Applications of Microlocal Sheaves of Spectra and K-Theory

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This research project concerns algebraic topology and symplectic geometry. Algebraic topology is the theory of those properties of space that are unchanged by large continuous deformations; for example, the property of having a hole. Algebraic topologists organize their subject using number systems called ring spectra or extraordinary homology theory. Symplectic geometry is a mathematical formalism for classical mechanics. Many questions in symplectic geometry have been resolved through use of a tool called Lagrangian Floer theory. This project aims to develop a single tool that generalizes both theories, forging new connections between the subjects.Floer homotopy theory is an emerging refinement of Floer homology. It bears the same relationship to Floer homology that extraordinary homology bears to ordinary homology. Ultimately it should attach a category, enriched in spectra, to a symplectic manifold. This project aims to construct this category using the microlocal theory of sheaves, to give new techniques for computing it, to use it to prove some conjectures about the algebraic topology of symplectic manifolds, and conversely to symplectically illuminate some aspects of the category of spectra.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.