Seminar

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For a Calabi-Yau threefold X and a stability condition, one expects to define a Hall algebra from (the cohomology of) the moduli spaces of semistable sheaves on X. This Hall algebra should be a quantum group constructed from a Lie algebra associated to X. The dimensions of the graded pieces of this Lie algebra should be BPS invariants of X, which are certain enumerative invariants of X which determine other enumerative invariants of interest, such as Gromov-Witten and Donaldson-Thomas.

Before attempting to realize these expectations in full generality, one can consider the local case of a quiver with potential, where such a Hall algebra was constructed by Kontsevich-Soibelman.

In these lectures I will explain some results and techniques used to study (Kontsevich-Soibelman) Hall algebras for quivers with potential. The tentative plan is as follows:



Lectures 1 and 2: definition of cohomological Hall algebras for quivers with potential; the Davison-Meinhardt PBW theorem; the BBDG decomposition theorem; dimensional reduction; examples.



Lecture 3: window categories; matrix factorizations.



Lecture 4 (about joint work with Yukinobu Toda): quasi-BPS categories for quivers with potential, with a focus on C^3.