Structure of Low-Dimensional Floer Homologies

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).

Floer homology theory, a technique using partial differential equations to study problems in smooth / symplectic topology, were first introduced in the 1980's. Since then, these theories have led to many dramatic discoveries, including the resolution of the Arnold conjecture in symplectic geometry and Gordon's conjecture on lens space surgeries, but their structure remains somewhat mysterious. The focus of this project is on better understanding the structure of certain Floer homology theories. One particular goal is to continue to develop Lipshitz-Ozsvath-Thurston's theory of 'bordered Floer homology,' an effort to axiomatize Heegaard Floer homology by a type of second-order degeneration. Another goal is to construct a 'Floer homotopy type' for Lagrangian intersection Floer homology, an idea suggested by Cohen-Jones-Segal and carried out in other contexts by Manolescu, Kronheimer-Manolescu, and Sarkar. A third goal is to associate maps on knot Floer homology to knot cobordisms, hopefully leading to deep structural results about 3- and 4-dimensional Heegaard Floer theory.

Most of mathematics falls into one of two categories. One category studies continuous problems, like fluid flow or minimal surface (bubble) formation, often through partial differential equations. Another studies more rigid, algebraic problems like hidden symmetries of real world objects (e.g., mosaics or particle physics theories) or mathematical objects (like groups and fields). Some of the most striking mathematics sits at the intersection of the two types, using algebraic methods to study qualitative properties of continuous systems which are quantitatively intractable. One powerful example of this method is Floer homology, which uses algebraic structures like chain complexes, Hochschild homology, and topological field theories to study certain families of partial differential equations coming from physics. Floer theory then re-interprets the results to answer questions in topology (large-scale or qualitative geometry). This project will extend several Floer homology theories by considering even deeper algebraic and topological structure to obtain stronger geometric results.