Motives, mapping class groups, and monodromy

TL;DR

This paper surveys recent advances connecting algebraic geometry, surface topology, and differential equations, focusing on mapping class groups, character varieties, and isomonodromy equations, highlighting open problems and conjectures.

Contribution

It explores the interplay between mapping class groups, character varieties, and isomonodromy equations, providing new perspectives and raising open questions in the field.

Findings

Analysis of mapping class group actions on character varieties

Connections between isomonodromy equations and algebraic geometry

Open conjectures on the interplay of topology and differential equations

Abstract

We survey some recent developments at the interface of algebraic geometry, surface topology, and the theory of ordinary differential equations. Motivated by "non-abelian" analogues of standard conjectures on the cohomology of algebraic varieties, we study mapping class group actions on character varieties and their algebro-geometric avatar -- isomonodromy differential equations -- from the point of view of both complex and arithmetic geometry. We then collect some open questions and conjectures on these topics. These notes are an extended version of my talk at the April 2024 Current Developments in Mathematics conference at Harvard.