Geometry, Representation theory, and Langlands duality

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This mathematics research project is in the general area of representation theory and geometry. It involves the study of basic symmetries that occur in nature, namely the continuous symmetries known as Lie groups, introduced by Sophus Lie in the 1880's. One of the main goals of the project is to understand the basic building blocks of the theory, the irreducible unitary representations of Lie groups, using geometric methods. In addition the PI will make use of geometric methods to attack several longstanding problems concerning systems of differential equations, modular representation theory, and dualities for Lie groups. In more detail, the PI and Wilfried Schmid have made far-reaching conjectures which put the problem of finding the irreducible unitary representations in a general mathematical context. The conjectures themselves go beyond their application to representation theory and involve the theory of mixed Hodge modules of Morihiko Saito. The Langlands program provides a 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. In this spirit the PI has initiated a collaboration with Geordie Williamson, whose goal is to understand modular representation theory. In particular, they want to settle the longstanding problem of understanding the irreducible characters. Also in this direction, the PI proposes, in joint work with Roman Bezrukavnikov, to prove a categorical Langlands duality for real groups. Many structures in mathematics can be modeled by systems of differential equations. Of particular interest are the maximally over-determined systems. 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 working towards a comprehensive understanding of holonomic regular microdifferential systems (these are the systems that most often come up in applications to other areas). To understand these issues better, the PI also plans to develop a relative version of the classical theory of several complex variables.