Geometry and Topology Seminar: Weinstein domains: a symplectic geometer's handlebodies

Abstract: Weinstein domains are symplectic analogs of smooth handlebodies and come equipped with decompositions with elementary symplectic pieces. As a result, they are easy to construct, have computable invariants, and include classical examples like cotangent bundles. After giving some background, I will survey several questions (and recent answers) about Weinstein domains, many of which are motivated by analogous questions in smooth topology and have categorical interpretations. For example, the minimal number of Weinstein handles in a Weinstein domain is related to the Grothendieck group of its Fukaya category, yielding new examples of categories with vanishing Grothendieck group. In joint work with Andrew Hanlon and Jeff Hicks, we proved that the minimal number of "elementary" Weinstein sectors needed to cover a Weinstein domain is related to the Rouquier dimension of its Fukaya category. In the case of the cotangent bundle of M, this Rouquier dimension is bounded by the LS-category of M, which was recently used to prove that the Rouquier dimension of the category of coherent sheaves of toric varieties is equal to their dimension.

Host: Ali Daemi