Moduli spaces of Galois representations

伽罗瓦表示的模空间

• 批准号: 2302619
• 负责人: Bao Le Hung
• 金额: $ 30.3万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302619&HistoricalAwards=false
• 关键词: Moduli; spaces; Galois; representations

Number Theory is the branch of mathematics which studies the properties and patterns of whole numbers. Despite this seemingly elementary premise, number theory has been at the the forefront of some of the most intricate structures discovered in mathematics, as well as underlying key practical applications (such as public key cryptography, which powers current secure communications over the internet). One fundamental idea of modern number theory is that collections of numbers sharing some common feature (such as being solutions of some list of equations) possess interesting emergent properties and symmetries. The most primordial of such emergent symmetry is the absolute Galois group of the rational numbers, and a large swath of number theory in the last few centuries concerns probing its (complicated) internal structure. In the 1970s, Langlands made a web of surprising predictions that this absolute Galois group is related to the (continuous) symmetry of vibrations on some highly symmetric geometric shapes (the automorphic representations). Such conjectures are known to have far reaching consequences: for instance, a proven special case was at the heart of the resolution of Fermat's Last Theorem. One promising approach to Langlands Conjectures that crystallized over the last few decades is the method of p-adic deformation, where one organizes the information on the two sides of the conjecture according to divisibility by powers of a given prime number p. The key point whose importance has only come into focus very recently is that this process reveals macroscopic/geometric features which make it easier to match the two sides, and the project aims to study exactly those features. Belonging to an emerging research direction, the project is a fertile ground for the discovery of and experimentation with new concrete phenomena, and thus create excellent opportunities for the training of students at both the graduate and undergraduate level. The PI also plans to disseminate the new geometric perspectives in the Langlands program to a broader audience through organizing summer schools and mini-courses.More specifically, the project studies the geometry of the moduli stack of representations of the Galois groups of p-adic fields, with focus on loci cut out by p-adic Hodge-theoretic conditions. These recently constructed spaces are expected to play a pivotal role in the nascent categorical p-adic Langlands program, which seeks to promote the (conjectural) relationship between individual smooth representations of p-adic Lie groups and individual local p-adic Galois representations to a relationship between the entire categories of such objects. The project aims to establish a bridge between these two categories, by relating both to categories of sheaves on some intermediate objects, moduli spaces of (semi-)linear algebraic objects, which are susceptible to analysis via methods of geometric representation theory. A sufficiently strong control on the geometry would lead to major progress on local questions such as the Breuil-Mezard conjecture as well as global questions such as Serre weight conjectures, automorphy lifting and the structure of mod p cohomology of locally symmetric spaces. The flow of information can also be reversed, namely one can predict new phenomena in geometric representation theory from arguments and heuristics with Galois representations. Furthermore, these linear algebraic moduli spaces are sufficiently concrete that one can experiment on them with computer algebra software, leading to many theoretical and computational projects accessible to undergraduate students.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.

数字理论是数学的分支，研究了整数的特性和模式。尽管存在这个看似基本的前提，但数字理论还是在数学中发现的一些最复杂的结构以及基本的关键实际应用（例如公共密钥密码学）的最前沿，这为当前的安全通信提供了通过Internet的动力）。现代数字理论的一个基本思想是，数字的集合共享了一些共同的特征（例如某些方程列表的解决方案）具有有趣的新兴属性和对称性。这种紧急对称性的最原始的是理性数字的绝对galois群，而在过去的几个世纪中，大量的数字理论涉及探测其（复杂）内部结构。在1970年代，兰兰兹（Langlands）做出了令人惊讶的预测，即这个绝对的galois群与某些高度对称的几何形状（自动形式表示）上的振动（连续）对称性有关。众所周知，这种猜想的影响很大：例如，一个事实证明的特殊情况是解决费马特最后一个定理的核心。在过去的几十年中，兰兰兹猜想的一种有希望的方法是P-ADIC变形的方法，在该方法中，人们会根据给定质量数字p的权力根据猜想的两侧组织信息。仅最近才成为重点的关键点是，此过程揭示了宏观/几何特征，从而更容易匹配双方，并且该项目旨在精确研究这些功能。该项目属于新兴的研究方向，是发现和实验新的混凝土现象的肥沃基础，因此为在研究生和本科水平上培训学生培训学生创造了绝佳的机会。 PI还计划通过组织暑期学校和迷你场来传播兰兰兹计划中的新几何观点，以向更广泛的受众传播。更具体地说，该项目研究了P-Adic领域的Galois代表群的几何形状，重点是通过P-Adadic Hodge Hodge Hodge hodge hodge hodge hodge hodge hodge shodge hodge hodge hodge hodge hodge hodge hodge hodge hodge hodge。预计这些最近构建的空间将在新生的分类P-Adic Langlands计划中发挥关键作用，该计划旨在促进P-Adic Lie群体的单个平滑表示与单个局部P-Adic Galois表示形式之间的（猜想）关系，以与此类对象的整个类别之间的关系之间的关系。该项目的目的是通过将这两个类别与某些中间对象，（半）线性代数对象的模量空间相关联，这两个类别之间建立了桥梁，这易于通过几何表示方法分析。对几何形状的足够强大的控制将导致在当地问题（例如Breuil-Mezard的猜想）以及全球性问题（例如Serre重量猜想，自动形成提升和MOD P共同体的结构）上的全球问题。信息流也可以逆转，即可以从Galois表示的参数和启发式学中预测几何表示理论中的新现象。此外，这些线性代数模量空间是足够的具体空间，可以通过计算机代数软件对其进行实验，从而导致许多理论和计算项目可访问本科生。该奖项反映了NSF的法定任务，并认为通过基金会的知识优点和广泛的crietia crietia criteria criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia均值得通过评估。