J. Kamnitzer will give a mini-course on representation theory, geometric Langlands duality and categorification.
Abstract:
The representation theory of reductive groups, such as the group GLn of invertible complex matrices, is an important topic, with applications to number theory, algebraic geometry, mathematical physics, and quantum topology. One way to study this representation theory is through the geometric Satake correspondence (also known as geometric Langlands duality). This correspondence relates the geometry of spaces called affine Grassmannians with the representation theory of reductive groups. This correspondence was originally developed from the viewpoint of the geometric Langlands program, but it has many other interesting applications. For example, this correspondence can be used to construct knot homology theories in the framework of categorification.
In my lecture, I will begin by explaining the representation theory of reductive groups, using GLn as a concrete example. I will then explain the geometric Satake correspondence. I will conclude by explaining how this can be used for the purposes of categorification.
Required background:
Basic graduate algebra and topology. Some knowledge of compact Lie groups or semisimple
Lie algebras would be helpful, but not required.
Literature:
W. Fulton and J. Harris, Representation theory: a first course.
J. Kamnitzer, Lectures on geometric constructions of the irreducible representations of GLn, arXiv: 0912.0569.
V. Ginzburg, Perverse sheaves on a loop group and Langlands duality, arXiv:alg-geom/9511007.
S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves II, sl(m) case, arXiv: 0710.3216.
Dates:
October 14, 16, 21 and 23 — 15:15 – 17:00
