Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Let $X_n$ be the set of conjugacy classes of n-tuples of 2x2 matrices, whose product is the identity matrix. There is a natural braid group action on $X_n$, whose study goes back to work of Markoff in the late 19th century. The most basic question one can ask about this action, which dates to work of Painlevé, Fuchs, Schlesinger, and Garnier in the beginning of the 20th century, is: what are the finite orbits of this action? I'll explain the history of this question, as well as some recent work (combining results of Bronstein-Maret with results obtained jointly with Josh Lam and Aaron Landesman), which answers it completely. Time permitting, I'll discuss "higher genus" variants of this question, whose answer relies on non-abelian Hodge theory and the Langlands program, and resolves conjectures of Esnault-Kerz, Budur-Wang, Kisin, and Whang.