Conférencier: Lucien Hennecart, University of Edinburgh 

Cohomological Hall algebras (CoHAs) were introduced around 15 years ago to define the BPS algebras of some quantum systems. Its existence was predicted by physics. CoHAs are now rigorously definedmathematical objects encoding enumerative invariants of the categories under consideration: representations of the preprojective algebra of aquiver, Higgs sheaves on a smooth projective curve, representations ofthe fundamental group of Riemann surfaces, and more generally 2Calabi—Yau Abelian categories. Locally, the stacks of objects of these categories and their good moduli space, a Poisson scheme, have the samegeometry. In this talk, we will introduce cohomological Hall algebras of 2-Calabi—Yau categories, the BPS (Lie) algebra, and present structural results (for totally negative categories): the BPS Lie algebra is freely generated by the intersection cohomology of some connected components ofthe good moduli space. The full CoHA is the Yangian of the BPS Liealgebra. This has striking consequences: a nonabelian Hodge isomorphismfor Dolbeault and Betti moduli stacks, and the positivity of cuspidal polynomials of quivers (a strengthening of Kac positivity conjecture).The talk is based on joint work with Ben Davison and Sebastian Schlegel Mejia.