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Explore the construction and fundamental results of BPS cohomology and quasi-BPS categories of symmetric stacks in this advanced mathematics lecture. Delve into the theoretical framework where BPS invariants, originally introduced as counts of objects in Calabi-Yau 3-categories generalizing Donaldson-Thomas invariants, are refined to BPS cohomology and quasi-BPS categories. Focus on specific applications to quivers with potential and Higgs bundles on curves, examining the construction of Hall products in singular or critical cohomology and for categories of coherent sheaves or matrix factorizations. Investigate cohomological integrality, semiorthogonal decompositions of relevant categories in terms of quasi-BPS categories, and fundamental properties of quasi-BPS categories. Discover the χ-independence phenomenon for moduli of sheaves supported on curves in Calabi-Yau 3-folds for both BPS cohomology and quasi-BPS categories, and understand its connections to mirror symmetry and Langlands duality for local curves. Learn about contributions from leading mathematicians including Kontsevich-Soibelman, Joyce, Meinhardt-Reineke, Davison-Meinhardt, and Toda, along with recent collaborative research findings.
