## University of Sydney Algebra Seminar

# John Rice (University of Sydney)

## Friday 4th September, 12.05-12.55pm, Carslaw 375

### An introduction to the Geometric Langlands Conjectures and their implications for representations of Kac-Moody algebras

Every one dimensional local system on a compact Riemann surface is the pull back, under the Abel-Jacobi embedding, of a unique such local system on its Jacobian variety. The latter can be extended in a natural way to the whole Picard variety. I will explain this in detail, and in what way it constitutes an analogue of Artin's Reciprocity Law in the case of curves over finite fields and of algebraic number fields. The global Geometric Langlands conjectures are a certain generalisation to bundles of higher rank. I will explain its general features, and illustrate its connection with finite field and number field analogues from whence the conjecture gets its name.

The Langlands conjectures come in local and global forms, as with class field theory, and enjoy a similar complex relationship. The local Geometric Langlands conjectures posit a parametrisation of the 'representations' of complex loop groups by local systems on the formal punctured disc. In a second talk I will explain the ideas of Frenkel and Gaitsgory about what 'representation' should mean in this context, and the implications which flow from this for representations of Kac-Moody algebras at the critical level.