The main purpose of this project is to allow the PI to continue to train graduate students in exciting, but very technically demanding, research in arithmetic geometry. This research concerns an extraordinary web of conjectures, including the Langlands conjectures, connecting algebra (i.e. the theory of polynomial equations) with analysis and geometry (particularly the study of geometric symmetries). Whenever these conjectures can be established, they provide a powerful tool which can reduce very hard problems in one domain to much easier problems in the other. The most celebrated example was Andrew Wiles' proof, after over 300 years of effort, of Fermat's last theorem. Wiles, partly in conjunction with the PI, reduced this algebraic problem to a much easier problem concerning symmetries of the hyperbolic plane (a sort of geometry popularized in Escher's "Circle Limit" woodcuts). The field of arithmetic geometry has seen extraordinary progress in recent decades. Among the many consequences flowing from this area are new error correcting codes which are essential for both modern computers (hard disks) and compact disks.
In addition to supporting graduate students, the PI will continue his work on automorphy lifting theorems in the regular, but non-self-dual, case. Recently a group of 10 mathematicians, including the PI, made the first serious progress on this problem: They proved the potential automorphy of all elliptic curves over CM fields and the Ramanujan conjecture for the cohomology of Bianchi 3-manifolds. Further progress seems possible. One of the PI's students is also working on related issues and the PI will continue himself to think about these questions. In addition the PI will continue his attempts to prove that the eigenvalues of Hecke operators acting on algebraic, but non-regular, automorphic forms are algebraic.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
