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Explore the advanced mathematical theory of BPS cohomology and quasi-BPS categories in this fourth and final lecture of a specialized series delivered at the M-Seminar at Kansas State University. Delve into the construction and fundamental results concerning BPS cohomology and quasi-BPS categories of symmetric stacks, with particular emphasis on quivers with potential and Higgs bundles on curves. Examine the construction of Hall products in singular or critical cohomology and for categories of coherent sheaves or matrix factorizations, while investigating cohomological integrality and semiorthogonal decompositions of relevant categories. Discover the properties of quasi-BPS categories and analyze the χ-independence phenomenon for moduli of sheaves supported on curves in Calabi-Yau 3-folds, exploring its connections to mirror symmetry and Langlands duality for local curves. Learn about foundational work by Kontsevich-Soibelman, Joyce, Meinhardt-Reineke, Davison-Meinhardt, and Toda, alongside recent collaborative research findings with Yukinobu Toda and other leading mathematicians in the field.
