Higgs Bundles, Real Quasi-Maps, and Automorphic L-Functions

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This mathematics research project naturally sits at the intersection of representation theory and geometry. Representation theory is a branch of mathematics studying symmetries of mathematical objects and structures. Methods from geometry have been very successful in answering questions in representation theory. The main goal of the project is to study open questions in representation theory using geometric methods. Specifically, the investigator plans to use and develop geometric tools to attack several longstanding problems in the study of representations of Lie groups, Higgs bundles and representations of fundamental groups, and the Langlands program. In more detail, the research centers on three projects: (1) Hitchin morphisms for higher dimensional varieties, (2) real quasi-maps and applications, and (3) nonlinear Fourier transforms and the Braverman-Kazhdan program. In project (1), the investigator will develop the theory of Hitchin morphisms for higher dimensional varieties and apply it to the study of the Simpson correspondence. This project is closely related to deep questions in invariant theory, higher-dimensional algebraic geometry, and representations of fundamental groups. In project (2), the investigator will explore the geometric structure of real quasi-maps and establish applications to the study of representation theory of real groups, the Kostant-Sekiguchi correspondence, and Springer theory for symmetric spaces. In project (3), the investigator will study the Braverman-Kazhdan approach to meromorphic continuation and functional equations of automorphic L-functions. Based on results on nonlinear Fourier transforms and the Braverman-Kazhdan conjecture in the finite field and D-modules setting, the project aims to construct nonlinear Fourier kernels in the local fields setting.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.