This award involves two main research projects: a long-term joint project with Wilfried Schmid (Harvard University) on automorphic distributions, and collaborations with Ramarathnam Venkatesan (Microsoft Research) on applications of number theory and discrete groups to cryptography. The first project involves the study of automorphic distributions, and their analytic properties. The PI and Schmid have developed an alternative method for proving the holomorphy of Langlands L-functions, which establishes their holomorphy in new cases. It also has been used to develop Voronoi-style summation formulas, which were later applied to obtain the first subconvexity results for GL(3) L-functions. The project involves extending their core analytic technique of pairing automorphic distributions to more general situations. The second project seeks to develop new algorithms and cryptographic primitives using information from modern number theory, ranging from discrete groups to L-functions.
L-functions are a central topic in modern number theory, arising from problems as diverse as classical questions about solving polynomial equations with integer solutions, to studying the properties of waves on curved surfaces. The proposed research mainly involves establishing further analytic properties of L-functions, primarily with applications to Langlands' holomorphy conjectures in mind. Such information can be used, as in the second project, to give explicit bounds and parameter estimates for cryptosystems. Work of the PI and Venkatesan on this topic has already found cryptographic applications in numerous Microsoft products. Indeed, modern cryptosystems are often based on hard mathematical problems such as factoring large integers. The proposal seeks to study these problems from the point of view of SAT solvers and coding theory, and to create cryptographic algorithms for other applications.
The research undertaken through this award concerned automorphic forms, a branch of modern number theory. Automorphic forms are highly symmetric functions in which fascinating information about number theory is embedded. One of the major research themes in this area is the Langlands program, which (among other things) asks about the properties of certain complex functions (known as "L-functions") formed from coefficients obtained from the automorphic forms. The main result of this project was to prove new cases of Langlands' conjectures about his L-functions. The full Langlands conjectures will solve many fundamental questions about the distributions of quantities coming from number theory problems, once proved. Even partial results thus far have been crucial, and this area has been identified by many experts as a fundamental area of investigation. Another fascinating aspect of automorphic forms is that they come up in many unexpected subjects. The PI has worked with string theorists Michael Green (Cambridge University) and Pierre Vanhove (IHES) on computing correction terms to the 4-graviton expansion in the low-energy limit of string theory. By virtue of how type II string theory is defined, these are automorphic functions; the theory of automorphic forms was used to obtain new information about them. The PI used ideas learned from string theorists to apply automorphic forms to a different area of mathematics, representation theory, and prove cases of a conjecture of J. Arthur that certain representations of exceptional groups can be realized as unitary operators on a Hilbert space. This is part of a much larger unsolved problem called the "Unitary dual" problem, a central problem in representaiton theory.
