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Speaker: Langwen Hui (University of Illinois)

Abstract:
I will explain the construction of a Fourier-Mukai transform in the setting of Lurie's spectral algebraic geometry, and discuss its fundamental properties. This can be regarded as a derived enhancement of the classical Fourier-Mukai transform for abelian varieties.

As an application, this yields a common refinement of three facets of the phenomenon known as "level–rank duality": (1) positive-energy representations of LU(n)_k and LSU(k)_n (representation theory), (2) the level-rank duality on equivariant TMF (homotopy theory), and (3) the "strange duality" of Verlinde bundles (algebraic geometry). This is joint work in progress with Daniel Berwick-Evans and Akira Tominaga.