Symplectic Topology, Symmetries, and Singularities

Columbia University, New York NY

Investigators

Abstract

This project pertains to an area of pure mathematics called symplectic geometry. Modern geometry is centered around the study of manifolds, smooth objects that at small enough scale look like the standard space of a fixed dimension; for instance, the surface of a ball is a two-dimensional manifold. Symplectic manifolds are equipped with extra structure that generalizes conservation laws from classical mechanics. Also, some models in string theory, a branch of physics, allow any symplectic manifold in lieu of space-time. This project studies the question: What are the transformations (that is, global symmetries) of a symplectic manifold? In particular, the investigator will use dualities predicted by string theory to further our understanding of these transformations. This project aims to investigate symmetries of symplectic manifolds, using tools from Floer theory, singularity theory, and mirror symmetry. The most naive such symmetries are symplectomorphisms, i.e. diffeomorphisms which preserve the symplectic form. These form a topological group, the connected components of which give the symplectic mapping class group. This generalizes the "usual" mapping class group for Riemann surfaces. This project studies structural properties of symplectic mapping class groups, with a focus on four-dimensional examples coming from singularity theory. The investigator will also aim to understand the mirror phenomena in algebraic geometry. Symplectomorphisms induce automorphisms of the Fukaya category, an algebraic strengthening of Lagrangian Floer theory. The project will also consider "hidden" symmetries of symplectic manifolds: automorphisms of the Fukaya category that are not induced by symplectomorphisms, such as C*-actions.