CAREER: The symplectic category, Floer field theory, and relations to gauge theory and topology

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

Abstract Award: DMS-0844188 Principal Investigator: Katrin Wehrheim This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This research project addresses several areas within symplectic topology and its interaction with gauge theory, complex geometry, and low-dimensional topology. Lagrangian morphisms in the symplectic category are viewed as morphisms between manifolds that are not necessarily symplectomorphic, but their geometric composition is generically singular. A previous project partially resolves this problem, with consequences including the construction of invariants for three-manifolds and knots; the next goal is to fit these invariants into a framework of topological quantum field theory, and another goal is to extend the existing framework to a refined Fukaya category, leading to a tool for mirror symmetry proofs. An ongoing project on the Atiyah-Floer conjecture will continue, and more speculative work will consider non-squeezing effects in PDE flows from an infinite-dimensional symplectic point of view. The research projects described in this proposal concern symplectic geometry, the geometric structure that lies behind the Hamiltonian formulation of mechanics. Recent applications of ideas from symplectic geometry include invariants that help to distinguish one three-dimensional space from another. Broader impacts of this project concentrate on promoting women in mathematics, and feature both a conference at MIT on women in mathematics that is aimed at junior researchers and Girls' Angle, a math club for girls that is based on close informal interaction with female mentors from mathematical areas. See http://www.girlsangle.org/ for more information.