The theory of Galois representations is a flourishing area of activity in the landscape of present-day arithmetic. Since Wiles' proof of Fermat's Last Theorem, the theory of automorphic forms has played an especially important role, via Langlands program and the connections it entails. There has also been a convergence of significant themes arising from non-abelian class field theory. The symposium will present an overview of current developments, focussing on some of the significant advances since the LMS Durham Symposium of 2004 such as:

• The proof of Serre's conjecture and most cases of the Fontaine-Mazur conjecture for GL(2);
• Higher-rank generalisations of modularity theorems, with applications such as the proof of the Sato-Tate conjecture;
• The development of the p-adic Langlands program and its applications to the classical Langlands program;
• The proof of the fundamental lemma on orbital integrals and its applications to Langlands functoriality and the cohomology of Shimura varieties.

There will be five one-hour lectures per day, some of which will constitute short courses of 3-5 lectures. A tentative list of these is:

1. The p-adic Langlands correspondence
2. Vector bundles and p-adic Hodge theory
3. The fundamental lemma and trace formulas
4. Automorphy of Galois representations
5. Fundamental groups

In addition to the main themes, there will be lectures on related topics of critical interest, such as p-adic Hodge theory, special values of L-functions, and non-abelian Iwasawa theory.

The aim of the lectures will be to provide to young researchers a bird's-eye view of the ideas and techniques, and to the experienced researchers the opportunity to learn of exciting new developments in neighbouring areas and to build bridges for collaboration.

Organising Committee:
Fred Diamond (King's College London), Payman Kassaei (King's College London), Minhyong Kim (University College London)