New Connections of Representation Theory to Algebraic Geometry and Physics

During the 2007-08 academic year, Roman Bezrukavnikov of MIT led the special program on algebraic geometry and physics in representation theory.

The focus of the year was on related recent developments in representation theory, algebraic geometry and physics.

The first conference, "Gauge Theory and Representation Theory", was held during the week of November 26-30, 2007.

The second conference, "Algebro-Geometric Derived Categories and Applications", was held during term II from March 10-14, 2008.

In the first term the emphasis was on mathematics connected to quantum field theory, in particular the new differential geometric approach to the geometric Langlands program. A part of the second term was devoted to absorbing the emerging new homotopy foundations of algebraic geometry, with a view towards applications.

One common feature of recent trends is 'categorification', often synonymous with 'geometrization'. Categorification stands for the passage from a traditional mathematical object to its higher categorical analogue, and, more loosely, for the emphasis on categories instead of particular objects. The categories involved are typically of geometric nature (categories of sheaves of certain kind) and are constructed in a homological framework, i.e., they are triangulated categories, or refinements of these. Examples in representation theory include geometric Langlands duality (a categorification of the theory of automorphic forms); character sheaves (a categorification of representation theory of finite Chevalley groups); localization techniques for modular representations; Nakajima's geometric construction of Kac-Moody Lie algebra representations etc. However, there are many examples in other fields which are relevant for representation theory: categories of D-branes in string theory; Fukaya categories (a categorical version of symplectic geometry); homological mirror symmetry and, more generally, focus on derived categories of coherent sheaves in algebraic geometry, which is a categorification of standard cohomology theories. The goal of the year is to explore these subjects and establish bridges to representation theory.

During the academic year, the School hosted two conferences related to the special program.