Singularities and symplectic mapping class groups

Introduction

Symplectic topology is a central part of modern geometry, with historical roots in classical mechanics. Symplectic structures also arise naturally in low-dimensional topology, in representation theory, in the study of moduli spaces of algebraic varieties, and in quantum mechanics. A fundamental question is to understand the automorphisms of a symplectic manifold. The most natural ones are symplectomorphisms, i.e., diffeomorphisms preserving the symplectic structure. I propose to study structural properties of their group of isotopy classes, called the symplectic mapping class group (SMCG).

Understanding SMCG

In dimension two, the SMCG agrees with the classical mapping class group; in higher dimensions, our understanding is very sparse. I propose to systematically study SMCGs for the family that I believe to be the key building blocks for developing a general theory: smoothings (i.e., Milnor fibres) of isolated singularities.

Project Proposals

I first propose to give complete descriptions of categorical analogues of SMCGs for two major, complementary families:

1. Milnor fibres of simple elliptic and cusp singularities (Project 1);
2. Stein varieties associated to two-variable singularities and quivers (Project 2).

These capture two different generation paradigms: one where the classical story generalizes, and one for which it systematically breaks. This will inform Project 3, in which I propose to describe the categorical SMCGs of universal Milnor fibres, introduced here. Progress on these projects will also bring questions about the dynamics of SMCGs within reach for the first time; Project 4 will study these applications.

Methodology

The proposed constructions combine insights from different viewpoints on mirror symmetry with ideas from representation theory and singularity theory. I also plan to apply symplectic ideas to answer classical questions in singularity theory. Beyond this, the proposal borrows ideas from, inter alia, geometric group theory, algebraic geometry, and homological stability.