Floer theories in symplectic geometry and low dimensional topology

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

A primary goal of this project is to finish the proof of the Atiyah-Floer conjecture by a version of the large structure limit in a formulation of mirror symmetry for Kaehler surfaces. This would relate the gauge theoretic Floer homology of a homology three-sphere to a Floer homology of Lagrangians which arise from moduli spaces of flat bundles associated to a Heegaard splitting. Another large part of the project aims to realize Lagrangian correspondences as composable functors on refined Donaldson-Fukaya categories. This should lead to topological invariants and TQFT's by using gauge theoretic moduli spaces to represent topological morphisms ( e.g. 3-cobordisms or tangles) as Lagrangian correspondences. The project belongs into the general realm of interaction between symplectic geometry and low dimensional topology. The construction of topological invariants via a symplectic category has been a guiding vision in this field although the geometric composition of Lagrangian correspondences is only partially defined. This project aims to realize this vision, based on a full algebraic definition of compositions. Moreover, a proof of the Atiyah-Floer conjecture would be an important step towards understanding the relations between different invariants of 3-manifolds. More generally, this project aims to further the understanding and exposition of the analytic foundations of gauge theory, pseudoholomorphic curves, and moduli spaces of nonlinear PDE's in general.