Moduli of Galois Representations and Applications

数学最古老的分支之一是数论，它研究整数的性质和模式，并且是重要的实际应用的基础，例如公钥密码术（广泛用于互联网上的安全通信）。尽管历史悠久，但我们对这一问题的理解仍然不能令人满意。19世纪，E.伽罗瓦引入了通过它们所满足的方程来研究数的想法，这些方程的（离散）对称性在伽罗瓦群的概念中被捕获。在1970年代，朗兰兹提出了一个意义深远的网络结构，将这些伽罗瓦群与一种完全不同的对称性联系起来，即在高度对称形状（自守形式）上的振动中看到的（连续）对称性。在某些特殊情况下，这些理论的进展已经导致了惊人的成就，如费马大定理的证明，Sato-Tate猜想的证明，许多新的丢番图方程的解决方案，等等。这些发展的一个新兴主题是，人们可以卓有成效地研究伽罗瓦群（因此整数）通过将它们包含的信息包装成连续的家庭，这些空间与量子物理学的思想有关。该项目旨在进一步推进这一思想圈，这对朗兰兹计划的进一步进展至关重要。更具体地说，该项目研究了p-adic域的伽罗瓦群的p-adic表示的模空间的结构。这是通过在几何表示理论中研究的空间，特别是仿射Springer纤维中，用p进霍奇理论条件对伽罗瓦表示的变形空间进行建模来实现的。这在两个不同的高度发展和活跃的数学领域之间建立了新的联系。因此，一方面，几何技术可以用来在模p朗兰兹程序中的主要问题，如Serre权矩阵，Breuil-Mezard猜想和局部-全局相容性问题上取得实质性进展;另一方面，来自伽罗瓦理论的见解和启发可以用来发现和研究模表示理论和几何表示理论中的新现象。此外，该项目还提出了许多具体的理论和计算问题，为本科生提供了绝佳的研究机会。该奖项反映了NSF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。

One of the oldest branch of mathematics is Number Theory, which studies the properties and patterns of whole numbers, and which underlies important practical applications such as public key cryptography (widely used for secure communication over the internet). Despite its long history, our understanding of this subject remains unsatisfactory. In the 19th century, E. Galois introduced the idea of studying numbers via the equations that they satisfy, whose (discrete) symmetries are captured in the notion of Galois groups. In the 1970s, Langlands proposed a far reaching web of conjectures which relates these Galois groups to a completely different kind of symmetry, namely the (continuous) symmetries seen in vibrations on highly symmetric shapes (automorphic forms). Progress on particular cases of these conjectures has already led to spectacular achievements, such as the proof of Fermat's Last Theorem, the proof of the Sato-Tate conjecture, solutions of many new Diophantine equations, etc. An emerging theme from these developments is that one can fruitfully study Galois groups (and hence whole numbers) by packaging the information they contain into continuous families, and strikingly these spaces are related to ideas from quantum physics. This project aims to further this circle of ideas, which is essential to make further progress in the Langlands program.More specifically, the project studies the structure of the moduli space of p-adic representations of Galois groups of p-adic fields. This is achieved by modelling deformation spaces of Galois representations with p-adic Hodge theory conditions in terms of spaces studied in Geometric representation theory, especially affine Springer fibers. This creates new links between two different highly-developed and active fields of mathematics. As a result, on one hand, geometric techniques can be used to make substantial progress on major problems in the mod p Langlands program such as Serre weight conjectures, the Breuil-Mezard conjecture, and local-global compatibility problems; on the other hand, insights and heuristics from Galois theory can be used to discover and study new phenomena in modular representation theory and geometric representation theory. Furthermore, the project opens up many concrete questions, both theoretical and computational, which make excellent research opportunities for 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.