CAREER: Trace Formula and Geometric Analysis of Automorphic Forms

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Many aspects of this research project are intimately related to establishing instances of randomness in number theory. Number theory is among the oldest branches of mathematics; its applications to technology are prevalent and vital for communication systems, data processing, and computational algorithms. The goals of this project are driven by landmark problems on arithmetic families. Families arise when assembling and studying together objects that share common features. Families are often crucial even if one is a priori interested in a single object and thereby are central to the recent resolution of certain difficult algebraic and asymptotic questions. These goals of the project are complemented by concrete initiatives targeted at undergraduate and graduate education that are centered on developing effective writing and communication skills. In collaboration with the Institute for Writing at Cornell University, the PI will organize a monthly seminar on writing, regular writing groups, and an online wiki that will serve as a communication platform and access to resources for the general public. The PI will continue to mentor undergraduate research projects, disseminating knowledge and discoveries while promoting learning through the investigation of open problems.This research project aims to develop a quantitative theory of the asymptotics of special functions, such as characters of representations. The long-term goal is to solve problems on automorphic periods, subconvexity and non-vanishing of L-functions, and arithmetic statistics of families. The trace formula is a fundamental tool in number theory and the development of the Langlands program in particular. Even though there has been enormous progress, important questions remain open, notably analytic aspects that are critical for many applications. These questions are now ripe for investigation following the works of Arthur and others. An immediate outcome of this research is a Sato-Tate equi-distribution theorem for families of Maass forms on GL(n), resolving a long-standing problem. The understanding of the absolute convergence of the geometric side of the trace formula is currently one of the most urgent problems in the subject. A second focus is on trace characters, which are a central concern in representation theory, such as the local Langlands correspondence and functorial transfers. The PI will work on quantitative aspects that have seen little progress since the seminal work of Harish-Chandra. Related to this, the PI will continue work on Whittaker periods, notably towards a conjecture of Zuckerman on the asymptotic behavior at infinity. The proposed activity is to bring methods from analysis, geometry, representation theory, and mathematical physics in their full strength, notably symplectic geometry and integrable systems; an immediate goal is the systematic study of the quantitative aspects of coadjoint orbits.