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

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AbstractAward: DMS-0844188Principal Investigator: Katrin WehrheimThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This research project addresses several areas within symplectictopology and its interaction with gauge theory, complex geometry,and low-dimensional topology. Lagrangian morphisms in thesymplectic category are viewed as morphisms between manifoldsthat are not necessarily symplectomorphic, but their geometriccomposition is generically singular. A previous projectpartially resolves this problem, with consequences including theconstruction of invariants for three-manifolds and knots; thenext goal is to fit these invariants into a framework oftopological quantum field theory, and another goal is to extendthe existing framework to a refined Fukaya category, leading to atool for mirror symmetry proofs. An ongoing project on theAtiyah-Floer conjecture will continue, and more speculative workwill consider non-squeezing effects in PDE flows from aninfinite-dimensional symplectic point of view.The research projects described in this proposal concernsymplectic geometry, the geometric structure that lies behind theHamiltonian formulation of mechanics. Recent applications ofideas from symplectic geometry include invariants that help todistinguish one three-dimensional space from another. Broaderimpacts of this project concentrate on promoting women inmathematics, and feature both a conference at MIT on women inmathematics that is aimed at junior researchers and Girls' Angle,a math club for girls that is based on close informal interactionwith female mentors from mathematical areas. See http://www.girlsangle.org/ for more information.