Dynamics on character varieties and Hodge theory

• Applicant: Dr. Yeuk Hay Joshua Lam
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541272769

The goal of this project is to study the dynamics of a rich class of group actions, known as the braid group action on character varieties. The main motivation lies in the relation to algebraic geometry; there are also close ties to Hodge theory and geometric topology. One of the main aims of this project is to prove new cases of an important conjecture known as Simpson's motivicity conjecture. The group action takes place on spaces known as the character varieties of surfaces. These are rather classical spaces in algebraic geometry and topology, but have also played a major role in recent years, for example in the geometric Langlands programme. Roughly, they parametrise representations of the fundamental group of a topological surface; these representations are also called local systems. There is a special class of local systems, known as the motivic ones: these are the ones coming from algebraic geometry, and carry much more structure than an arbitrary local system. Character varieties have a huge group of symmetries known as the mapping class group, which may be thought of as all the ways of cutting up a topological surface and gluing it back together. This project will focus on the most classical and arguably the richest case of this group action, namely the case of punctured spheres; in this case the mapping class group is the famous braid group. A specific goal is to classify the most ``symmetric" points of this dynamical system, namely those which have finite orbits: we refer to them as canonical points, and they are conjectured to be motivic. In a recent joint work with Landesman and Litt, we completely classified the canonical points in the case of rank two local systems, assuming a mild condition. This is a very classical problem, for which we developed a novel approach using Hodge theory. It is also the direct generalisation of the classification problem of algebraic solutions to Painlevé VI, which has attracted a lot of attention throughout the years. In this project, we will push our methods to study higher rank local systems. As mentioned above, this will prove new cases of Simpson's motivicity conjecture. This will involve developing new tools in Hodge theory. To understand more on the dynamics of the braid group action, I will also study special subvarieties, namely those whose orbit under the braid group is finite. I will then explore relations to other areas, such as the geometry of curves, as well as analogous problems in positive characteristic.

• DFG Programme: WBP Position