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  • Tudor Padurariu

    The commuting stack via quasi-BPS categories

    18 juin 2024 - 14:00Salle de séminaires IRMA

    For a Calabi-Yau threefold X, one expects, following Kontsevich-Soibelman, to define a Hall algebra on the moduli stacks of sheaves on X. Such algebras are a bridge between the enumerative geometry of X and quantum groups. The first example to study is C^3. In this case, the Kontsevich-Soibelman Hall algebras are the same as Hall algebras defined by Schiffmann-Vasserot and Porta-Sala. I will discuss the structure of the categorical Hall algebra of C^2. More precisely, I will discuss a semi orthogonal decomposition of this Hall algebra in quasi-BPS categories, compute these quasi-BPS categories, and relate them to a Bridgeland-King-Reid-Haiman theorem for the Hilbert scheme of points on C^3. I will also explain a (heuristic) connection between the categorical Hall algebra of C^2, the Betti Langlands program for an elliptic curve and the Fukaya category of the torus (following Schiffmann-Vasserot). This is based on joint work in progress with Sabin Cautis and Yukinobu Toda.