CAREER: Models for Galois deformations and Applications

数论是数学的一个分支，研究整数的算术模式。例如，一个基本问题是，一个特定的方程组有多少个整数解？这通常是一个非常困难的问题，涉及到各种不同数学领域的技术。伽罗瓦理论，介绍了在19世纪，研究群对称的多项式方程，是中央现代数论。在20世纪70年代，朗兰兹引入了一个意义深远的结构网络，将伽罗瓦对称性与分析中出现的其他对称结构（模形式）联系起来。在朗兰兹结构框架下建立算术世界（伽罗瓦理论）和复分析世界之间的桥梁一直是现代数论的一项艰巨任务和推动力。该项目旨在证明朗兰兹的新实例。该项目将开辟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，这是通过证明一个将算术对象（椭圆曲线）与分析对象（模形式）联系起来的二维猜想来完成的。在费马大定理的证明和许多其他重要结果中的一个关键思想是研究模形式之间的模同余。这个提议将系统地构造更高维度的同余，将它们与伽罗瓦理论中的新几何结构联系起来。这一思想循环导致了21世纪初塞尔模形式猜想的解决，这是朗兰兹纲领进展中的另一项重大成就。塞尔猜想的权部分，它分类模形式之间的同余，已被推广到许多方向作为增长领域的p进朗兰兹计划的一部分。该项目将解决这些概括进展中的主要差距，包括缺乏对广泛分歧表示的猜想和大于3维的潜在结晶变形环的分支问题。该项目还将通过研究中等权重的伽罗瓦表示的模空间的几何来证明新的模块化提升定理。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值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.