Langlands duality and higher traces
A central theme in arithmetic geometry over finite fields is the passage from geometric invariants to arithmetic information by taking the trace of Frobenius. I will describe a higher version of this procedure with a particular focus on applications to the Langlands correspondence over function fields. In this case, this procedure relates the geometric Langlands correspondence with the classical one. Specifically, we obtain that the space of automorphic functions is the categorical trace (aka Hochschild homology) of Frobenius acting on an appropriate version of the automorphic category. This leads to a localization of the space of automorphic functions on a moduli space of Langlands parameters, giving a refinement of V. Lafforgue's spectral decomposition. This is based on joint works with Arinkin, Gaitsgory, Kazhdan, Raskin, and Varshavsky.
Dr. Rozenblyum received his PhD from the Massachusetts Institute of Technology in 2011 and is currently an Assistant Professor at the University of Chicago.
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