Galois Representations and Automorphic Forms

这个项目的主要目的是让PI继续在令人兴奋但技术要求很高的算术几何研究中培养研究生。这项研究涉及一个非凡的猜想网，包括朗兰兹猜想，将代数（即多项式方程理论）与分析和几何（特别是几何对称的研究）联系起来。只要这些猜想能够建立起来，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容易得多的问题。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。怀尔斯部分地与PI合作，将这个代数问题简化为一个更简单的关于双曲平面对称性的问题（一种在埃舍尔的“圆极限”木刻中普及的几何）。近几十年来，算术几何领域取得了非凡的进步。在这一领域产生的许多结果中，包括对现代计算机（硬盘）和光盘都必不可少的新的纠错码。除了支持研究生之外，PI还将继续研究正则但非自对偶情况下的自同构提升定理。最近，一个由10名数学家组成的小组，包括PI，在这个问题上取得了第一个重大进展：他们证明了CM场上所有椭圆曲线的潜在自同态，以及Bianchi 3-流形上同调的Ramanujan猜想。进一步的进展似乎是可能的。PI的一名学生也在研究相关问题，PI将继续自己思考这些问题。此外，PI将继续尝试证明作用于代数但非正则自同构形式的Hecke算子的特征值是代数的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.