The project is in the realm of Number Theory. More specifically, it is in the area of automorphic forms and representation theory. The most ambitious enterprise in automorphic forms is the Langlands program, and, especially, the issue of functorial lifting. Loosely speaking, it postulates the formation of automorphic forms on a bigger group from those on a smaller group. This bounds together some of the most outstanding open problems in Number Theory, such as the Artin's Conjecture and the Ramanujan Hypothesis. So far, the advance in the Langlands program was accomplished along three major themes which are inter-related: explicit constructions of automorphic forms (e.g. using theta-kernels or Fourier coefficients of residues of Eisenstein series), the theory of L-functions combined with converse theorems, and the trace formula in its various guises. The focus in this project is on the trace formula approach. All trace formulas have their origin in expressing the kernel function in two different ways -- geometrically and spectrally. The application to functoriality comes about when trace formulas of two different groups are compared. The project deals with analytical aspects of the spectral expansion of a trace formula for {em symmetric spaces} (also called a relative trace formula), inaugurated by Jacquet. This is a major step in establishing functoriality in such instances as quadratic base change, and characterizing cusp forms having a non-zero period integral over certain period subgroups. The project has 3 parts and it is a collaboration with Jonathan Rogawski from UCLA (first 2 parts), and Steve Rallis and Herve Jacquet (third part).
The project is a basic research in the realm of Number Theory. The more specific area is called automorphic forms and representation theory. The most ambitious enterprise in this field is the Langlands program. In a nutshell, the goal is to establish relationships between various objects which "live" on completely different worlds, and seemingly have little in common. Such relations are extremely deep. A spectacular and relatively recent example is a correspondence between modular forms and elliptic curves, which was a keystone in Wiles' proof of Fermat's Last Theorem. Roughly speaking, an elliptic curve is a "doughnut" which is described as the locus of two equations of degree 3 in 4 variables. A modular form on the other hand can be thought of, in the simplest cases, as a sequence which is obtained by assigning to each integer the sum of all its divisors (or powers of them). Beside their inherent significance in many branches of mathematics, modular forms and their related objects have found applications in physics, cryptography and communication systems.
