2007 LMS Lectures on Geometric Langlands

Abstracts, Notes and Videos

Here are abstracts and links to the notes and videos of the 2007 LMS Lectures on Geometric Langlands. I plan to write up the lectures as a book in fall 2007. Many thanks to Charles Beesley, Head of Media Production, University of Oxford for the taping and to Mike DeLeon, UT Austin Division of Instructional Innovation and Assesment, for the transcription.

Notes for other lectures at the meeting: Constantin Teleman: Examples and Counterexamples in Geometric Langlands. Dmitri Rumynin: D-modules in positive characteristic. Luis Alvarez-Consul: Moduli of coherent sheaves and Kronecker modules.

In this lecture series we will explore the geometric Langlands program, a subject of much exciting recent activity at the interface of representation theory, algebraic geometry and quantum field theory. The lectures will assume a familiarity with first courses on Lie groups and algebraic varieties; graduate students are strongly encouraged to attend. In broad outline, the talks will cover the following topics:

I. Geometric Function Theory.

Notes and Video: Part 1 and Part 2.

Algebraic geometry affords a rich analog of classical questions in harmonic analysis, in which function spaces are replaced by derived categories of sheaves of various types and the Fourier transform is replaced by the Fourier-Mukai transform. We'll present this dictionary and its motivations from representation theory and noncommutative geometry, with an emphasis on the geometry of D-modules.

II. Moduli of Bundles.

Notes and Video: Part 1 and Part 2.

The setting for the geometric Langlands program is the geometry of moduli spaces of bundles on an algebraic curve. Beginning with the more familiar setting of Picard groups, we will explore the principal structures on these spaces, the loop group uniformization and Hitchin's integrable system. The latter provides a crucial "abelianization" of nonabelian moduli spaces and suggests a duality between geometric function spaces on moduli of bundles for dual groups - the geometric Langlands correspondence.

III. Hecke Operators.

Notes and Video: Part 1 and Part 2.

What acts on functions on a quotient space, or on spaces of invariant functions? A general answer is provided by Hecke algebras. We'll investigate the symmetries of the geometric Langlands function spaces through the study of the affine Grassmannian. Miraculously, the Hecke operators we encounter all commute, and the geometric Langlands program emerges naturally as an attempt to simultaneously diagonalize commuting Hecke operators. The geometric origin of the commutativity will also be profitably linked to the notion of a vertex algebra.

IV. Topological Field Theory.

Notes and Video: Part 1 and Part 2.

An exciting recent development is the emergence of a profound connection between the geometric Langlands program and topological quantum field theories, due to Kapustin and Witten. In their approach, the geometric Langlands duality is described as a two-dimensional manifestation of the electric-magnetic duality of supersymmetric four-dimensional gauge theories. Starting from the axiomatic definition of topological field theories we will see the basic structures of geometric Langlands emerge. The finer features come out of detailed examination of observables in gauge theories, mirror symmetry for hyperk"ahler manifolds and the categories of boundary conditions (D-branes).

V. Applications.

Notes and Video: Part 1 and Part 2.

In the final unit we will explain some applications of the geometric Langlands program to representation theory. We will begin with a crash course in geometric representation theory, focussing on the Beilinson-Bernstein localization and the appearance of braid group actions. Next we'll outline Bezrukavnikov's geometric theory of affine Hecke algebras (the "tamely ramified" geometric Langlands program), a new fundamental tool in representation theory with wide-ranging implications. Finally we'll see how geometric Langlands can be applied to provide a new approach to the representation theory of real Lie groups (joint work with David Nadler).