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The stack "Broken" and associative algebras

Presented by: 
Hiro Tanaka Harvard University
Date: 
Tuesday 30th October 2018 - 14:00 to 15:00
Venue: 
INI Seminar Room 2
Abstract: 
 After reviewing some aspects of Morse theory, I'll talk about "Broken," the moduli stack of constant Morse trajectories (possibly broken) on a point. Surprisingly, this stack has the following property: Factorizable sheaves on it are the same thing as (possibly non-unital) associative algebras. We all know that having geometric descriptions of algebraic structures should buy us mileage; so what mileage does this property buy us? If time allows, I'll try to explain why this theorem leads to a roadmap for constructing Morse chain complexes, and in fact, for constructing the stable homotopy type of a compact manifold with a Morse function. (That is, this gives a different way to realize ideas of Cohen-Jones-Segal.) The motivation is to construct a stable homotopy type for Lagrangian Floer Theory--the latter is an important invariant in symplectic geometry and mirror symmetry. This is all joint work with Jacob Lurie.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons