This research project concerns work in number theory, a branch of mathematics that has seen applications in cryptography and in coding theory and that has connections with physics. There have been many fruitful interactions between number theory and other areas of mathematics, as exemplified by the Langlands program and the proof of Fermat's Last Theorem. The proof of Fermat's last theorem included verification of a small part of the Langlands program, a vast web of conjectures involving disparate areas of mathematics, which, in this case, connected Galois representations and automorphic forms and the L-functions of elliptic curves. The investigator will pursue research directions that involve strong global techniques to study automorphic forms and Galois representations. These may include methods such as the trace formula, Shimura varieties, p-adically completed cohomology, or the Taylor-Wiles-Kisin patching construction.
This research project will address the following topics. (1) The Langlands-Kottwitz approach to the cohomology of Shimura varieties of abelian type with good reduction in full generality; (2) Arithmetic statistics for families of automorphic representations and their L-functions; (3) The endoscopic classification for representations of local and global unitary groups that are not quasi-split; and (4) p-adic Langlands program beyond GL(2,Qp). The common theme of the investigator's research is to obtain strong consequences from these techniques via a clear understanding of interactions between local and global theories. The Langlands-Kottwitz method is a fundamental component of the Langlands program, and its completion for Shimura varieties of abelian type will not only be a milestone but also lead to further arithmetic applications. The study of families of L-functions makes a connection with random matrix theory and would shed light on families of algebraic varieties. The project on unitary groups settles an interesting case of Langlands functoriality as well as the local Langlands classification with several expected applications in arithmetic, which rely essentially on the use of (often non-quasi-split) unitary groups. The continued effort on the p-adic Langlands program aiming at general groups would have growing impact and open up new research directions.
