Equivariant homotopy theory and geometric Langlands

Sanath Devalapurkar, Harvard University

Online Talk 

Let G be a connected compact Lie group. The "geometrization" of the Langlands program relates the category of constructible sheaves of complex vector spaces on the space of G-bundles on a Riemann surface C to the category of (quasi)coherent sheaves on the stack of G^-local systems on C. Here, G^ is the Langlands dual group of G. 

In this talk, I would like to discuss an approach to understanding this conjecture from the perspective of homotopy theory; for instance, if ku denotes connective complex K-theory, how might the story change if one studies the category of constructible sheaves of ku-modules (instead of complex vector spaces)? The main step, as I hope to explain, is almost always to compute certain equivariant homology groups in combinatorial terms; I will illustrate this through key examples. If time permits, I hope to share some ideas about the story obtained by studying sheaves of spectra (i.e., modules over the sphere spectrum).