Geometry, Representation theory, and the Langlands program

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This project makes use of and develops geometric methods to attack several longstanding problems on: systems of differential equations, representation theory of Lie groups, and the Langlands program. It has been understood for a long time that (special) functions, in generalized sense, can be studied in terms of the systems of differential equations that they satisfy. To this end, a general theory of systems of linear differential equations was developed by the Sato school in Kyoto. This point of view, in its various incarnations, is now ubiquitous in mathematics. The PI, jointly with Masaki Kashiwara, has solved the key longstanding problem in this area, the codimension three conjecture. Kashiwara and the PI will continue their collaboration towards a comprehensive understanding of holonomic regular microdifferential systems. These are the systems that most often come up in applications to other areas. The Langlands program provides means of relating areas of mathematics that often do not have a straightforward direct relationship. It implements this relationship via the symmetries of the theories by exhibiting a relationship between their representations. A major area of this proposal is the theory of real groups. Real groups are the fundamental symmetries that occur both in number theory and physics. A key outstanding question in representation theory of real groups is the determination of the particularly important class of unitary representations . Wilfried Schmid and the PI have made far-reaching conjectures which put this question in a general mathematical context phrasing the problem in terms of Hodge theory. They are on their way to settling these conjectures and the structures of the unitary dual. The PI also proposes, in joint work with Roman Bezrukavnikov, to prove a categorical Langlands duality for real groups. Differential equations are used to model various phenomena in nature. One major aspect of this project aims at a comprehensive understanding of an important class of systems of differential equations. Lie groups are the fundamental symmetries that occur in nature. They are important both in physics and number theory. A key question is to understand where these symmetries occur and it what form they occur. The second major aspect of this project will answer this question in terms of basic geometric structures called Hodge structures.