How to Construct Topological Invariants via Decompositions and the Symplectic Category

Members Seminar
Topic:How to Construct Topological Invariants via Decompositions and the Symplectic Category
Speaker:Katrin Wehrheim
Affiliation:Massachusetts Institute of Technology; Member, School of Mathematics
Date:Monday, October 17
Time/Room:2:00pm - 3:00pm/S-101
Video Link:https://video.ias.edu/members/wehrheim

A Lagrangian correspondence is a Lagrangian submanifold in the product of two symplectic manifolds. This generalizes the notion of a symplectomorphism and was introduced by Weinstein in an attempt to build a symplectic category. In joint work with Chris Woodward we define such a category in which all Lagrangian correspondences are composable morphisms. We extend it to a 2-category by extending Floer homology to cyclic sequences of Lagrangian correspondences. This is based on counts of 'holomorphic quilts' - a collection of holomorphic curves in different manifolds with 'seam values' in the Lagrangian correspondences. A fundamental isomorphism of Floer homologies ensures that our constructions are compatible with the geometric composition of Lagrangian correspondences. This provides a general prescription for constructing topological invariants by decomposition into simple pieces and a partial functor into the symplectic category (which need only be defined on simple pieces; with moves corresponding to geometric composition).