Séminaires à venir

The celebrated Skolem–Mahler–Lech Theorem asserts that the zero set of a linear recurrence sequence has a remarkably rigid structure: it is a finite union of arithmetic progressions together with a finite set of additional “exceptional” zeros. While the periodic zeros can be described relatively explicitly, the structure of the exceptional ones remains mysterious. In particular, obtaining effective upper bounds for these exceptional zeros is a major open problem.
In this talk, we approach the problem from a probabilistic perspective. Fixing a non-degenerate recurrence relation over a number field, we consider the family of algebraic linear recurrence sequences obtained by varying the initial conditions subject to a bounded height condition. We then estimate how many such sequences admit an exceptional zero exceeding a given explicit bound. In this way, we establish a form of pseudo-effectivity: although absolute effectivity appears to be out of reach, effective bounds may be obtained for almost all sequences with respect to a natural height-based density.
We present results for linear recurrences of order three, based on the construction of suitable auxiliary polynomials, and, time permitting, outline ongoing work aimed at extending these methods to higher orders.

The representation theory of the finite group GLn(Fq) is well understood, thanks to its combinatorial description (given by Green) and its geometric interpretation due to the results of Deligne and Lusztig. Still, we have very little understanding of the multiplicities, i.e. of the decomposition of tensor product of irreducible representations. Hausel, Letellier and Rodriguez-Villegas gave a combinatorial description of the multiplicities in the generic case. This description relates these multiplicities to the cohomology of complex character varieties for GLn(C). In a joint work with Emmanuel Letellier, we try to generalize these results to other groups of type A. In particular, we study multiplicities for characteristic functions of character sheaves of SLn(Fq), rather than irreducible characters, and relate them to the cohomology of complex character varieties for PGLn(C). We expect more generally that , for a finite reductive G(Fq), multiplicities should be related to the character varieties for its complex Langlands dual.

https://www.mathconf.org/agqt2026

We introduce an interacting particle system originating from a nucleation process and investigate the nucleation time as a function of the interaction strength, ranging from weak to strong. Using (uniform) propagation of chaos, we study the non-linear mean-field limit. A standard analysis yields a Yaglom limit conditionned on non-nucleation and its associated tails for the distribution of the nucleation time. The most surprising result arises in the strong interaction regime: the tails follow a decay, where denotes the nucleus size. This result is obtained through an application of the centre manifold theorem. This is a joint work with Frédéric Paccaut.

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