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Join Fabian Haiden from the Center for Quantum Mathematics in this lecture from the M-Seminar at Kansas State University as he explores a replacement for homotopy cardinality in contexts where it is inherently undefined, particularly in Z/2-graded dg-categories. Learn about Calabi-Yau structures and their relative generalizations as key components of this approach. Discover applications including obtaining Hall algebras for many pre-triangulated dg-categories and providing intrinsic replacements for ad-hoc constructions like the elliptic Hall algebra via the Drinfeld double. Understand how this work proves a conjecture by Ng-Rutherford-Shende-Sivek about expressing the ruling polynomial of Z/2m-graded Legendrian knots in terms of the homotopy cardinality of augmentation categories. This lecture presents joint research with Mikhail Gorsky (arxiv:2409.10154).
