CAREER: Hitchin morphisms, relative Langlands duality, and automorphic L-functions

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This project sits naturally at the intersection of representation theory and geometry. Representation theory is a branch of mathematics devoted to the study of symmetries that occur throughout mathematics and science. For example, the study of symmetries in three-dimensional space or more generally the study of continuous symmetries of mathematical objects and structures, known as representation theory of Lie groups, or the study of symmetries of solutions of polynomial equations, known as Galois theory. Methods from geometry have been very successful in solving problems in representation theory. One of the main goals of the project is to study questions in representation theory using geometric methods. In this project the PI will develop and use geometric tools to attack several longstanding problems on: Higgs bundles and representations of the fundamental group, representations of Lie groups and symmetric varieties, and the Langlands program. The education component of the project will create a vertically integrated learning environment for representation theory, number theory, and algebraic geometry at the University of Minnesota, from which students and researchers at all levels will benefit. The plans include supports for outreach activities, development of courses and learning seminars, and summer programs. In more detail, the PI will conduct research on the following projects: (1) Hitchin morphisms for higher dimensional varieties, (2) Real groups, symmetric varieties, and the relative Langlands duality, and (3) Fourier transforms, automorphic L-functions, and the Braverman-Kazhdan program. In project (1), the PI will develop the theory of Hitchin morphisms for higher dimensional varieties. This project is closely related to deep questions in invariant theory, algebraic geometry, and representations of the fundamental group. In project (2), the PI will explore connections between the geometry of real reductive groups and the algebraic geometry of symmetric varieties and apply them to the study of geometric Satake equivalence and the geometric Langlands correspondence for real groups and the relative Langlands duality conjectures. In project (3), the PI will systematically study the Braverman-Kazhdan program on meromorphic continuation and functional equations of automorphic L-functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.