Automorphic Representations

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Automorphic representations are abstract objects from mathematics that carry concrete information. The information is of great significance. It is believed to hold the keys for a grand unification of many areas of mathematics, and possibly also fundamental areas of physics. However the information is well hidden. It is embedded at the centre of what has become known as the Langlands program.***The heart of the Langlands program is a very deep yet quite explicit conjecture, known as the principle of functoriality, which represents a fundamental organizing principle for the data in automorphic representations. There has been significant progress on this conjecture in recent years. It comes as a biproduct of a classification of many automorphic representations through what is known as the theory of endoscopy. However, the complementary cases of functoriality are deeper, and have always been regarded as intractible. They fall into a domain Langlands has called "beyond endoscopy", in suggesting a remarkable but rather speculative strategy for attacking them in 2000.***I am proposing to work on general cases of functoriality, and in particular, Langlands' ideas in "beyond endoscopy". Taken as a whole, this problem includes some of the most difficult questions in mathematics. Success is by no means answered. However, any progress that can be made would have an enormous impact on many broad areas of mathematics. ***I will also work on extending the actual theory of endoscopy. This is a very large problem in its own right, but it is more accessible. It contains many interesting questions, which are a rich source of problems for graduate students and young mathematicians. Further progress in the theory of endoscopy should eventually lead to a classification of automorphic representations for a very wide class of groups.**