Moduli spaces of Galois representations

数论是研究整数的性质和模式的数学分支。尽管有这个看似基本的前提，但数论一直处于数学中发现的一些最复杂结构的前沿，以及潜在的关键实际应用(如为当前互联网上的安全通信提供动力的公钥密码学)。现代数论的一个基本思想是，具有某些共同特征(如某些方程的解)的数的集合具有有趣的涌现性质和对称性。这种新出现的对称性最原始的是有理数的绝对伽罗瓦群，而过去几个世纪的一大批数论都涉及到探索它(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测，认为这个绝对伽罗瓦群与一些高度对称的几何形状(自同构表示)上振动的(连续)对称性有关。众所周知，这样的猜想具有深远的影响：例如，费马大定理解决方案的核心是一个已被证明的特例。在过去的几十年里形成的朗兰兹猜想的一个很有前途的方法是p-进变形法，其中一个人根据给定素数p的幂的整除来组织猜想两侧的信息。其重要性直到最近才被关注的关键点是，这个过程揭示了宏观/几何特征，使得匹配两侧变得更容易，该项目的目的正是研究这些特征。该项目属于一个新兴的研究方向，是发现和试验新的具体现象的肥沃土壤，因此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迷你课程向更广泛的受众传播朗兰兹计划中的新几何观点。更具体地说，该项目研究p-adad场的Galois群表示的模堆栈的几何，重点是p-adadyHodge理论条件所切出的轨迹。这些新构造的空间有望在新出现的范畴p-adic朗兰兹计划中发挥关键作用，该计划试图将p-adic李群的个体光滑表示和个体局部p-adic Galois表示之间的(猜想)关系推广到此类对象的整个范畴之间的关系。该项目的目的是在这两个范畴之间建立一座桥梁，通过将这两个范畴都与一些中间对象上的轮子范畴、(半)线性代数对象的模空间联系起来，这些范畴易于通过几何表示理论的方法进行分析。对几何的足够强的控制将导致在局部问题上的重大进展，如Breuil-Mezard猜想，以及全局问题，如Serre重量猜想，自同构提升和局部对称空间的mod p上同调的结构。信息流也是可以逆转的，即人们可以通过伽罗瓦表示的论证和启发式来预测几何表示理论中的新现象。此外，这些线性代数模空间是足够具体的，人们可以在它们上用计算机代数软件进行实验，导致本科生可以接触到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 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.