The Poisson summation conjecture predicts that suitable spaces admit Schwartz spaces, Fourier transforms and a Poisson summation formula generalizing the familiar Poisson summation formula on a vector space. It implies the expected analytic properties of Langlands $L$-functions and thereby much of Langlands functoriality.
The lectures will focus on the following work, which are aimed at reducing a key case of the conjecture to local assertions. Several methods that can be used to construct new Poisson summation formulae from old ones will be isolated and explained.
Another relevant work in preparation is the following:
The Langlands program is a web of far-reaching and influential conjectures about connections between number theory, representation theory and geometry. Within this program, the relative Langlands program has emerged as one of its most important and productive branches. In these talks, I will give an overview of key problems in the relative Langlands program, with a focus on the elegant theory of relative Langlands duality, recently developed by Ben-Zvi, Sakellaridis, and Venkatesh.
