Galois Representations and Automorphic Forms

这个项目的主要目的是让PI继续培养研究生在令人兴奋的，但非常技术要求，研究算术几何。这项研究涉及一个非凡的网络结构，包括朗兰兹结构，连接代数（即多项式方程理论）与分析和几何（特别是几何对称性的研究）。只要建立了这些结构，它们就提供了一个强大的工具，可以将一个领域中非常困难的问题简化为另一个领域中更容易的问题。最著名的例子是安德鲁·怀尔斯经过300多年的努力证明了费马最后定理。怀尔斯，部分与PI，减少了这个代数问题，以一个更容易的问题，有关对称性的双曲平面（一种几何普及在布里尔的“圆极限”木刻）。算术几何领域在近几十年来取得了非凡的进展。在许多后果从这一领域的流动是新的纠错码是必不可少的现代计算机（硬盘）和光盘。除了支持研究生，PI将继续他的工作在自同构提升定理的定期，但非自对偶，情况。最近，包括PI在内的10位数学家在这个问题上取得了第一个重大进展：他们证明了CM域上所有椭圆曲线的潜在自同构和比安奇三维流形上同调的Ramanujan猜想。似乎有可能取得进一步进展。PI的一名学生也在研究相关问题，PI将继续思考这些问题。此外，PI将继续努力证明Hecke算子作用于代数但非正则自守形式的特征值是代数的。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.