Speaker:   Fabian Haiden   (Center for Quantum Mathematics)

Abstract: I will discuss a replacement for homotopy cardinality in situations where it is a priori ill-defined, including Z/2-graded dg-categories. A key ingredient are Calabi-Yau structures and their relative generalizations. As an application we obtain a Hall algebra for many pre-triangulated dg-categories for which it was previously undefined. This also gives an intrinsic replacement for many ad-hoc constructions, such as that of the elliptic Hall algebra via the Drinfeld double. Another application is the proof of a conjecture of Ng-Rutherford-Shende-Sivek expressing the ruling polynomial of a Z/2m-graded Legendrian knot (which is part of the HOMFLY polynomial if m=1) in terms of the homotopy cardinality of its augmentation category. This is joint work with Mikhail Gorsky, arxiv:2409.10154.