Modern Mathematics Lecture Series | Dynamics on character varieties and Hodge theory

Abstract

Let X_n be the set of tuples of 2x2 matrices (A_1, A_2, ..., A_n) such that the product A_1...A_n is the identity matrix, and considered up to simultaneous conjugation. On each X_n, there is a very classical and explicit action of the so-called braid group B_n. This is an elementary case of the so-called mapping class group action on the character varieties of surface groups, and was studied by Markoff and Painlevé, for example.

In recent joint works with Litt and Landesman-Litt, for each n, we classified all points of X_n with finite orbit under this braid group action, assuming one of the A_i's has infinite order. I will discuss why this classification problem, which seems purely group theoretic or topological, may be of interest to algebraic and arithmetic geometers. I'll indicate how Hodge theory plays a crucial role in our work, some surprising relations to complex reflection groups, and many open questions in this area.

About the speaker

Josh Lam is currently a Dirichlet Fellow at Humboldt University in Berlin. He graduated from Harvard University with a PhD in Mathematics in May 2021, where he was fortunate to work with Mark Kisin, and after that he spent a year as a postdoc at IHES. He is interested in various aspects of number theory and algebraic geometry, and most recently in some interactions with low dimensional topology.