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In this lecture from the M-Seminar at Kansas State University, Fabian Haiden from the Center for Quantum Mathematics explores a replacement for homotopy cardinality in contexts where it is initially ill-defined, particularly in Z/2-graded dg-categories. Discover how Calabi-Yau structures and their relative generalizations serve as essential components in developing Hall algebras for many pre-triangulated dg-categories that previously lacked definition. Learn about the intrinsic replacement this approach provides for various ad-hoc constructions, including the elliptic Hall algebra via the Drinfeld double. The lecture also covers the proof of a conjecture by Ng-Rutherford-Shende-Sivek regarding the ruling polynomial of Z/2m-graded Legendrian knots and their relationship to the HOMFLY polynomial when m=1, expressed through the homotopy cardinality of augmentation categories. This presentation is based on joint work with Mikhail Gorsky (arxiv:2409.10154) and was delivered on March 13, 2025.
