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.
ANALYSIS/MATHEMATICAL PHYSICS SEMINAR
Observations of ongoing climate change, paleoclimate data, and climate simulations concur: human-made greenhouse gases have set Earth on a path to climate change with potentially dangerous consequences for humanity. James Hansen, climatologist and Adjunct Professor in the Department of Earth and Environmental Sciences at Columbia University, explains the urgency of the situation and discusses why he believes it is a moral issue that pits the rich and powerful against the young and unborn, against the defenseless, and against nature. He explores available options to avoid morally unacceptable consequences.
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.