Cohomological Hall Algebras of 2-Calabi-Yau Categories and Applications - Lecture 1

Explore the first lecture in a four-part series on Cohomological Hall algebras (CoHAs) of 2-Calabi-Yau categories and their applications, delivered by Lucien Hennecart from the University of Edinburgh. Delve into the interactions between CoHAs and key questions in algebraic geometry and representation theory, including enumerative geometry and Kac-Moody algebras. Discover how CoHAs provide associative algebra structures on the Borel-Moore homology of object stacks in certain Abelian categories, such as sheaves on surfaces, quiver representations, and fundamental group representations. Learn about the role of CoHAs in understanding the cohomology of stacks and moduli spaces, and their applications to various conjectures like cohomological integrality, positivity, and purity. Gain insights into the geometric construction of generalised Kac-Moody algebras and prepare for subsequent lectures covering applications to quiver varieties and nonabelian Hodge theory. This lecture, part of the M-Seminar at Kansas State University, offers a comprehensive introduction to the subject based on joint work with Ben Davison and Sebastian Schlegel Mejia.