Spring School: Geometric Langlands and Derived Algebraic Geometry
March 30 – April 3, 2015
The first purpose of our school is to introduce young researchers to the geometric Langlands program in its most recent categorical version. The latter was formulated in the paper  by Arinkin, Gaitsgory, « Singular support of coherent sheaves and the geometric Langlands conjecture ». It provides a deep interrelation between derived algebraic geometry, geometric representation theory, theory of automorphic forms, and representation theory of p-adic groups. To this end, there will be 3 morning mini-courses by B. Toen, D. Arinkin, D. Ben-Zvi oriented to young researchers. In the afternoon there will be research talks on the recent advances in the geometric Langlands program by leading specialits in the field.

Geometric Langlands is a very active area of research in mathematics and physics. Important breakthrough developments have occurred recently in this field, in particular, B-C. Ngo’s proof of the fundamental lemma (which was one of the central problems in the classical Langlands program) via geometric methods. Another major development was a deep interconnection established between the geometric Langlands program and four-dimensional gauge theory due to Witten, Kapustin and others. (Conformal field theories, the AdS/CFT correspondence, and integrability are some of the most active areas in modern theoretical physics). There are also continously developing interactions with other areas of representation theory, number theory and algebraic geometry, such as affine and double affine Hecke algebras and their relations to integrable models, geometry of various homogeneous spaces, Knizhnik-Zamolodchikov equations, Verlinde algebras, Gromov-Witten invariants of flag varieties, categorification and canonical bases in representation theory, quantization of certain algebraic varieties, vertex algebras, etc. We think that this unifying structure of the geometric Langlands program would be a motivation to learn it for young researchers.
Scientific & Organizing committee

Sergey Lysenko (Université de Lorraine)
Ivan Mirkovic (University of Massachusetts)
Simon Riche (Université Blaise Pascal Clermont-Ferrand)

Three mini-courses

Dima Arinkin (University of Wisconsin-Madison)
David Ben-Zvi (University of Texas, Austin)
Bertrand Toen (Université de Montpellier)


Geometric Langlands Correspondence and Topological Field Theory (pdf)

The Extended Whittaker Category (pdf)

Towards a Cluster Structure on Trigonometric Zastava (pdf)

The Category of Singularities as a Crystal and Global Springer Fibers

Betti Langlands in Genus One

Spectral Decomposition of the Principal Series Category (pdf)
QCoh and IndCoh in Derived Algebraic Geometry (preparatory talk for Gaitsgory’s mini-course) (pdf)

Higher Differential Operators and Applications (pdf)

Infinitesimal Aspects of Derived Algebraic Geometry (pdf)

A Simple Case of Ramified Geometric Langlands (pdf)