Moduli of Galois Representations and Applications

伽罗瓦模表示及应用

基本信息

批准号：
1802037
负责人：
Bao Le Hung
金额：
$ 18万
依托单位：
Northwestern University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802037&HistoricalAwards=false
关键词：
Moduli Galois Representations Applications

项目摘要

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.

数学理论是数学最古老的分支之一，它研究了整数的属性和模式，并构成了重要的实际应用，例如公共密钥密码学（广泛用于互联网上的安全通信）。尽管历史悠久，但我们对这个主题的理解仍然不令人满意。在19世纪，E. Galois通过他们满足的方程式研究数字的想法，其（离散的）对称性是在Galois组的概念中捕获的。在1970年代，兰兰兹（Langlands）提出了一个遥远的猜想，将这些galois组与完全不同的对称性联系起来，即在高度对称形状（自动形式）上看到的（连续的）对称性。这些猜想的特定情况的进展已经导致了壮观的成就，例如Fermat的最后一个定理的证明，Sato-Tate猜想的证明，许多新的Diophantine方程式的解决方案等。这些发展的主题来自这些发展的主题，即通过将整个群众置于范围中，可以将整体范围融合到范围内，并将整体范围纳入这些信息，并将这些范围融合到这些信息中，并构成了这些范围，这些范围都可以使这些范围融合了这些范围，这些范围都可以使这些范围融合了这些范围。该项目的目的是进一步，这对于在兰兰兹计划中取得进一步的进步至关重要。更具体地说，该项目研究了P-ADIC领域Galois组的P-ADIC代表模量的结构。这是通过在几何表示理论中研究的空间（尤其是仿射弹簧纤维中研究的空间）对GALOIS表示的变形空间进行建模来实现的。这在两个数学的两个不同发达和活跃的领域之间创建了新的链接。结果，一方面，几何技术可用于在Mod P Langlands计划中取得重大进展，例如Serre重量猜想，Breuil-Mezard猜想和局部 - 全球兼容性问题；另一方面，Galois理论的见解和启发式方法可用于在模块化表示理论和几何表示理论中发现和研究新现象。此外，该项目开辟了许多具体问题，包括理论和计算，这为本科生提供了极好的研究机会。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子的评估来获得支持的，并具有更广泛的影响。