Programme :
10h30-11h30 Quentin Gazda (École polytechnique, Palaiseau) : Some thoughts on Zagier's Conjecture: From Functions to Numbers [SLIDES]
Abstract. Zagier's conjecture is a certain formulation of the following slogan: linear relations among polylogarithms evaluated at algebraic numbers arise from relations between K-theory symbols. By identifying the different attributes of this conjecture, one can state and prove a similar version in arithmetic of function fields. Classical polylogarithms are then replaced by those of Carlitz. The proof, very different from the techniques developed so far, uses ingredients from the theory of difference equations. It involves deformations of Carlitz polylogarithms where a new variable \(t\) appears. This results from a joint work with A. Maurischat. While we currently lack the technology to reproduce this argument in number theory, it is amusing to speculate on a hypothetical transcription. \(q\)-deformations of polylogarithms then replace these « \(t\)-deformations." With T. Bouis, we recently encountered a \(q-Li_1\) in the syntomic Chern class introduced by Bhatt-Lurie. This is encouraging...! I will mention these works in a second part of the presentation.
11h30-12h30 Jehanne Dousse (Université de Genève) : q-difference equations and computer algebra for partition identities [SLIDES]
Abstract. A partition of a positive integer \(n\) is a non-increasing sequence of positive integers whose sum is \(n\). A partition identity is a theorem stating that for all \(n\), the number of partitions of \(n\) satisfying some conditions equals the number of partitions of \(n\) satisfying some other conditions. In this talk, we will show how \(q\)-difference equations, recurrences and computer algebra can be used to prove such identities.
Repas
15h-16h Maxim Kontsevich (IHES, Bures-sur-Yvette) : p-Determinants and monodromy of differential operators
Abstract. We prove that \(p\)-determinants of a certain class of differential operators can be lifted to power series over \(\mathbb{Q}\). We compute these power series in terms of monodromy of the corresponding differential operators.
