School of Mathematics

Modularity of Galois Representations

Chandrashekhar Khare
University of California, Los Angeles
November 22, 2010

In this expository talk, I will outline a plausible story of how the study of congruences between modular forms of Serre and Swinnerton-Dyer, which was inspired by Ramanujan's celebrated congruences for his tau-function, led to the formulation of Serre's modularity conjecture. I will give some hints of the ideas used in its proof given in joint work with J-P. Wintenberger. I will end by pointing out just one of the many interesting obstructions to generalising the strategy of the proof to get modularity results in more general situations.

Potential Automorphy for Compatible Systems of l-Adic Galois Representations

David Geraghty
Princeton University; Member, School of Mathematics
November 18, 2010

I will describe a joint work with Barnet-Lamb, Gee and Taylor where we establish a potential automorphy result for compatible systems of Galois representations over totally real and CM fields. This is deduced from a potential automorphy result for single l-adic Galois representations satisfying a `diagonalizability' condition at the places dividing l.

Planar Convexity, Infinite Perfect Graphs and Lipschitz Continuity

Menachem Kojman
Ben Gurion University of the Negev; Member, School of Mathematics
November 16, 2010

Infinite continuous graphs emerge naturally in the geometric analysis of closed planar sets which cannot be presented as countable union of convex sets. The classification of such graphs leads in turn to properties of large classes of real functions - e.g. the class of Lipschitz continuous functions - and to meta-mathematical properties of sub-ideals of the meager ideal (the sigma-ideal generated by nowhere dense sets over a Polish space) which reduce to finite Ramsey-type relations between random graphs and perfect graphs.