2007 Special Year on Mathematical Physics

Topical Program on the Geometric Langlands Conjecture

Lectures: Titles and Abstracts

Speaker: Dennis Gaitsgory (Harvard University)
Title: Local geometric Langlands correspondence

Speaker: Ian Grojnowski (Cambridge University)
Title: Geometric Satake correspondence and all that ...

Content: Geometric Satake, defining spaces (affine Grassmanians, loop groups, etc etc), proof of the theorem in its various forms, Hecke operators ("chiral category"), and the vague form of Geometric Langlands (O-mods on deRham stack of Bun_G vs O-mods on Bun_deRham).

Speaker: Tony Pantev (University of Pennsylvania)
Title: Geometric Langlands conjecture - classical limit and quantization

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Speaker: Siye Wu (University of Hong Kong)
Title: Gauge theory and Langlands duality

Abstract: These lectures are an introduction to the recent work of Gukov, Kapustin and Witten on the relation of S-duality in physics and the geometric Langlands programme. It covers various aspects of quantum field theory such as magnetic monopoles, quark confinement, Wilson and 't Hooft operators in gauge theory and mirror symmetry, branes in sigma models. It explains how the electric-magnetic duality (S-duality) in four dimensions reduces in two dimensions to the geometric Langlands programme. Topics to be covered are listed below:

gauge theory and electric-magnetic duality
supersymmetry and topological field theory
topological sigma model and mirror symmetry
branes, loop operators, and Hecke operators
relation to geometric Langlands programme

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Talks: Titles and Abstracts

Speaker: Jarah Evslin (Université Libre de Bruxelles)
Title: Twisted Homology and Calibrations

Abstract: We present two closely related twisted homology theories, one of which is an integral theory that approximates twisted K-theory, and the other of which is rational and classifies calibrated cycles in twisted complex manifolds. We find a spectral sequence that calculates the former. We demonstrate that both homology groups are conserved charges of D-branes in type II string theories, and that in this context the twisted boundary operator calculates an anomaly.

Speaker: Jonathan Rosenberg (University of Maryland)
Title: An analogue of the Novikov conjecture in complex algebraic geometry

Abstract: We introduce an analogue of the Novikov Conjecture on higher signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain "higher Todd genera" are birational invariants. This implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). Unlike the usual Novikov Conjecture, this statement is actually a theorem, not just a conjecture, thanks to work of several other authors. We also show that, in a certain sense, our statement is best possible.

Speaker: Hisham Sati (Yale University)
Title: Loop groups and elliptic curves in string theory

Abstract: We survey the appearance of loop groups and elliptic curves in ten-dimensional string theories and then highlight their relation to elliptic cohomology, sigma models, and S-duality.

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