The Beauville conjecture predicts that the number of deform ation types of compact irreducible hyperkähler manifolds is finite in any dimension. In this talk I will briefly discuss some basic notions of the theory\, explain why hyperkähler manifolds play a very important role in classification of complex manifolds\, and then explain what are the eviden ces for Beauville's conjecture. X-ALT-DESC:Hyperkähler manifolds are higher-dimensional generalizations of K3 surfaces. The Beauville conjecture predicts that the number of deforma tion types of compact irreducible hyperkähler manifolds is finite in any dimension. In this talk I will briefly discuss some basic notions of the t heory\, explain why hyperkähler manifolds play a very important role in c lassification of complex manifolds\, and then explain what are the evidenc es for Beauville's conjecture. DTEND;TZID=Europe/Zurich:20181120T120000 END:VEVENT BEGIN:VEVENT UID:news361@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181204T082628 DTSTART;TZID=Europe/Zurich:20181113T103000 SUMMARY:Seminar Algebra and Geometry: Lukas Lewark (University of Bern) DESCRIPTION:This talk will explore different notions of positivity of knots \, how to encode such knots as graphs\, and how to unknot them. Joint resu lts with Baader/Liechti and Feller/Lobb will make appearances. No prerequi sites in knot theory will be necessary. X-ALT-DESC:This talk will explore different notions of positivity of knots\ , how to encode such knots as graphs\, and how to unknot them. Joint resul ts with Baader/Liechti and Feller/Lobb will make appearances. No prerequis ites in knot theory will be necessary. DTEND;TZID=Europe/Zurich:20181113T120000 END:VEVENT BEGIN:VEVENT UID:news340@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181204T082628 DTSTART;TZID=Europe/Zurich:20181106T103000 SUMMARY:Seminar Algebra and Geometry: Juliette Bavard (University of Rennes ) DESCRIPTION:The plane minus a Cantor set and its mapping class group appear in many dynamical contexts\, including group actions on surfaces\, the study of groups of homeomorphisms on a Cantor set\, and complex dynamics. In this talk\, I will motivate the study of this 'big mapping class grou ps'. I will then present the 'ray graph'\, which is a Gromov-hyperbolic graph on which this group acts by isometries. X-ALT-DESC:The plane minus a Cantor set and its mapping class group appear in many dynamical contexts\, including \;group \;actions on surfac es\, the study of groups of homeomorphisms on a Cantor set\, and complex d ynamics. \;In this talk\, I will motivate the study of this 'big mappi ng class groups'. I will then present the 'ray \;graph'\, which is a G romov-hyperbolic \;graph \;on which this \;group \;acts by isometries. \; \;

DTEND;TZID=Europe/Zurich:20181106T120000 END:VEVENT BEGIN:VEVENT UID:news347@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181204T082628 DTSTART;TZID=Europe/Zurich:20181016T103000 SUMMARY:Seminar Algebra and Geometry: Pierre-Marie Poloni and Richard Griff on (University of Basel) DESCRIPTION:TBA X-ALT-DESC:TBA DTEND;TZID=Europe/Zurich:20181016T120000 END:VEVENT BEGIN:VEVENT UID:news324@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181204T082628 DTSTART;TZID=Europe/Zurich:20181002T103000 SUMMARY:Seminar Algebra and Geometry: Immanuel van Santen\, Anne Lonjou and Jeremy Blanc (University of Basel) DESCRIPTION:Immanuel van Santen will speak about „Embeddings and tame aut omorphisms in affine geometry"\, \\r\\nAnne Lonjou will explain some link between "Cremona group and geometric group theory"\, and \\r\\nJérémy Blanc will present us "Birational geometry of surfaces and threefolds". X-ALT-DESC:\nImmanuel van Santen will speak about „Embeddings and tame au tomorphisms in affine geometry"\;\, \;\nAnne Lonjou will explain s ome link between "\;Cremona group and geometric group theory"\;\, and \;\nJérémy Blanc will present us "\;Birational geometry of s urfaces and threefolds"\;. DTEND;TZID=Europe/Zurich:20181002T120000 END:VEVENT BEGIN:VEVENT UID:news322@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181204T082628 DTSTART;TZID=Europe/Zurich:20180925T103000 SUMMARY:Seminar Algebra and Geometry: Julia Schneider\, Egor Yasinsky and P hilipp Mekler (University of Basel) DESCRIPTION:Julia will speak about "A_k-singularities of plane curves of fi xed bidegree".\\r\\nEgor will answer to "What transformation groups in alg ebraic\, differential and metric geometry have in common?". \\r\\nPhilipp will speak about "Algebraic Statistics: Gaussian Mixtures and Beyond". X-ALT-DESC:\nJulia will speak about "\;A_k-singularities of plane curve s of fixed bidegree"\;.\nEgor will answer to "\;What transformatio n groups in algebraic\, differential and metric geometry have in common?&q uot\;. \;\nPhilipp will speak about "\;Algebraic Statistics: Gauss ian Mixtures and Beyond"\;. DTEND;TZID=Europe/Zurich:20180925T120000 END:VEVENT BEGIN:VEVENT UID:news644@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190125T172712 DTSTART;TZID=Europe/Zurich:20180529T103000 SUMMARY:Seminar Algebra and Geometry: Jérémy Blanc (Basel) DESCRIPTION:I will describe a joint work with Stéphane Lamy and Susanna Zi mmermann which provides abelian quotients of the Cremona groups in high dimension. X-ALT-DESC: I will describe a joint work with Stéphane Lamy and Susanna Zi mmermann which provides abelian quotients of the Cremona groups in high dimension. DTEND;TZID=Europe/Zurich:20180529T120000 END:VEVENT BEGIN:VEVENT UID:news643@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190125T172506 DTSTART;TZID=Europe/Zurich:20180522T103000 SUMMARY:Seminar Algebra and Geometry: Andriy Regeta (Köln) DESCRIPTION:In this talk we are going to discuss the following question: t o which extent a so-called Danielewski surface is determined by its autom orphism group seen as an abstract group or as an ind-group? X-ALT-DESC: In this talk we are going to discuss the following question: t o which extent a so-called Danielewski surface is determined by its autom orphism group seen as an abstract group or as an ind-group? DTEND;TZID=Europe/Zurich:20180522T120000 END:VEVENT BEGIN:VEVENT UID:news642@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190125T172321 DTSTART;TZID=Europe/Zurich:20180515T103000 SUMMARY:Seminar Algebra and Geometry: Peter Feller (ETHZ) DESCRIPTION:We start by recalling that a smooth algebraic curve of degree d in CP2 is a genus (d-1)(d-2)/2 surface (read `smooth 2-manifold'). The ` Thom Conjecture'\, proven by Kronheimer and Mrowka\, asserts that such al gebraic curves have a surprising minimizing property. We derive consequenc es of the Thom Conjecture for transversal intersections of algebraic curv es with round spheres\, describe the knots one finds as such intersection s following Rudolph\, and give precise instances of the sentiment that th ese intersections constitute very special elements in the so-called smoot h concordance group. In contrast\, in the topological category\, we prove that all knots are topological concordant to such an intersection. Based on joint work with Maciej Borodzik. No knowledge about knot theory and co ncordance theory---the study of 1-manifolds in the 3-dimensional sphere a nd surfaces in 4-dimensional ball bounding them---will be assumed. X-ALT-DESC: We start by recalling that a smooth algebraic curve of degree d in CP

In the cu rrent joint project with Laurent Gruson and Nicolas Perrin\, we study some examples of Fano varieties in low dimension via explicit birational metho ds. DTEND;TZID=Europe/Zurich:20180417T120000 END:VEVENT BEGIN:VEVENT UID:news633@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190125T172302 DTSTART;TZID=Europe/Zurich:20180403T103000 SUMMARY:Seminar Algebra and Geometry: Pierre-Marie Poloni (Bern) DESCRIPTION:An $\\mathbb{A}^2$-fibration is a flat morphism between complex affine varieties whose fibers are isomorphic to the complex affine plane . In this talk\, we study explicit families $f:\\mathbb{A}^4\\to\\mathbb{ A}^2$ of $\\mathbb{A}^2$-fibrations over the affine plane. The famous Dol gachev-Weisfeiler conjecture predicts that such fibrations are in fact is omorphic to the trivial bundle. We will show that this holds true in some particular examples. For instance\, we will recover a result of Drew Lew is 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 conside red fibrations have a fiber bundle structure when restricted over the pun ctured affine plane. This is a joint work in progress with Jérémy Blanc. X-ALT-DESC:An $\\mathbb{A}^2$-fibration is a flat morphism between complex affine varieties whose fibers are isomorphic to the complex affine plane. In this talk\, we study explicit families $f:\\mathbb{A}^4\\to\\mathbb{A }^2$ of $\\mathbb{A}^2$-fibrations over the affine plane.

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

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

This is a joint work in progress wit h Jérémy Blanc. DTEND;TZID=Europe/Zurich:20180403T120000 END:VEVENT BEGIN:VEVENT UID:news632@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190121T142447 DTSTART;TZID=Europe/Zurich:20180313T103000 SUMMARY:Seminar Algebra and Geometry: Konstantin Shramov (HSE Moscow) DESCRIPTION:I will speak about finite groups acting by birational automorph isms of surfaces over algebraically non-closed fields\, mostly function f ields. One of important observations here is thata smooth geometrically rational surface S is either birational to a product of a projective line and a conic (in particular\, S is rational provided that it has a point) \, or finite subgroups of its birational automorphism group are bounded. We will also discuss some particular types of surfaces with interesting au tomorphism groups\, including Severi-Brauer surfaces. X-ALT-DESC: I will speak about finite groups acting by birational automorph isms of surfaces over algebraically non-closed fields\, mostly function f ields. One of important observations here is thata smooth geometrically rational surface S is either birational to a product of a projective line and a conic (in particular\, S is rational provided that it has a point) \, or finite subgroups of its birational automorphism group are bounded.< br /> We will also discuss some particular types of surfaces with interest ing automorphism groups\, including Severi-Brauer surfaces. DTEND;TZID=Europe/Zurich:20180313T120000 END:VEVENT BEGIN:VEVENT UID:news631@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190121T141926 DTSTART;TZID=Europe/Zurich:20180306T103000 SUMMARY:Seminar Algebra and Geometry: Hans-Christian Graf v. Bothmer (Hambu rg) DESCRIPTION:I will review some of the history and recent developments of th e rationality question for conic bundles over projective spaces. I will t hen explain our contribution to this question in the case of conic bundle s over IP^3 (joint work with Asher Auel\, Christian Boehning and Alena Pi rutka). X-ALT-DESC: I will review some of the history and recent developments of th e rationality question for conic bundles over projective spaces. I will t hen explain our contribution to this question in the case of conic bundle s over IP^3 (joint work with Asher Auel\, Christian Boehning and Alena Pi rutka). DTEND;TZID=Europe/Zurich:20180306T120000 END:VEVENT BEGIN:VEVENT UID:news652@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190126T193831 DTSTART;TZID=Europe/Zurich:20171212T103000 SUMMARY:Seminar Algebra and Geometry: Sara Durighetto (Ferrara) DESCRIPTION:Let C\, D be two birational subvarieties of the projective spac e Pn. I am interested in understanding when there exists a Cremona modif ication f: Pn --->Pn such that f(C) = D. For the sake of this talk I will restrict to a configuration of lines in P2. In this special case I'll suggest a classification of configurations of lines that are c ontractible to a bunch of points. In doing this I will propose an unexpec ted configuration that seems to violate an established conjecture. X-ALT-DESC: Let C\, D be two birational subvarieties of the projective spac e \;

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 n onsingular liftings and allows us to calculate the homology (and thus\, i n particular\, the number of liftings up to rigid isotopy). We will show that\, up to degree 5\, this hyperplane arrangement is a rigid isotopy in variant (of planar curves) and can provide real algebraic analogues of th e classical Reidemeister moves. Obstructions in the case of higher degree s will be discussed. The talk should be accessible to nonspecialists. DTEND;TZID=Europe/Zurich:20150313T120000 END:VEVENT BEGIN:VEVENT UID:news695@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190202T194348 DTSTART;TZID=Europe/Zurich:20150220T103000 SUMMARY:Seminar Algebra and Geometry: Jörg Winkelmann (Ruhr-Universität B ochum) DESCRIPTION:Conjecturally abundance of entire curves is closely related to abundance of rational points on projective varieties defined over a numbe r field.This is discussed in connection with algebraic groups and prinici pal bundles. More precisely\, let X be a projective manifold\, G an algeb raic group and E a G-principal bundle on X\, all defined over some number field K. Then E admits a Zariskidense set of L-rational points for some finite extension field iff X does. This corresponds to the homotopy lift ing property whose complex analytic analogue allows to lift entire curves . X-ALT-DESC: Conjecturally abundance of entire curves is closely related to abundance of rational points on projective varieties defined over a numbe r field.

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-rationa l points for some finite extension field iff X does. This corresponds to the homotopy lifting property whose complex analytic analogue allows to l ift entire curves. DTEND;TZID=Europe/Zurich:20150220T120000 END:VEVENT BEGIN:VEVENT UID:news716@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190207T232809 DTSTART;TZID=Europe/Zurich:20141219T103000 SUMMARY:Seminar Algebra and Geometry: Amos Turchet (Chalmers Universitet\, Göteborg) DESCRIPTION:Lang-Vojtas Conjectures are a set of deep and far reaching conj ectures\, formulated by Paul Vojta using ideas of Lang\, which embrace the distribution of solutions to Diophantine equations over number fields\, t he behaviour of holomorphic maps into complex manifolds and of algebraic c urves into algebraic varieties. In the (split) function field case the co njecture predicts (weak) algebraic hyperbolicity for log-general type vari eties.When the completion of the variety is the projective plane the conje cture is known both if the divisor at infinity consits of four lines in ge neral position (Brownawell-Masser and\,independently\, Voloch) and for a c onic and two lines with five singular points (Corvaja and Zannier). With different methods Chen and Pacienza-Rousseau proved that the conjecture ho lds 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 fascinati ng subject\, we will show how to prove the conjecture in general for the c omplement 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 the ory of logarithmic stable maps as defined by Abramovich-Chen (and independ ently by Gross and Siebert) which extends usual stable maps to the logarit hmic category (in the sense of Kato and Illusie). X-ALT-DESC: Lang-Vojtas Conjectures are a set of deep and far reaching conj ectures\, formulated by Paul Vojta using ideas of Lang\, which embrace the distribution of solutions to Diophantine equations over number fields\, t he behaviour of holomorphic maps into complex manifolds and of algebraic c urves into algebraic varieties.

In the (split) function field case the conjecture predicts (weak) algebraic hyperbolicity for log-general typ e 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 f or a conic and two lines with five singular points (Corvaja and Zannier). With different methods Chen and Pacienza-Rousseau proved that the conject ure 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 th is fascinating subject\, we will show how to prove the conjecture in gener al 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 an d 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). DTEND;TZID=Europe/Zurich:20141219T120000 END:VEVENT BEGIN:VEVENT UID:news715@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190207T232307 DTSTART;TZID=Europe/Zurich:20141212T103000 SUMMARY:Seminar Algebra and Geometry: Salomon Vishkautsan (ERC postdoc at S cuola Normale Superiore di Pisa) DESCRIPTION:In this talk we present a local-global property in arithmetic d ynamics called strong residual periodicity\, as defined by Bandman\, Grune wald and Kunyavskii in 2010. We start with a dynamical system induced by an endomorphism of a quasi projective variety defined over a number field. This system can be reduced mod p for “almost all” primes in the ring of integers of the number field. We can then ask how the dy namics of the global system relate to the dynamics of the system reduce d mod p for almost all primes p. Strong residual periodicity occurs when points of small period exist modulo almost every prime\,but “cannot be e xplained” by the dynamics of the global system. The aim of this talk is to present many motivating examples and raise some interesting questions to encourage further research on this topic. X-ALT-DESC: In this talk we present a local-global property in arithmetic d ynamics called

This is a joint work in progress with Ciro Cilib erto (Univ. Roma "\;Tor Vergata"\;). DTEND;TZID=Europe/Zurich:20141121T120000 END:VEVENT BEGIN:VEVENT UID:news711@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190207T230340 DTSTART;TZID=Europe/Zurich:20141114T103000 SUMMARY:Seminar Algebra and Geometry: Immanuel Stampfli (Jacobs University\ , Bremen) DESCRIPTION:This is joint work with Peter Feller (University of Bern). Let X be a smooth affine algebraic variety. A natural question is\, whe ther two algebraic embeddings f\, g : X → Cm are algebraically equiva lent\, i.e. whether there exists an algebraic automorphism φ of Cm such that φ◦f=g. Kaliman\, Nori and Srinivas gave an affirmative an swer\, provided that 2 dimX+ 2≤m. In this talk we discuss the followi ng one-dimensional improvement under a relaxed equivalence condition.\\r\\ nTheorem. If f\, g : X → Cm are algebraic embeddings and 2 dimX+ 1≤m\, then there exists a holomorphic automorphism φ of Cm such that φ◦f=g. \\r\\nIn fact\, the proof is based on an idea of Kaliman\, with which he proved that two algebraic embeddings of C into C3 are holom orphically equivalent. In the course of this talk\, we discuss this idea . Moreover\, we provide examples of algebraic embeddings into Cm that ar e holomorphically non-equivalent. X-ALT-DESC: This is joint work with Peter Feller (University of Bern). Let X be a smooth affine algebraic variety. A natural question is\, whe ther two algebraic embeddings f\, g : X →

After a brief historic al 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 appli cation to the study of finite simple groups of birational automorphisms of

\; \; \; (

with

In recent wo rk by Hauenstein\, Rodriguez\, and Sturmfels the maximum-likelihood degre e of determinantal varieties was studied. Extensive computations using nu merical algebraic geometry led to the conjecture that the maximum-likelih ood degree of the variety of rank-r matrices whose entries add up to 1 eq uals 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 Rodrig uez. DTEND;TZID=Europe/Zurich:20130419T120000 END:VEVENT BEGIN:VEVENT UID:news742@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190211T221148 DTSTART;TZID=Europe/Zurich:20130412T103000 SUMMARY:Seminar Algebra and Geometry: Antoine Ducros (Université Paris 6) DESCRIPTION:If one tries to mimic naively in the p-adic setting what is u sually done in complex analytic geometry\, one immediately faces big probl ems\, because p-adic fields are totally disconnected. Hence in order to de velop a relevant p-adic geometry\, a more subtle approach is needed. In this talk\, I will begin with a presentation of that of Berkovich. It r oughly consists of 'adding plenty of points' to the usual p-adic spaces s o that they get good topological properties (like local path-connectedne ss). I will describe some Berkovich spaces associated with simple variet ies (the projective line\, the algebraic curves....). \\r\\nThen I will try to illustrate the following slogan: 'to see the good analog of a comp lex object in the p-adic world\, one often has to work with Berkovich spa ces'\, through three examples: spectral theory\; dynamical systems\; and the theory of real (p\,q)-forms and related notions (integrals\, bounda ry integrals\, curvature forms of metrized line bundles) that as been r ecently developped in the framework of Berkovich spaces by Chambert-Loir and myself\, and that I will try to describe in some detail. X-ALT-DESC:If one tries to mimic naively in the \;

In this talk\, I will begin with a pres entation of that of Berkovich. It roughly consists of 'adding plenty of points' to the usual

\nThen I will try to illustrate the follo wing slogan: 'to see the good analog of a complex object in the

The group of automorphisms

I shall present a joint work with William Gignac\, on the growth of attrac ting rates for iterates of a superattracting germ in C

If time allows\, I will present other applications and examples. DTEND;TZID=Europe/Zurich:20121005T120000 END:VEVENT BEGIN:VEVENT UID:news754@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T001338 DTSTART;TZID=Europe/Zurich:20120601T103000 SUMMARY:Seminar Algebra and Geometry: Danny de Jesus Gomez (Universität Os nabrück) DESCRIPTION:Let R be a commutative ring with unity\; f\,f1\,...\,fn elemen ts of R (data) and I=(f1\,...\,fn)\, the corresponding ideal of R. The F orcing Algebra A:=R[T1\,...\,Tn]/(f1T1+...+fnTn-f) for the data f\,f1\,.. .\,fn is the most "natural" R-algebra such that f belongs to the expansio n of I in A. On this talk we present criteria and properties over the ba se ring R and the data in order to guarantee the connectedness of the cor responding forcing scheme Spec A. In particular\, we present an arithme tical Criteria for Principal Ideals Domains. Besides\, we rewrite the con dition of belonging to the integral closure of I by means of the univers al connectedness of the canonical morphism between the forcing scheme and the base space Spec R. Finally\, we study the local nature (over the bas e) of the connectedness in a very general setting. X-ALT-DESC:Let

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 unde r 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 wil l describe some conditions for the existence of an edge between two given vertices. \nThis is joint work with Diane Maclaga. DTEND;TZID=Europe/Zurich:20120525T120000 END:VEVENT BEGIN:VEVENT UID:news756@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T002832 DTSTART;TZID=Europe/Zurich:20120518T103000 SUMMARY:Seminar Algebra and Geometry: Vladimir Tsygankov\, Gutenberg Univer sität\, Mainz) DESCRIPTION:I will survey briefly known results about the finite subgroup s in the plane Cremona group over field of complex numbers. The emphasis will be done on the so called ''geometrical method''. By the method the study of the finite subgroups is reduced to study of so called the ''mi nimal'' finite subgroups of automorphism groups of nonsingular del Pezzo s urfaces and conic bundles. \\r\\nThe description by equations of nonsingu lar Del Pezzo surfaces in weighted projective spaces and description of t heir automorphism groups is a very old completed problem. The description of the ''minimal'' finite subgroups of automorphism groups of nonsingula r del Pezzo surfaces was done by I.V. Dolgachev and V.A. Iskovskikh in 20 07. I constructed a method that allows us to describe by equations in weighted projective spaces the nonsingular conic bundles with an action o f ''minimal'' subgroup of automorphisms . I applied the method in case\, when the ''minimal'' subgroup of automorphisms is nonsolvable. X-ALT-DESC:I will survey briefly known results about \;the finite subg roups in the plane Cremona group over field of complex numbers. The empha sis will be done on the so called ''geometrical method''. By the method t he study of the finite subgroups is reduced to \;study of so called the ''minimal'' finite subgroups of automorphism groups of

nonsingu lar del Pezzo surfaces and conic bundles. \nThe description by equations of nonsingular Del Pezzo surfaces in weighted projective spaces and descr iption 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. Iskov skikh in 2007. I constructed a method that allows us to describe \;b y equations in weighted projective spaces the nonsingular conic bundles w ith an action of ''minimal'' subgroup of automorphisms . I applied the me thod in case\, when the ''minimal'' subgroup of automorphisms is nonsolva ble. DTEND;TZID=Europe/Zurich:20120518T120000 END:VEVENT BEGIN:VEVENT UID:news757@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T003452 DTSTART;TZID=Europe/Zurich:20120511T103000 SUMMARY:Seminar Algebra and Geometry: Rosa M. Miró-Roig (Universitat de Ba rcelona) DESCRIPTION:In my talk\, I will construct families of non-isomorphic Arithm etically Cohen Macaulay (ACM) sheaves (i.e.\, sheaves without intermediat e cohomology) on projective varieties. Since the seminal result by Horro cks characterizing ACM bundles on $\\mathbb{P}^n$ as those that split int o a sum of line bundles\, an important amount of research has been devote d to the study of ACM sheaves on a given variety. ACM sheaves also prov ide a criterium to determine the complexity of the underlying variety. Mo re concretely\, this complexity can be studied in terms of the dimension and number of families of indecomposable ACM sheaves that it supports\, n amely its \\emph{representation type}. Along this line\, a variety that a dmits only a finite number of indecomposable ACM sheaves (up to twist and isomorphism) is called of finite representation type. These varieties ar e completely classified: They are either three or less reduced points in $\\mathbb{P}^2$\, a projective space $\\mathbb{P}_k^n$\, a smooth quadric hypersurface $X\\subset\\mathbb{P}^n$\, a cubic scroll in $\\mathbb{P}_k ^4$\, the Veronese surface in $\\mathbb{P}_k^5$ or a rational normal curv e. On the other extreme of complexity\, we would find the varieties of wild representation type\, namely\, varieties for which there exist r-dime nsional families of non-isomorphic indecomposable ACM sheaves for arbitra ry large r. In the case of dimension one\, it is known that curves of wil d representation type are exactly those of genus larger or equal than two . In dimension greater or equal than two few examples are known ans in ma y talk\, I will give a brief account of the known results. X-ALT-DESC: In my talk\, I will construct families of non-isomorphic Arithm etically Cohen Macaulay (ACM) sheaves (i.e.\, sheaves without intermediat e cohomology) on projective varieties. Since the seminal result by Horro cks characterizing ACM bundles on $\\mathbb{P}^n$ as those that split int o a sum of line bundles\, an important amount of research has been devote d to the study of ACM sheaves on a given variety.

\;

ACM sheaves also provide a criterium to determine the complexity of the under lying variety. More concretely\, this complexity can be studied in terms of the dimension and number of families of indecomposable ACM sheaves tha t 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 com plexity\, we would find the varieties of

This talk is based on joint work with Tom Leinster (Glasgow) . DTEND;TZID=Europe/Zurich:20120504T120000 END:VEVENT BEGIN:VEVENT UID:news759@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T003913 DTSTART;TZID=Europe/Zurich:20120420T103000 SUMMARY:Seminar Algebra and Geometry: Tommy Wuxing Cai (Universität Basel) DESCRIPTION:We give a formula generalizing Newton's formula on symmetric fu nctions. As an application of it\, we give a simple proof of the existenc e of Macdonald symmetric functions and compute the Macdonald functions of shapes of two-row partitions. X-ALT-DESC: We give a formula generalizing Newton's formula on symmetric fu nctions. As an application of it\, we give a simple proof of the existenc e of Macdonald symmetric functions and compute the Macdonald functions of shapes of two-row partitions. DTEND;TZID=Europe/Zurich:20120420T120000 END:VEVENT BEGIN:VEVENT UID:news760@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T004214 DTSTART;TZID=Europe/Zurich:20120413T103000 SUMMARY:Seminar Algebra and Geometry: Kevin Langlois (Institut Fourier\, Gr enoble) DESCRIPTION:Let A=C[f1\,...\, fr] be an integral algebra of finite type ove r the field of complex numbers. Using the elements f1\,..\,fr it is diff icult in general to describe the normalization of A. In this talk\, we p rovide some examples whenever A is a multigraded algebra. Consider the gro up T=C*×...×C*=(C*)n given by the componentwise multiplication. We say that T is an algebraic torus of dimension n. Let M be the character latti ce of T. Then a T-action on X=Spec A is equivalent to endow A with a M-gra duation. We classify the M-graded algebras A by a number called complexity . Geometrically\, it corresponds to the codimension of general T-orbits in X. Algebraically\, the complexity is somehow "the thickness of gra ded pieces" of the algebra A. The problem of normalization for complexity zero case is well known (monomial or toric case). For the complexity o ne\, the normalization of A admits a construction due to Timashev and Altmann-Hausen in terms of polyhedral divisors over an algebraic smoot h curve. Taking homogeneous generators\, we will explain how to build t he polyhedral divisor corresponding to the normalization of A. Assume that A is normal. Then A is given by a polyhedral divisor. A similar p roblem arises for the integral closure of homogeneous ideals. We will g ive an answer for the complexity one case. We will provide also a classifi cation of homogeneous integrally closed ideals of A. X-ALT-DESC: Let

In thi s talk\, we provide some examples whenever

Consider the group \;

We classify the

The prob lem of normalization for complexity zero case is well known (monomial or toric case). \; For the complexity one\, the \; normalization of < i>A

Assume that

In this talk we introduce a theoretical definition for 'minimal' representatives in any even linkage class. We show that these ideals

exist under reas onable 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 the y enjoy the best homological and local properties among all the ideals in their even linkage class. This justifies why they are\, in some sense\, th e `best' possible ideals in the even linkage class.

We provide several classes of ideals that are the minimal representatives of their ev en linkage classes (including determinantal ideals) and\, if time permits\ , show an easy application to produce more evidence towards the Buchsbaum- Eisenbud-Horrocks Conjecture. DTEND;TZID=Europe/Zurich:20111216T120000 END:VEVENT BEGIN:VEVENT UID:news766@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T170052 DTSTART;TZID=Europe/Zurich:20111209T103000 SUMMARY:Seminar Algebra and Geometry: Volkmar Welker (Philipps-Universität Marburg) DESCRIPTION:Generalized Inversion Statistics X-ALT-DESC:Generalized Inversion Statistics DTEND;TZID=Europe/Zurich:20111209T120000 END:VEVENT BEGIN:VEVENT UID:news767@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T170340 DTSTART;TZID=Europe/Zurich:20111202T103000 SUMMARY:Seminar Algebra and Geometry: Andriy Regeta (Technische Universitä t Kaiserslautern) DESCRIPTION:(joint work with A. Petravchuk and O. Iena)Let us consider the Lie algebra Wn=Wn(K) of all K-derivations of the polynomial ring K[x1\,... \,xn]. We discuss here a classof subalgebras in W2(K) over an algebraicall y closed field of characteristic zero. \\r\\nLet us consider natural actio n of the Lie algebra W2(K) on the field of rational functions K(x\,y). Rec all that every derivation Dof W2(K) of the ring K[x\,y] can be uniquely ex tended to a derivation of the field K(x\,y). It is natural to consider for afixed rational function u in K(x\,y)\\K the set A_{W2}(u) of all derivat ions D of W2 such that D(u)=0. This set is called the annihilatorof u in W 2(K). It is a Lie subalgebra of W2(K) and at the same time a K[x\,y]-submo dule of the K[x\,y]-module W2(K).We show that it is a free K[x\,y]-module of rank 1 and describe centralizers of elements and the maximal abelian su balgebras of theLie algebra A_{W2}(u). X-ALT-DESC:(joint work with A. Petravchuk and O. Iena)

Let us co nsider the Lie algebra W

of subalgebras in W

\nLet us consider natural actio n of the Lie algebra W

of W

fixed rational function u in K(x\,y)\\ K the set A_{W

of u in W

We show that it is a free K[x\,y]-module of rank 1 and describe centralizers of el ements and the maximal abelian subalgebras of the

Lie algebra A_{W

In case of a Danielewski surface the so-called overshear group is dense in the group of holomorp hic automorphisms. We describe the group structure of the overshear grou p with the help of Nevanlinna theory. DTEND;TZID=Europe/Zurich:20111118T120000 END:VEVENT BEGIN:VEVENT UID:news769@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T171244 DTSTART;TZID=Europe/Zurich:20111111T103000 SUMMARY:Seminar Algebra and Geometry: Winfried Bruns (Universität Osnabrü ck) DESCRIPTION:We report on joint work with Chr. Krattenthaler and J. Uliczka. Stanley decompositions of multigraded modules M over polynomials rings hav e been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition\, called the Stanley depth. Stanley conjectured that the Stanley depth of a module M is always at leas t the (classical) depth of M. We introduce a weaker type of decomposition\ , which we call Hilbert decomposition\, since it only depends on the Hilbe rt function of M\, and an analogous notion of depth\, called Hilbert depth . Since Stanley decompositions are Hilbert decompositions\, the latter set upper bounds to the existence of Stanley decompositions.The advantage of Hilbert decompositions is that they are easier to find.We test our new not ion on the syzygy modules of the residue class field of K[X1\,...\,Xn] (as usual identified with K). Writing M(n\,k) for the k-th syzygy module\, we show that the Hilbert depth of M(n\,1)\, namely the irrelevant maximal id eal\, is $\\lfloor(n+1)/2\\rfloor$. Furthermore\, we show that\, for $n > k \\ge \\lfloor n/2\\rfloor$\, the (multigraded) Hilbert depth of M(n\,k) is equal to n-1. We conjecture that the same holds for the Stanley depth. For the range n/2 > k > 1\, it seems impossible to come up with a compact formula for the Hilbert depth. Instead\, we provide very precise asymptoti c results as n becomes large.Related ideals are the powers of the irreleva nt maximal ideal. For them the (standard graded) Hilbert depth can be comp utedprecisely. X-ALT-DESC: We report on joint work with Chr. Krattenthaler and J. Uliczka.

Stanley decompositions of multigraded modules M over polynomia ls rings have been discussed intensively in recent years. There is a natur al 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 power s of the irrelevant maximal ideal. For them the (standard graded) Hilbert depth can be computed

precisely. DTEND;TZID=Europe/Zurich:20111111T120000 END:VEVENT BEGIN:VEVENT UID:news770@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T171622 DTSTART;TZID=Europe/Zurich:20111104T000000 SUMMARY:Seminar Algebra and Geometry: Hanspeter Kraft and Maria Fernanda Ro bayo (Universität Basel)\, Two 30 minutes talks DESCRIPTION:T.B.A. X-ALT-DESC:T.B.A. END:VEVENT BEGIN:VEVENT UID:news771@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T171920 DTSTART;TZID=Europe/Zurich:20111028T000000 SUMMARY:Seminar Algebra and Geometry: Naihuan Jing (North Carolina State Un iv. and South China Univ. of Technology) DESCRIPTION:Exactly 100 years ago Schur constructed irreducible characters of the spin groups of the symmetric group and introduced the famous Schur Q-functions. Using ideas from affine Lie algebras and McKay corresponde nce\, Frenkel\, Jing and Wang fixed the irreducible characters for the sp in group of the wreath products for the symmetric group without knowing t he complete character table. In this work we will generalize Schur's work and completely determine all character values for the spin wreath produc ts in type A case. This is joint work with X. Hu. X-ALT-DESC: Exactly 100 years ago Schur constructed irreducible characters of the spin groups of the symmetric group and introduced the famous Schur Q-functions. Using ideas from affine Lie algebras and McKay corresponde nce\, Frenkel\, Jing and Wang fixed the irreducible characters for the sp in group of the wreath products for the symmetric group without knowing t he complete character table. In this work we will generalize Schur's work and completely determine all character values for the spin wreath produc ts in type A case. This is joint work with X. Hu. END:VEVENT BEGIN:VEVENT UID:news772@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T172107 DTSTART;TZID=Europe/Zurich:20111021T000000 SUMMARY:Seminar Algebra and Geometry: Émilie Dufresne and Jérémy Blanc ( Universität Basel)\, Two 30 minutes talks DESCRIPTION:T.B.A. X-ALT-DESC:T.B.A. END:VEVENT BEGIN:VEVENT UID:news773@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T172233 DTSTART;TZID=Europe/Zurich:20111014T000000 SUMMARY:Seminar Algebra and Geometry: Alexandru Constantinescu and Pierre-M arie Poloni (Universität Basel)\, Two 30 minutes talks DESCRIPTION:T.B.A. X-ALT-DESC:T.B.A. END:VEVENT BEGIN:VEVENT UID:news774@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T172423 DTSTART;TZID=Europe/Zurich:20111007T000000 SUMMARY:Seminar Algebra and Geometry: Bruno Duchesne (The Hebrew University of Jerusalem) DESCRIPTION:Symmetric spaces of non-compact type (which are simply-connecte d Riemannian manifolds of non-positive curvature with a geodesic symmetr y at each point) are classical and useful geometric tools to understand finite-dimensional linear representations of groups.\\r\\nWe 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 spac e has dimension at most p.\\r\\nThe talk will be focused on the properti es of these spaces and some group actions which come from (non-unitary) infinite-dimensional representations. X-ALT-DESC:Symmetric spaces of non-compact type (which are simply-connected Riemannian manifolds of non-positive curvature with a geodesic symmetry at each point) are classical and useful geometric tools to understand finite-dimensional linear representations of groups.\nWe will look at s ome infinite dimensional symmetric spaces of non-positive curvature whic h have a remarkable property : they have finite rank. There exists a pos itive integer p such that any isometrically embedded Euclidean space has dimension at most p.\nThe talk will be focused on the properties of the se spaces and some group actions which come from (non-unitary) infinite- dimensional representations. END:VEVENT BEGIN:VEVENT UID:news775@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T172550 DTSTART;TZID=Europe/Zurich:20110930T000000 SUMMARY:Seminar Algebra and Geometry: Tommy Wuxing Cai (Universität Basel) DESCRIPTION:We first realize Jack function of rectangular shapes. Then we introduce an iterative method to realize Jack functions of general shape s from those of rectangular shapes. As a byproduct\, we prove some speci al cases of Richard P. Stanley's conjecture about the positivity of the Littlewood-Richardson coefficients of Jack functions.Contents: 1. partiti ons 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 X-ALT-DESC:We first realize Jack function of rectangular shapes. Then we i ntroduce an iterative method to realize Jack functions of general shapes from those of rectangular shapes. As a byproduct\, we prove some specia l cases of Richard P. Stanley's conjecture about the positivity of the Littlewood-Richardson coefficients of Jack functions.

Contents:

1. partitions and symmetric functions

2. Some history and que stions about symmetric functions

3. Vertex operator realization of r ectangular Jack functions

4. A special case of Stanley's conjecture and the realization of general Jack functions

5. Open questions END:VEVENT BEGIN:VEVENT UID:news776@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T172827 DTSTART;TZID=Europe/Zurich:20110527T000000 SUMMARY:Seminar Algebra and Geometry: Olivier Serman (Université Lille 1) DESCRIPTION:Moduli spaces of principal bundles on curves\, which have been introduced as non-abelian analogs of the Jacobian variety\, carry a very rich geometry. In this talk I will focus on the singularities of these v arieties. I will show how local factoriality of GIT quotients can be used to give a description of their singular locus\, and then compute the fun damental group of the smooth locus of these moduli spaces. X-ALT-DESC: Moduli spaces of principal bundles on curves\, which have been introduced as non-abelian analogs of the Jacobian variety\, carry a very rich geometry. In this talk I will focus on the singularities of these v arieties. I will show how local factoriality of GIT quotients can be used to give a description of their singular locus\, and then compute the fun damental group of the smooth locus of these moduli spaces. END:VEVENT BEGIN:VEVENT UID:news777@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T180652 DTSTART;TZID=Europe/Zurich:20110520T000000 SUMMARY:Seminar Algebra and Geometry: Émilie Dufresne (Universität Basel) DESCRIPTION:Nagata's famous counterexample to Hilbert's fourteenth problem shows that the ring of invariants of an algebraic group action on an aff ine algebraic variety is not always finitely generated. In some sense\, h owever\, invariant rings are not far from being affine. Indeed\, invarian t rings are always quasi-affine\, and finite separating sets always exist . We give a new method for finding a quasi-affine variety on which the ri ng of regular functions is equal to a given invariant ring\, and give a criterion to recognize separating algebras. We use the method and criteri on to construct new examples. X-ALT-DESC: Nagata's famous counterexample to Hilbert's fourteenth problem shows that the ring of invariants of an algebraic group action on an aff ine algebraic variety is not always finitely generated. In some sense\, h owever\, invariant rings are not far from being affine. Indeed\, invarian t rings are always quasi-affine\, and finite separating sets always exist . We give a new method for finding a quasi-affine variety on which the ri ng of regular functions is equal to a given invariant ring\, and give a criterion to recognize separating algebras. We use the method and criteri on to construct new examples. END:VEVENT BEGIN:VEVENT UID:news778@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T180831 DTSTART;TZID=Europe/Zurich:20110516T160000 SUMMARY:Seminar Algebra and Geometry: Aldo Conca (University of Genova) DESCRIPTION:The goal of the talk is to explain recent results and conjectur es regarding Koszul algebras and their syzygies. Koszul algebras are g raded K-algebras R such that the residue filed K has a linear R-free res olution. Koszul algebras are defined by quadrics. But not all algebras defined by quadric are Koszul. However many classical algebras defined b y quadrics (e.g. the coordinate ring of the Grassmannian in its standa rd embedding) are Koszul.The main idea I will discuss is that the sy zygies of Koszul algebras have some properties in common with the syzygi es of algebras defined by monomialsof degree two. X-ALT-DESC: The goal of the talk is to explain recent results and conjectur es regarding Koszul algebras and their syzygies. Koszul algebras are \; graded K-algebras R such that the residue filed K has a linear R-free resolution. Koszul algebras are defined by quadrics. \; But not all algebras defined by quadric are Koszul. However many classical algebras defined by quadrics \; (e.g. the coordinate ring of the Grassmannian in its standard embedding) \; are Koszul.

The main idea I will discuss is that \; the syzygies of Koszul algebras have some proper ties in common with the syzygies of algebras defined by monomials

of degree two. DTEND;TZID=Europe/Zurich:20110516T170000 END:VEVENT BEGIN:VEVENT UID:news779@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T181018 DTSTART;TZID=Europe/Zurich:20110513T000000 SUMMARY:Seminar Algebra and Geometry: Ivan Arzhantsev (Moscow State Univers ity and Institut Fourier) DESCRIPTION:(joint work with Devrim Celik and Juergen Hausen)We begin with a survey of known results concerning categorical quotients.Given an actio n of an affine algebraic group with only trivial characters on a factoria l variety\, we characterize existence of categorical quotient in the cate gory of algebraic varieties. Moreover\, allowing constructible sets as qu otients\, 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 categor ial quotients for actions on\, e.g. complete\, varieties with finitely ge nerated Cox ring via lifting to the universal torsor. X-ALT-DESC: (joint work with Devrim Celik and Juergen Hausen)

We begin with a survey of known results concerning categorical quotients.

Given an action of an affine algebraic group with only trivial charact ers on a factorial variety\, we characterize existence of categorical quo tient in the category of algebraic varieties. Moreover\, allowing constr uctible sets as quotients\, we obtain a more general existence result\, w hich\, for example\, settles the case of a finitely generated algebra of invariants. As an application\, we provide a combinatorial GIT-type const ruction of categorial quotients for actions on\, e.g. complete\, varietie s with finitely generated Cox ring via lifting to the universal torsor. END:VEVENT BEGIN:VEVENT UID:news780@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T181256 DTSTART;TZID=Europe/Zurich:20110506T000000 SUMMARY:Seminar Algebra and Geometry: Andreas Maurischat (Universität Heid elberg) DESCRIPTION:In Hopf-Galois theory one can attach a Hopf algebra H (resp. an affine group scheme G=Spec(H) ) to any finite field extension. In gene ral\, however\, this Hopf-algebra is not unique\, but in some cases - e.g . normal separable extensions - there are natural choices. In this talk\ , we will show how one can obtain "natural choices" for finite extensions (even inseparable ones) by imposing an extra structure on the field\, na mely a so-called iterative derivation. X-ALT-DESC: In Hopf-Galois theory one can attach a Hopf algebra H (resp. &n bsp\;an affine group scheme G=Spec(H) ) to any finite field extension. In general\, however\, this Hopf-algebra is not unique\, but in some cases - e.g. normal separable extensions - there are natural choices. In this talk\, we will show how one can obtain "\;natural choices"\; for f inite extensions (even inseparable ones) by imposing an extra structure o n the field\, namely a so-called iterative derivation. END:VEVENT BEGIN:VEVENT UID:news781@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T181459 DTSTART;TZID=Europe/Zurich:20110429T000000 SUMMARY:Seminar Algebra and Geometry: Matteo Varbaro (University of Genova) DESCRIPTION:T.B.A. X-ALT-DESC:T.B.A. END:VEVENT BEGIN:VEVENT UID:news782@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T181619 DTSTART;TZID=Europe/Zurich:20110415T000000 SUMMARY:Seminar Algebra and Geometry: Abraham Broer (Université de Montré al) DESCRIPTION:Cohomology of line bundles on the cotangent bundle of a complet e homogeneous space X-ALT-DESC:Cohomology of line bundles on the cotangent bundle of a complete homogeneous space END:VEVENT BEGIN:VEVENT UID:news783@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190212T181851 DTSTART;TZID=Europe/Zurich:20110408T000000 SUMMARY:Seminar Algebra and Geometry: Enrico Schlesinger (Politecnico di Mi lano) DESCRIPTION:Grothendieck has constructed the Hilbert scheme that parametr izes all subschemes of PN with a given Hilbert polynomial\, and Hartshorne in his thesis has shown that it is connected. For a curve in P3 the Hil bert polynomial is given by the degree d and by the arithmetic genus g. I will explain why\, even if one is primarily interested in smooth curves\ , the correct class of curve to look at is that of locally Cohen Macaulay curves. These are parametrized by an open subscheme Hd\,g of the full Hil bert scheme.It is an open question whether Hd\,g is connected whenever non empty. This question was motivated by a result by Martin-Deschamps and Per rin: they showed that Hd\,g always has an irreducible component made up by "extremal curves"\; these curves have the largest cohomology among curves in Hd\,g. So there is no obstruction from semicontinuity that prevents t he possibility that any smooth curve be specialized to an extremal curve. I will discuss the state of affairs about this question\, and briefly de scribe 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 compon ent of extremal curves\; this problem was raised in Hartshorne's papers "On the connectedness of the Hilbert scheme of curves in P3" Comm. Al g. 28\, 2000 and "Questions of connectedness of the Hilbert scheme of cu rves in P3" in the volume for Abhyankar's 70th (2004)\, and was still on the open problems list of the Workshop "Components of the Hilbert Schemes " (AIM Palo Alto 2010). X-ALT-DESC: Grothendieck has constructed the Hilbert scheme \; that par ametrizes all subschemes of P

It is an open question whether H

I will discuss the state of affairs about this question\, and brief ly 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 Hartshorn e's papers \; \; "\;On the connectedness of the Hilbert schem e of curves in P