Recordings of past talks and presentation slides will be available here.
Abstract: This talk is based on joint work with Serge Cantat, Hanspeter Kraft, and Andriy Regeta. We investigate the following question for a variety X: given an irreducible family of automorphisms of X that contains the identity, under which conditions does this family generate an algebraic subgroup of Aut(X)?
When the members of the family commute pairwise and X is affine, a result of Serge Cantat, Andriy Regeta, and Junyi Xie shows that such a family indeed generates an algebraic subgroup. We extend this theorem to the case where the family generates a solvable subgroup and X is only assumed to be quasi-affine. We also present several applications of this result, for example to Borel subgroups in Aut(X).
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).