Geometric harmonic analysis is a new chapter in geometric representation theory, in which techniques from homotopical algebra and unifying structures from topological field theory are combined to describe the decomposition of categories in the presence of symmetries. The PI proposes a variety of projects (joint with David Nadler) that further the foundations of the subject and apply it to resolve classical problems in representation theory. The most far-reaching project aims to develop a geometric analog of the Arthur-Selberg trace formula, consistently enhancing vector spaces of functions to categories of sheaves (in particular providing a natural context for the work of Ngo). Other projects seek to prove general Langlands dualities (or "nonabelian Fourier transforms") for categories of representations of real groups and for cohomologies of character varieties (conjectures of Soergel and Hausel--Rodriguez-Villegas, respectively). The PI also proposes (with Jonathan Block and Nigel Higson) to bridge two disparate approaches to harmonic analysis of Lie groups --- noncommutative algebraic geometry (D-modules and Beilinson-Bernstein localization) and noncommutative topology (C*-algebras and the Baum-Connes conjecture). The PI is highly committed to mathematical exposition at a variety of levels, and endeavors to convey the intuitions behind sophisticated mathematical ideas to a broad range of audiences. The significant educational component of the proposal builds on the PI's experience and enthusiasm as an expositor.
The Langlands program is one of the fundamental organizing principles in modern mathematics, predicting that diverse phenomena can be understood by the systematic exploitation of hidden symmetries. Among its successes in number theory are the solution of Fermat's Last Theorem, while in physics it underlies the symmetry between electricity and magnetism and its theoretical generalizations to other fundamental forces. Recently Ngo proved the Langlands Fundamental Lemma, one of Time Magazine's Top Ten Scientific Discoveries of 2009. This proof suggests a new link between the apparently unrelated number theoretic and physical settings for the Langlands program. The current proposal is aimed at developing these connections, finding physical analogues of classical results in number theory while using the rich geometric intuition behind the physics to suggest new patterns in the application of symmetries. An important component is the dissemination of the exciting but often inaccessible developments in this field through development of online resources, lecture series, books and courses.
