CAREER: Models for Galois deformations and Applications

数论是研究整数算术模式的数学分支。例如，一个基本的问题是，一个特定的方程系统有多少个整数解？这通常是一个非常困难的问题，涉及到数学中各种不同领域的技巧。伽罗瓦理论于19世纪引入，研究多项式方程组的对称性，是现代数论的核心。在20世纪70年代，朗兰兹引入了一个影响深远的猜想网络，它将伽罗瓦对称与分析中出现的其他对称结构(模形式)联系起来。在算术世界(伽罗瓦理论)和遵循朗兰兹猜想框架的复杂分析世界之间架起桥梁一直是现代数论的一项艰巨的任务和推动力。这个项目旨在证明朗兰兹猜想的新实例。这个项目将开放和参与一些新的问题和具体的问题，这将为研究生和博士后提供研究机会。该项目还将为本科生和研究生提供指导和在线培训机会。更详细地说，朗兰兹项目的一个重大突破是费马大定理的证明，该证明是通过证明一个连接算术对象(椭圆曲线)和分析对象(模形式)的二维猜想来实现的。在费马大定理的证明和许多其他重要结果中，一个关键思想是研究模形式之间模素数p的同余。这一提议将通过将高维同余连接到伽罗瓦理论中的新几何结构来系统地构建高维同余。这一思想圈导致了Serre关于模形式的猜想在21世纪初被解决，这是朗兰兹计划进展中的另一个主要成就。Serre猜想的权重部分，它对模形式之间的同余进行分类，已经在许多方向上被推广，作为p-adic朗兰兹计划不断增长的领域的一部分。这个项目将解决在这些概括方面的主要进展差距，包括缺乏对广泛分支表示的猜想，以及维度大于3的潜在晶体变形环的分支问题。该项目还将通过研究中等权重的Galois表示的模空间的几何来证明新的模提升定理。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Number theory is the branch of mathematics that studies patterns in the arithmetic of whole numbers. A fundamental question, for example, is how many whole number solutions does a particular system of equations have? This is often a very difficult question and engages with techniques from a wide variety of different areas of mathematics. Galois theory, introduced in the 19th century, studies groups of symmetries of polynomial equations and is central to modern number theory. In the 1970s, Langlands introduced a far-reaching web of conjectures which connects Galois symmetries with other symmetric structures appearing in analysis (modular forms). Building bridges between the arithmetic world (Galois theory) and the complex analytic world following Langlands conjectural framework has been a formidable task and a driving force in modern number theory. This project aims to prove new instances of Langlands’ conjectures. This project will open up and engage a number of new questions and concrete problems that will provide research opportunities for graduate students and postdocs. The project will also provide mentoring and online training opportunities for undergraduate and graduate students.In more detail, a major breakthrough in the Langlands program was the proof of Fermat’s Last Theorem which was accomplished by proving a two-dimensional conjecture connecting arithmetic objects (elliptic curves) to analytic objects (modular forms). A key idea in the proof of Fermat’s Last Theorem and in many other important results is the study of congruences modulo a prime p between modular forms. This proposal will systematically construct congruences in higher dimensions by connecting them to new geometric structures in Galois theory. This circle of ideas led to the resolution of Serre’s conjecture for modular forms in the early 2000s, another major achievement in the progression of the Langlands program. The weight part of Serre’s conjecture, which classifies congruences between modular forms, has been generalized in a number of directions as part of the growing field of the p-adic Langlands program. This project will resolve major gaps in progress on these generalizations including the lack of a conjecture for wildly ramified representations and the branching problem for potentially crystalline deformation rings in dimension greater than three. The project will also prove new modularity lifting theorems by studying the geometry of moduli spaces of Galois representations in intermediate weight.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.