31.10.2024 14:15 - 31.10.2024 15:15

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Abstract: In this talk we address an open question in arithmetic dynamics regarding the frequency of primes modulo which all the iterates of an integer polynomial remain irreducible. More precisely, for a class of integer polynomials $f$, which in particular includes all quadratic polynomials, we show that, under some natural conditions, the set of primes $p$ such that all iterates of $f$ are irreducible modulo $p$ is of relative density zero. Our results rely on a combination of analytic (Selberg's sieve) and Diophantine (finiteness of solutions to certain hyperelliptic equations) tools, which we will briefly describe. Joint wok with Laszlo Mérai and Igor Shparlinski (2021, 2024).

Spiegelgasse 5, Seminarraum 05.002

]]>24.10.2024 14:15 - 24.10.2024 15:15

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Abstract: Define, for a positive integer~$d$, $S(d)$ to be the set of all primes $p$ that occur as the order of a point $P \in E(K)$ on an elliptic curve $E$ defined over a number field $K$ of degree $d$. We discuss how some plausible conjectures on the sparsity of newforms with certain properties would allow us to deduce a fairly precise result on the asymptotic behavior of $\max S(d)$ as $d$ tends to infinity.

This is joint work with Maarten Derickx.

Location: Spiegelgasse 5, Seminarraum 05.002

]]>10.10.2024 10:45 - 10.10.2024 12:00

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Abstract: We present new extension theorems for differential forms on singular complex spaces and explain their use in the study of minimal varieties. We survey a number of applications, pertaining to classification and characterisation of special varieties, non-Abelian Hodge Theory in the singular setting, and quasi-étale uniformization.

Location: Hörsaal 114, Kollegienhaus

**Please carefully note the unusual time and location.**

11.04.2024 09:30 - 11.04.2024 16:10, Kollegienhaus, Hörsaal 120

Fifth Meeting

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

10.11.2023 14:00 - 11.11.2023 11:30, Philipp Habegger (Unibas), Özlem Imamoglu (ETHZ), Philippe Michel (EPFL), Maryna Viazovska (EPFL) , HS -101, Alte Universität, Rheinsprung 9, Universität Basel

Speakers: Anna Cadoret (Sorbonne Université), Pierre Colmez (CNRS, Sorbonne Université), Vesselin Dimitrov (Georgia Institute of Technology), Javier Fresán (Sorbonne Université), Kaisa Matomäki (University of Turku)

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

Registration is free but mandatory. To register, please fill out the registration.

]]>26.05.2022 17:00 - 26.05.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Applications of arithmetic holonomicity theorems

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

This is joint work with Frank Calegari and Vesselin Dimitrov.

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

]]>19.05.2022 17:00 - 19.05.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Schinzel's determinant inequality and a conjecture of F. Rodriguez Villegas

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

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

]]>12.05.2022 17:00 - 12.05.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Generalizations of the Montgomery-Hooley asymptotic formula; A survey.

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

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

]]>05.05.2022 17:00 - 05.05.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

On integers which are(n't) the sum of two rational cubes

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

I will explain joint work with Bhargava and Shnidman in which we show:

1. At least a sixth of integers are not the sum of two rational cubes,

and

2. At least a sixth of odd integers are the sum of two rational cubes!

(--- with 2. relying on new 2-converse results of Burungale-Skinner.)

The basic principle is that "there aren't even enough 2-Selmer elements to go around" to contradict e.g. 1., and we show this by using the circle method "inside" the usual geometry of numbers argument applied to a particular coregular representation. Even then the resulting constant isn't small enough to conclude 1., so we use the clean form of root numbers in the family x^3 + y^3 = n and the p-parity theorem of Nekovar/Dokchitser-Dokchitser to succeed.

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

]]>28.04.2022 17:00 - 28.04.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Linear Divisibility sequences

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

In a sequence of papers in the 1930s, Marshall Hall further developed several of Lucas' themes, including studying and trying to classify third order linear divisibility sequences; that is, linear recurrences like the Fibonacci numbers which have the additional property that $F_m$ divides $F_n$ whenever $m$ divides $n$. Because of many special cases, Hall was unable to even conjecture what a general theorem should look like, and despite developments over the years by various authors, such as Lehmer, Morgan Ward, van der Poorten, Bezivin, Petho, Richard Guy, Hugh Williams,... with higher order linear divisibility sequences, even the formulation of the classification has remained mysterious.

In this talk we present our ongoing efforts to classify all linear divisibility sequences, the key new input coming from a wonderful application of the Schmidt/Schlickewei subspace theorem from the theory of diophantine approximation, due to Corvaja and Zannier.

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

]]>21.04.2022 17:00 - 21.04.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Short exponential sums of the primes

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

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

]]>14.04.2022 17:00 - 14.04.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Probability Theory and the Riemann Hypothesis

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

]]>07.04.2022 17:00 - 07.04.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

On the cohomology of Shimura varieties with torsion coefficients

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

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

]]>31.03.2022 17:00 - 31.03.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Markoff triples and connectivity of Hurwitz spaces

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

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

]]>24.03.2022 17:00 - 24.03.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Group based zeta functions

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

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

]]>17.03.2022 17:00 - 17.03.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Diophantine Approximation for Closed Subschemes

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

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

]]>10.03.2022 17:00 - 10.03.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Shrinking targets on homogeneous spaces and improving Dirichlet's Theorem

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

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

]]>03.03.2022 17:00 - 03.03.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Modular Curves and Asymptotic Solutions to Fermat-type Equations

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

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

]]>24.02.2022 17:00 - 24.02.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Sums of Kloosterman and Salie Sums and Moments of L-functions

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

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

]]>17.02.2022 17:00 - 17.02.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Counting rational points and lower bounds for Galois orbits for special points on Shimura varieties

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

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

]]>10.02.2022 17:00 - 10.02.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Beyond uniform distribution

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

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

]]>03.02.2022 17:00 - 03.02.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

L^p-norm bounds for automorphic forms

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

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

]]>27.01.2022 17:00 - 27.01.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Reflections on the proof(s) of the Vinogradov mean value conjecture

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

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

]]>20.01.2022 17:00, Zoom: register here: https://www.ntwebseminar.org/

Rank of matrices with entries from a multiplicative group

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

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

]]>13.01.2022 17:00 - 13.01.2022 18:00, Zoom: register here: https://www.ntwebseminar.org/

Irreducibility of random polynomials

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

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

]]>16.12.2021 17:00 - 16.12.2021 18:00, Zoom: register here: https://www.ntwebseminar.org/

Euler systems and the Birch—Swinnerton-Dyer conjecture for abelian surfaces

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

I will recall a series of recent works (variously joint with Loeffler, Pilloni, Skinner) giving rise to an Euler system in the cohomology of Shimura varieties for GSp(4), and an explicit reciprocity law relating the Euler system to values of L-functions. I will then recent work with Loeffler, in which we use this Euler system to prove new cases of the BSD conjecture for modular abelian surfaces over Q, and modular elliptic curves over imaginary quadratic fields.

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

]]>09.12.2021 17:00 - 08.12.2021 18:00, Zoom: register here: https://www.ntwebseminar.org/

The Fermat equation and the unit equation

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

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

]]>02.12.2021 17:00 - 02.12.2021 18:00, Zoom: register here: https://www.ntwebseminar.org/

Orders of abelian varieties over $\mathbb{F}_2$

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

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

]]>25.11.2021 17:00 - 25.11.2021 18:00, Zoom: register here: https://www.ntwebseminar.org/

On uniformity conjectures for abelian varieties and K3 surfaces

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

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

]]>18.11.2021 17:00 - 18.11.2021 18:00, Zoom: register here: https://www.ntwebseminar.org/

Towards uniformity in the dynamical Bogomolov

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

In this talk we discuss uniform versions of the dynamical Bogomolov conjecture across 1-parameter families of certain split maps. To this end, we establish an instance of a 'relative dynamical Bogomolov'. This is work in progress joint with Harry Schmidt (University of Basel).

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

]]>11.11.2021 17:00 - 11.11.2021 18:00, Zoom: register here: https://www.ntwebseminar.org/

Randomness

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

Humanity has pondered the meaning and utility of randomness for millennia.

There is a remarkable variety of ways in which we utilize perfect coin tosses to our advantage: in statistics, cryptography, game theory, algorithms, gambling... Indeed, randomness seems indispensable! Which of these applications survive if the universe had no (accessible) randomness in it at all? Which of them survive if only poor quality randomness is available, e.g. that arises from somewhat "unpredictable" phenomena like the weather or the stock market?

A computational theory of randomness, developed in the past several decades, reveals (perhaps counter-intuitively) that very little is lost in such deterministic or weakly random worlds. In the talk I'll explain the main ideas and results of this theory, notions of pseudo-randomness, and connections to computational intractability.

It is interesting that Number Theory played an important role throughout this development. It supplied problems whose algorithmic solution make randomness seem powerful, problems for which randomness can be eliminated from such solutions, and problems where the power of randomness remains a major challenge for computational complexity theorists and mathematicians. I will use these problems (and others) to demonstrate aspects of this theory.

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. ]]>

29.11.2018 14:15 - 29.11.2018 15:15

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

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]]>

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 ${\bf C}^{28}$. ]]>

01.11.2018 14:15 - 01.11.2018 15:15

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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. ]]>

27.09.2018 14:15 - 27.09.2018 15:15

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

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

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