Noncommutative geometry, smoothness, and Fukaya categories

Noncommutative geometry, smoothness, and Fukaya categories - Sheel Ganatra

Sheel Ganatra
Member, School of Mathematics
November 30, 2016
Noncommutative geometry, as advocated by Konstevich, proposes to replace the study of (commutative) varieties by the study of their (noncommutative) dg/A-infinity categories of perfect complexes. Conveniently, these techniques can then also be applied to Fukaya categories. In this mini-course, we will review some basic properties and structures in noncommutative geometry, with an emphasis on the notion of "smoothness" of a category and its appearance in topology and both sides of homological mirror symmetry. Then, we will introduce a non-commutative version of the integration pairing defined by Shklyarov, as well as a new twisted version, and examine their interaction with smoothness. We will also study some applications to symplectic geometry, stemming from the compatibility of these structures with Floer-theoretic open-closed maps. Part of this is joint work with Perutz and Sheridan.