CAREER: Galois Representations: Deformation Theory and Motivic Origins

几个世纪以来，对称性在我们的数学和物理世界的研究中一直扮演着中心组织角色。也许一开始只是出于简单的好奇心——例如，2500年前构造和分类正多面体的冲动——已经一次又一次地被证明是研究一些最基本的数学和物理对象的最有效的方法，无论是通过更具体的几何物体的对称性，还是通过更抽象的（但对我们的数字世界至关重要的）量子力学的对称性。也许对称在科学中最令人惊讶的作用是在素数的研究中，素数是算术的基本对象，在这里没有像几何那样明显的“对称”。然而，我们对素数最深刻的认识大多来自于它们与多项式方程的“对称性”的联系，这一主题被称为伽罗瓦理论。这个研究项目将研究伽罗瓦表示，它是关于素数的代数编码信息的自然包。现代数论的中心项目之一，提出了通过将伽罗瓦表示与几何中出现的显著不同的数学对象联系起来，来“解开”伽罗瓦表示的新方法。PI将继续沿着这个方向进行研究，以描述这些神秘但绝对基本的关系。作为CAREER项目教育部分的一部分，PI将承担一系列服务于各种受众的教育项目。他将继续为犹他州的高中生开设一个密集的夏季数论课程，向他们介绍数学作为实验和发现的对象，从而鼓励他们养成创造性智力工作所必需的思维习惯。该项目包括研究生和当地高中教师作为合作教师，他们可以将其独特的教学模式带到其他教育机构。为了激发更广泛的数学公众，PI还将与犹他大学的数学历史课程教学一起，开发数学历史的课程材料，特别是视频，并在网上传播。最后，他将继续从事培养博士研究生的工作。更详细地说，朗兰兹纲领是指导当代数论工作的一系列猜想，特别是为伽罗瓦理论和素数有关的问题提供了最深刻的猜想答案。在此过程中，他们从数论延伸到代数几何、表示理论等领域。PI在朗兰兹计划的广泛范围内完成了两个主要项目。第一个涉及伽罗瓦表示的变形理论，这是建立费马大定理证明的两大支柱之一，也是代数数论的中心研究领域之一。这里PI将研究伽罗瓦表示在一般约化群中的变形理论；从广义上讲，这项工作旨在推广Serre著名的模块化猜想。第二个主要项目涉及伽罗瓦表示和动机之间的关系，后者在某种意义上是代数变量类别的最佳线性近似，其本身具有根本的兴趣，但在推测上也是伽罗瓦表示的代数几何对应。在这里，PI将研究与建立伽罗瓦表示的动机起源有关的各种问题。这些包括与PI的广义Kuga-Satake理论相关的Fontaine-Mazur猜想的实例；模空间的可abel性质研究以及几何表征理论中基本对象的动机结构。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。

For centuries, symmetry has played a central organizing role in the study of our mathematical and physical world. What may have begun as simple curiosity--for instance, the impulse to construct and classify the regular polyhedra nearly two and a half thousand years ago--has turned out time and again to be the most effective way to study some of the most fundamental objects of mathematics and physics, whether through the more concrete symmetries of geometrical objects or the more abstract (but essential to our digital world) symmetries of quantum mechanics. Perhaps the most surprising role of symmetry in the sciences is in the study of the prime numbers, the fundamental objects of arithmetic, where there are no manifest "symmetries" such as one encounters in geometry. Nevertheless, much of our deepest knowledge of prime numbers comes from their connections with the "symmetries" of polynomial equations, a subject known as Galois theory. This research project will study Galois representations, which are natural packages for algebraically encoding information about prime numbers. One of the central programs of modern number theory, proposes new ways to "unpack" Galois representations by relating them to remarkably different mathematical objects arising in geometry. The PI will continue his research in this direction to describe these mysterious but absolutely fundamental relationships. As part of the educational component of this CAREER project, the PI will undertake a series of educational projects serving a variety of audiences. He will continue to run an intensive summer number theory program for Utah high school students, introducing them to mathematics as an object of experimentation and discovery, and thereby encouraging them to develop the habits of mind essential to creative intellectual work. This program involves both graduate students and local high school teachers as co-teachers, who can then carry its distinctive pedagogical model with them to other educational settings. With a view toward exciting a broader mathematical public, the PI will also, in conjunction with teaching a history of mathematics course at The University of Utah, develop curricular materials, particularly videos, to be disseminated online, in the history of mathematics. Finally, he will continue his work training PhD students. In more detail, the Langlands program is a series of conjectures that guide much contemporary work in number theory, and in particular provide the deepest conjectural answers to problems relating Galois theory and prime numbers. In doing so, they reach out from number theory to algebraic geometry, representation theory, and beyond. The PI complete two main projects within the very broad purview of the Langlands program. The first concerns the deformation theory of Galois representations, one of the two pillars on which the proof of Fermat's Last Theorem was built, and ever since one of the central research areas within algebraic number theory. Here the PI will study the deformation theory of Galois representations valued in general reductive groups; broadly, this work aims at generalizations of Serre's famous modularity conjecture. The second main project concerns the relationship between Galois representations and motives, the latter being in some sense the best linear approximation to the category of algebraic varieties, of fundamental interest in its own right, but also conjecturally the algebro-geometric counterpart of Galois representations. Here the PI will study a variety of problems concerned with establishing the motivic origin of Galois representations. These include instances of the Fontaine-Mazur conjecture related to the PI's generalized Kuga-Satake theory; study of anabelian properties of moduli spaces; and motivic constructions underlying fundamental objects of geometric representation theory.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.