# Symmetry, Duality and Cinema with Edward Frenkel

**Edward Frenkel, recipient of the Foundation's Research Chaire in 2008 has organized a conference on thursday june 17th at the IHP (Amphithéâtre Hermite)**

With lectures by Edward Frenkel, David Hernandez (Ecole Polytechnique), Vincent Lafforgue (IMJ) and Nikita Nekrasov (IHES) and a projection of Rites d'amour et de maths (click here to read more about this film).

Program

**-** Edward Frenkel, "Symmetry and Duality in Mathematics and Physics"

**- **Vincent Lafforgue (IMJ), "On the geometric Langlands conjecture, Arthur's SL(2), and the case of the projective line"

** -** David Hernandez (Paris-Diderot Paris 7), "Langlands duality for quantum affine algebras and simple tensor products"

**-**Nikita Nekrasov (IHES), "Quantum integrability, instantons,and Langlands duality"

**-**Screening of the film "Rites of Love and Math" by Reine Gravesand Edward Frenkel

Abstracts

Symmetry and Duality in Mathematics and Physics

by Edward Frenkel

Symmetry plays an important role in geometry, number theory, and quantum physics. I will discuss the links between these areas from the vantage point of the Langlands Program. In this context "duality" means that the same theory, or category, may be described in two radically different ways. This leads to many surprisingconsequences. Moreover, Langlands type dualities in Mathematics and Physics turn out to be closely related to each other.

On the geometric Langlands conjecture, Arthur's SL(2), and the case of the projective line

by Vincent Laforgue

We first recall the conjectural geometric Langlands correspondence of Beilinson and Drinfeld, and in particular the compatibility with Hecke functors and the Whittaker normalization condition. Then we recall the work of Ginzburg, Bezrukavnikov and Finkelberg on the derived geometric Satake equivalence and we give a candidate for the geometric Langlands correspondence in the case of the projective line. Then we give, for any curve, a conjecture about the O-module on the moduli space of local ystems which should correspond to the constant D-module on the moduli space of torsors on the curve.

Langlands duality for quantum affine algebras and simple tensor products

by David Hernandez

*In this talk, we will first discuss results obtained in collaboration with E. Frenkel : we establish a correspondence (or duality) between representations of quantum (affine) algebras associated to Langlands dual complex Lie algebras. To prove the duality at the level of quantum affine algebras, we construct "(q,t)-characters", depending on two parameters and interpolating between Frenkel-Reshetikhin q-characters attached to a quantum affine algebra and its Langlands dual. In the second part of this talk, we will explain how in this Langlands duality framework we can use the recent proof by the speaker of the following conjecture : a tensor product of N simple finite dimensional representations of a quantum affine algebra is simple if and only if the tensor products of the representations are simple two by two. *

Quantum integrability, instantons,and Langlands duality

by Nikita Nekrasov

*I will review the recent developments in the study of Bethe/Gauge correspondence with the emphasis on the relation between the four dimensional N=2 supersymmetric theories in the Omega-background and the quantization of Hitchin systems and their degenerate limits. Based on the joint work with Samson Shatashvili.*