UC Berkeley

Multivariate orthogonal polynomials of Macdonald are an important tool to study a variety of topics in modern mathematical physics, such as chiral algebras, three-dimensional topological theories of Chern-Simons type, five-dimensional supersymmetric Yang-Mills theories, and others. We describe several recent applications of Macdonald polynomials, based on original research contributions. Introduction gives an overview of Macdonald theory, with a view towards applications. In Chapter 2, we discuss a Macdonald deformation of three-dimensional Chern-Simons topological field theory and construct it explicitly in the case of Heegaard splitting of genus one. The resulting knot invariants turn out to be related to the recently developed theory of knot homology. In Chapter 3, we show that Macdonald ensembles are natural integral representations for the Nekrasov functions -- important special functions in the context of five-dimensional supersymmetric Yang-Mills theories. This allows us to prove, in vast generality, a conjecture that Nekrasov functions are equal to the chiral blocks of W-algebras.