COHA of zero-dimensional sheaves on a smooth surface

Nakajima and Grojnowski independently demonstrated that the cohomology group of the Hilbert scheme of points on a smooth quasi-projective complex surface \(S\) is isomorphic to the Fock space of a (super) Heisenberg algebra, modeled on the cohomology of \(S\). The main tool employed in constructing the action of the (super) Heisenberg algebra was the 1-step Hecke correspondence, which modifies sheaves at a single point. These operators can be realized as elements of the cohomological Hall algebra of zero-dimensional sheaves on a smooth surface. The thesis will concentrate on the general construction of this algebra and provide an explicit description in terms of generators and relations, as given in arXiv:2311.13415.

Prerequisites: Familiarity with scheme theory, basic representation theory, and homology theory.

References: The papers arXiv:1901.07641 and arXiv:2311.13415.