Recordings of past talks and presentation slides will be available here.
2025-12-01 — Rafael Andrist (University of Ljubljana)
Title: Holomorphic symplectic automorphisms of Markov Surfaces
Abstract: The Diophantine solutions of the equation x² + y² + z² = 3xyz were originally considered by Markov, and are now called Markov triples.
Later, this equation was studied over the complex numbers and considered as an algebraic surface. The group of algebraic automorphisms of the Markov surface is discrete and acts transitively on the Markov triples. The Markov surface admits a natural meromorphic symplectic form which is preserved by all algebraic automorphisms.
We describe the identity component of the group of holomorphic symplectic automorphisms of the Markov surface. In contrast to the algebraic case, this group is infinite-dimensional and interpolates any permutation of (ordered) Markov triples. The results can be extended to so-called Markov-type surfaces of the form x² + y² + z² - 3xyz - Ax - By - Cz - D = 0.
2025-11-03 — Serge Cantat (CNRS, Universite de Rennes)
Abstract: Choose a compact Lie group G, say the group SO_k, and an integer n > 2. We will be interested in n-tuples (g_1, …, g_n) of elements in G and in the smallest closed subgroup containing such a tuple. The product replacement algorithm can be considered as a group action on G^n generated by the following simple moves:
(a) permuting the g_i,
(b) changing one of them into its inverse, and
(c) changing some g_j into g_jg_i with i distinct from j.
Such ``moves’’ do not change the group generated by (g_1, …, g_n). The problem is to describe the orbits of the product replacement algorithm in G^n. I will describe this problem and explain some results that hold when n is large. (based on a joint work with C. Dupont and F. Martin-Baillon).