Speaker:	Ben Davison (University of Edinburgh)
Title:	BPS cohomology in geometry and representation theory 1 (introductory talk in a four-lecture series)
Date (JST):	Thu, Mar 26, 2026, 15:30 - 17:00
Place:	Seminar Room A
Abstract:	I will introduce the notion of BPS cohomology of 1) smooth, 2) (-1)-shifted and 3) (0)-shifted symplectic stacks, as well as surveying some applications of this theory to problems in quantum groups, positivity problems in combinatorics, and enumerative algebraic geometry.
Remarks:	This seminar serves as the introductory talk in a four-lecture series on the same topic. Prof. Ben Davison will give three additional lectures at 13:30 on BPS cohomology of (0)-shifted symplectic stacks at the following venues. Seminar Room A on March 30 (Mon) Seminar Room B on April 1 (Wed) Seminar Room B on April 2 (Thu) In this series of talks I will focus on the theory of cohomological BPS invariants for (0)-shifted symplectic stacks. Although this represents only one of the three options above, there are highly developed applications in geometry, representation theory, and nonabelian Hodge theory, and I will be very happy to take input on which of these directions I should lean towards this second week. Stacks of objects in 2-Calabi-Yau categories are a rich source of (0)-shifted symplectic stacks. The Borel-Moore homology of such stacks carry a cohomological Hall algebra (CoHA) product, and the resulting algebras can be often be described in terms of intersection cohomology of coarse moduli spaces. I will review this theory for Nakajima quiver varieties, where this CoHA can be fairly explicitly described in terms of Borcherds algebras. This leads naturally to a comparison result with the Maulik-Okounkov Yangian, defined in terms of (cohomological) stable envelopes. Focusing more on general theory, there is now (due to joint work with Bu, Ibañéz-Nuñéz, Kinjo and Padurariu) a more general theory of BPS cohomology for (0)-shifted symplectic stacks admitting a good moduli space. This provides the required language for talking about topological mirror symmetry and nonabelian Hodge theory for stacks outside of type A, for example. Finally, BPS cohomology provides the framework for proving cohomological chi-independence type conjectures for enumerative invariants of K3 surfaces. A precise version of this conjecture states that the Gopakumar-Vafa invariants of moduli spaces of 1-dimensional coherent sheaves with primitive Mukai vector on K3 surfaces defined in work of Maulik and Toda, and for generally in further work of Toda, do not depend on the Euler characteristic of the sheaves under consideration. This work requires us to step beyond considering BPS cohomology for fixed abelian categories possessing good moduli spaces, and also provides the natural generalisation of Markman's theorem on tautological generation of the cohomology of moduli spaces of semistable coherent sheaves for primitive Mukai vector.

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