14.00 – 14.45 : Xenia Flamm (MPI Leipzig – IHES) 

Title: Positive representations and non-Archimedean ordered fields

Summary: Positive representations were introduced by Olivier Guichard and Anna Wienhard in their foundational work on higher Teichmüller theory. In this talk, we propose a notion of positive representation from the fundamental group of a (possibly non-closed) hyperbolic surface into the k-points of a reductive algebraic group, where k is an ordered field. Our approach provides a common framework that simultaneously extends the classical theory for closed surfaces and real Lie groups. We focus on the non-Archimedean case, where new phenomena arise, and explain how such representations can be understood as limits of real positive representations. This is joint work with Nicolas Tholozan, Tianqi Wang and Tengren Zhang.

15.00 – 15.45 : Vlerë Mehmeti (IMJ-PRG)

Title: Variation of the Hausdorff dimension and degenerating Schottky groups

Summary: I will talk about the continuity of the Hausdorff dimension of limit sets of Schottky groups defined over arbitrary complete valued fields.The common ambient topology allowing one to vary at the same time the Schottky group and the base field is induced by analytic spaces over Banach rings (in the sense of Berkovich). As an application, one can obtain information on the asymptotic behavior of families of degenerating complex Schottky groups, which can be extended to a continuous family by a non-Archimedean counterpart. No non-Archimedean prerequisites will be necessary. This is based on joint work with Nguyen-Bac Dang.

16.15 – 17.00 : Anna Ben-Hamou (Sorbonne Université)

Title: Mixing time of a random walk on binary matrices

Summary: In this talk, we will consider a Markov chain on n by n invertible binary matrices, which moves by picking an ordered pair of distinct rows and adding one to the other mod 2. This chain was first studied by Diaconis and Saloff-Coste (1996), who showed that the mixing time was O(n^4). Then Kassabov (2003) improved it to O(n^3). Using this last result, we will show that the logarithmic Sobolev constant is O(n^2), which yields an upper bound of O(n^2 log n) on the mixing time. Up to logarithmic terms, this matches the lower bound.