The investigator proposes to study arithmetic questions arising in the theory of automorphic forms in the context of the Langlands program. One such problem is the construction and classification of arithmetically interesting cusp forms on reductive groups, and relating them to certain influential conjectures of Arthur. The investigator intends to exploit the exceptional theta correspondence, as well as some recent generalizations of theta correspondences, for the construction of some of these cusp forms. He is also interested in exploiting p-adic methods for the construction of classical automorphic forms.
The theory of automorphic forms is one of the main themes of modern mathematical research. Besides being a beautiful subject in its own right, it is intimately connected with other important areas of mathematics such as number theory (the study of properties of numbers, with applications to cryptography) and representation theory (the study of symmetries, with applications to physics). Often, it has surprising applications to concrete problems, such as the construction of Ramanujan graphs which are of great interest in communication and network theory. The investigator's research is aimed at elucidating the connections of automorphic forms with number theory and representation theory.
