Traces of Partition Eisenstein series (Ken Ono, University of Virginia)

Integer partitions are ubiquitous in mathematics, arising in subjects as disparate as algebraic combinatorics, algebraic geometry, number theory, representation theory, to mathematics physics. Many of the deepest results on partitions have their origin in the work of Ramanujan. In this lecture, we will describe a completely new and unexpected role for partitions that also arises from the mysterious “lost notebook” of Ramanujan. We discover and explain the role of new q-series called “partition Eisenstein series”. These functions magically pop up as the key device for solving a conjecture of Andrews and Berndt, for studying symmetric functions of 2-dimensional lattice sums, for determining the properties of Andrews-Garvan “crank statistic”, and for representing the Taylor coefficients of virtually every interesting Jacobi automorphic form. This talk will tell the story of the recent discovery of these functions, and will offer a brief tour of these applications.