The elliptic Hall algebra

Since its introduction by Burban and Schiffmann, the elliptic Hall algebra has played a prominent role in algebraic geometry, particularly in connection with the study of moduli spaces (such as the K-theory of Hilbert schemes of points on the affine plane), and in representation theory concerning quantum groups. The thesis will introduce the construction of the elliptic Hall algebra and its two presentations: the original one stemming from the theory of Hall algebras, and the second resembling Drinfeld’s new realization of quantum groups. Prospective students should have a background in the theory of abelian categories and Lie theory, as well as an interest in geometric realizations of associative algebras.

Prerequisites: Familiarity with sheaf theory and the theory of abelian categories.

References: The first two chapters of Schiffmann’s lecture notes on Hall algebras, Burban and Schiffmann’s paper, and Schiffmann’s paper.