# Generalizations of Hamiltonian varieties in relative geometric Langlands

A recent article by Ben-Zvi, Sakellaridis, and Venkatesh proposes a framework for duality between automorphic periods and L-functions via a duality between Hamiltonian G- and G^-spaces, where G is a reductive group and G^ is its Langlands dual. In this talk, I will review some of this picture (mainly through examples), and describe some of my recent work on using homotopy-theoretic tools to understand this duality. These tools suggest a generalization of (relative) Langlands duality to more "exotic" coefficients like K-theory, elliptic cohomology, complex cobordism, and framed cobordism, which I will explain. Some concrete applications, for instance, include placing the theory of "quasi-Hamiltonian spaces"(and a certain generalization thereof) into this framework. Time permitting, I hope to describe how to "deformation quantize" this picture.