Bonn Topology Group - Abstracts

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Talk

November 5, 2024
Nathaniel Bottman (Max Planck Institute for Mathematics): Building symplectic invariants from J-curves

Abstract

In favorable geometric settings, work of Floer, Donaldson, Fukaya, and others allows one to define the Fukaya A-infinity category, Fuk. Fuk is an invariant of a symplectic manifold M, whose objects are Lagrangian submanifolds of M, whose morphisms are sums of intersections of Lagrangians, and whose composition operations are defined by counts of J-holomorphic polygons. Fuk plays a starring role in modern symplectic geometry, and enabled Kontsevich's Homological Mirror Symmetry conjecture. After sketching the definition of Fuk, I will explain the goal of my research: to equip Fuk with a good notion of functoriality by constructing the Symplectic (A-infinity,2)-Category. Two key ingredients are the 2-associahedra (combinatorial objects that track collisions of lines and points in R^2) and a certain adiabatic limit of elliptic PDEs. If time permits, I will mention new work with Abouzaid and Niu that aims to endow the Fukaya category with a monoidal structure in the context of SYZ mirror symmetry.