Grenoble, Lyon et Saint-Etienne (France)

Journée 24

Programme 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
  
10h30-11h20,  Johannes Sprang (Regensburg), Some new irrationality results for (p-adic) zeta values
  
    It has already been known to Euler that the values of the Riemann zeta function at positive even integers are non-zero 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 Q-vector 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 p-adic variant of the theorem of Rivoal and Ball.
 
11h30-12h20, Daniel Fiorilli (Paris), Low-lying zeros of quadratic Dirichlet L-functions: the transition 
 
    I  will discuss recent joint work with James Parks and Anders Södergren.  Looking at the one-level density of low-lying zeros of quadratic Dirichlet L-functions, 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 power-saving error term, we show that such a transition is also present in lower-order 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 

14h-14h50, 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.
 
 15h-15h50, 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 Baik-Deift-Johansson 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 fast-growing 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.
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