

Journée 24Programme de la Journée 24
Jeudi 15 novembre 2018
Les exposés auront lieu à l'Institut Camille Jordan, en salle Fokko du Cloux.
10h : café en salle de lecture au premier étage de l'ICJ
10h3011h20, Johannes Sprang (Regensburg), Some new irrationality results for (padic) zeta values
It has already been known to Euler that the values of the Riemann zeta function at positive even integers are nonzero rational multiples of powers of pi. Much less is known about the values at positive odd integers. The irrationality of zeta(3) has been proven by Apéry and a celebrated theorem of Rivoal and Ball shows that the dimension of the Qvector space spanned by 1, zeta(3), zeta(5),..., zeta(s) for an odd positive integer s is at least C log(s) for an absolute and explicit constant C. In particular, at least C log(s) among these numbers are irrational. In our recent work with Fischler and Zudilin, we improve this lower bound on the number of irrational zeta values to 2^{(1+o(1)) log(s) / loglog(s)}. The main ingredient is the construction of sufficiently many linear independent families of linear forms in zeta values with related coefficients. If time permits, we will explain how a related construction can be used to prove a padic variant of the theorem of Rivoal and Ball.
11h3012h20, Daniel Fiorilli (Paris), Lowlying zeros of quadratic Dirichlet Lfunctions: the transition
I will discuss recent joint work with James Parks and Anders Södergren. Looking at the onelevel density of lowlying zeros of quadratic Dirichlet Lfunctions, Katz and Sarnak predicted a sharp transition in the main terms when the support of the Fourier transform of the implied test functions reaches the point 1. By estimating this quantity up to a powersaving error term, we show that such a transition is also present in lowerorder terms. In particular this answers a question of Rudnick coming from the function field analogue. We also show that this transition is also present in the Ratios Conjecture's prediction. 12h30 : repas
14h14h50, Lucia Mocz (Bonn), On Northcott Properties for the Faltings Height In this talk we explore different Northcott properties for the Faltings height of abelian varieties. We begin with the work of Faltings, whose Northcott property played a key part in the proof of the Shafarevich conjecture for (the first cohomology of) abelian varieties over number fields. We review then the work of the speaker, which shows a Northcott property in an orthogonal direction for CM abelian varieties (a priori) over the complex numbers. With time permitting, we introduce some work in progress which unites these two viewpoints and express a conjectural new Northcott property for the Faltings height. 15h15h50, Jérémie Bouttier (Paris), Schur processes For X and Y two collections of variables, the Schur measure of parameters (X,Y) is the probability measure over integer partitions which assigns to a partition λ a weight proportional to s_λ(X) s_λ(Y), where s_λ denotes a Schur function. It was introduced by Okounkov in 2001 as a generalization of the Plancherel measure, which is itself related to the celebrated Ulam’s problem on the length of the longest increasing sequence (LLIS) in a random permutation. In fact, the techniques introduced by Okounkov yield a much simpler proof of the BaikDeiftJohansson theorem (1999) concerning the fluctuations of the LLIS. Shortly after, in 2003, Okounkov and Reshetikhin gave a further generalization called the Schur process, a random sequence of integer partitions where the marginal law of any element is a Schur measure. Their motivating application was the study of random plane partitions, but due to its generality the Schur process has now become one of the fundamental models in the fastgrowing field of ``integrable probability’’. The purpose of this talk is to give a pedagogical introduction to the Schur process, and also mention some of my own[*] recent contributions to the subject, concerning its periodic and free boundary variants. They can be viewed as elliptic deformations of the original process. [*] In collaboration with Dan Betea, Peter Nejjar and Mirjana Vuletić. 