Geometric representations of the elliptic Hall algebra

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 extension of this result to K-theory was initially explored by Schiffmann and Vasserot for the affine plane. Here, the elliptic Hall algebra takes the role of the (super) Heisenberg algebra. This result was later generalized to any smooth projective complex surface by Neguţ, who also categorized such a construction within the bounded derived categories. The thesis will focus on Neguţ’s construction within the bounded derived categories of coherent sheaves, which provides a “weak” categorification of the elliptic Hall algebra. Prospective students should have a background in algebraic geometry and Lie theory, as well as an interest in the geometric realizations of associative algebras and the study of the geometry of moduli spaces through representation theory techniques.

References: Sections 1, 2, and 3 of Schiffmann and Vasserot’s paper and Neguţ’s paper.