Classical and quantum geometric Langlands via quantization in positive characteristic

Classical and quantum geometric Langlands via quantization in positive characteristic

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Roman Travkin , IAS
Fine Hall 314

Geometric Langlands is an algebro-geometric and categorified analog of the Langlands program, introduced by A.Beilinson and V.Drinfeld. It can be understood as a certain non-commutative Fourier-Mukai transform between two moduli spaces associated to an algebraic curve and a Langlands dual pair of algebraic groups. We will describe the approach to geometric Langlands via quantization in characteristic p, implemented by R.Bezrukavnikov and A.Braverman for the GL_N case (and generalized by T.-H. Chen and X.Zhu togeneral groups). In this context, due to a large center of algebras of differential operators in characteristic p, an (appropriately localized) version of the geometric Langlands equivalence can be established via an essentially commutative Fourier-Mukai transform for torsors over abelian schemes. Time permitting, we will finish by a discussion of an application of characteristic p methods to a further deformation of this equivalence called _quantum geometric Langlands_, as developed in my thesis.