Noncommutative geometry, smoothness, and Fukaya categories

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.