Singularities and symplectic mapping class groups

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).In dimension two, the SMCG agrees with the classical mapping class group; in higher dimensions, our understanding is very sparce. 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. I first propose to give complete descriptions of categorical analogues of SMCGs for two major, complementary families: - Milnor fibres of simple elliptic and cusp singularities (Project 1); - Stein varieties associated to two-variable singularities and quivers (Project 2).These capture two different generation paradigms: one where the classical story generalises, 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. The proposed constructions combine insights from different viewpoints on mirror symmetry with ideas from representation theory and singularity theory, and 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.