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Explore the foundational concepts of BPS cohomology and quasi-BPS categories in this first lecture of a four-part series delivered at the M-Seminar at Kansas State University. Delve into the construction and fundamental results surrounding BPS invariants, which originated as counts of objects in Calabi-Yau 3-categories and generalize Donaldson-Thomas invariants for ideal sheaves and stable sheaves on Calabi-Yau 3-folds. Learn how these invariants extend to symmetric stacks and can be refined to BPS cohomology and quasi-BPS categories. Focus on 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. Discover cohomological integrality, semiorthogonal decompositions of relevant categories in terms of quasi-BPS categories, and key properties of quasi-BPS categories. Investigate 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 its connections to mirror symmetry and Langlands duality for local curves. The lecture draws from extensive research by Kontsevich-Soibelman, Joyce, Meinhardt-Reineke, Davison-Meinhardt, Toda, and collaborative work with Yukinobu Toda, Chenjing Bu, Ben Davison, Andrés Ibáñez Núñez, and Tasuki Kinjo.
