Moduli spaces of Galois representations

数论是数学的分支，研究整数的性质和模式。尽管这个看似基本的前提，数论一直处于数学中发现的一些最复杂结构的最前沿，以及潜在的关键实际应用（例如公钥密码学，它为当前互联网上的安全通信提供了动力）。现代数论的一个基本思想是，具有某些共同特征（例如是某些方程列表的解）的数的集合具有有趣的涌现性质和对称性。这种涌现对称性中最原始的是有理数的绝对伽罗瓦群，在过去的几个世纪里，大量的数论都在探索它的（复杂的）内部结构。在20世纪70年代，朗兰兹做出了一系列令人惊讶的预言，认为这个绝对伽罗瓦群与某些高度对称的几何形状（自守表示）上的振动（连续）对称性有关。这样的证明被认为具有深远的影响：例如，一个被证明的特例是费马大定理的核心。在过去的几十年里，朗兰兹猜想的一个有希望的方法是p-adic变形的方法，其中一个根据给定素数p的幂的整除性来组织猜想两边的信息。其重要性最近才成为焦点的关键点是，这个过程揭示了宏观/几何特征，使其更容易匹配两边，这个项目的目的就是研究这些特征。属于一个新兴的研究方向，该项目是发现和实验新的具体现象的沃土，从而为研究生和本科生的培训创造了极好的机会。PI还计划通过组织暑期学校和小型课程向更广泛的受众传播Langlands计划中的新几何观点。更具体地说，该项目研究p-adic域的Galois群表示的模堆栈的几何，重点关注p-adic Hodge理论条件切割的轨迹。这些最近构建的空间预计将在新生的范畴p进朗兰兹计划中发挥关键作用，该计划旨在促进p进李群的个体光滑表示与个体局部p进伽罗瓦表示之间的关系，以促进此类对象的整个范畴之间的关系。该项目旨在建立这两个类别之间的桥梁，通过将两者与一些中间对象上的层类别，（半）线性代数对象的模空间联系起来，这些对象易于通过几何表示理论的方法进行分析。一个足够强大的控制几何将导致重大进展的地方问题，如Breuil-Mezard猜想以及全球性的问题，如塞尔重量结构，自同构提升和结构的模p上同调的局部对称空间。信息的流动也可以被逆转，即人们可以从伽罗瓦表示的论证和推理中预测几何表示论中的新现象。此外，这些线性代数模空间足够具体，可以用计算机代数软件对其进行实验，从而为本科生提供了许多理论和计算项目。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 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.