Geometry of moduli stacks of Galois representations

数论中的一个中心问题涉及在整数和密切相关的数制中求解多项式方程。求解这些所谓的丢番图方程通常非常困难，这种困难是加密算法的安全性来源，例如，保证互联网上的私人通信。此外，这些丢番图问题的丰富性与数学的其他分支有着深刻的联系。人们可以通过其几何和动力学来富有成效地研究这些方程的解集，将它们重新包装为伽罗瓦表示。在20世纪70年代，罗伯特·朗兰兹（Robert Langlands）做出了影响深远的预测，将伽罗瓦表示与数学物理和谐波分析联系起来。朗兰兹定理的p-adic版本，其中包含了素数的幂整除性，导致了朗兰兹原始定理的令人兴奋的进展，因此，一系列经典数论问题，如费马大定理。在其他背景下，朗兰兹定理的进一步类似物刺激了表示论、代数几何和现代物理学的活动。最近，一些数学家提出了一个统一的范畴框架，这些不同版本的朗兰兹定理。这个项目的目的是通过对伽罗瓦表示空间进行建模，通过范畴框架来研究朗兰兹图的p-adic版本。伽罗瓦表示空间的几何是一个新的和肥沃的研究方向，提供了机会，培养年轻的研究人员和连接社区的数论和几何表示理论。PI还计划通过外展招募和培训研究人员，通过传播针对初入研究生的数论背景材料，并组织当地和区域研讨会和会议。该项目的目标是调查假设的mod p和p-adic Langlands对应关系，这些对应关系将Galois表示与p-adic约化群的模和积分表示联系起来。这种对应关系在理论上是由伽罗瓦表示的层所介导的。主要研究者将构建伽罗瓦表示堆栈的局部模型，以分析这些结构层，并回答围绕模块化提升，Breuil-Mezard猜想，Serre猜想的权重部分以及使用Taylor-Wiles方法的局部对称空间的mod p上同调的问题。这些调查将允许应用程序和从模块化和几何表示理论。该项目为本科生和研究生提供了通过组合方法和计算机代数软件包探索本地模型的机会。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。

A central problem in number theory concerns solving polynomial equations in the whole numbers and closely related number systems. Solving these so-called Diophantine equations is typically very difficult, and this difficulty is a source of security for encryption algorithms that, for instance, guarantee private communication on the internet. Moreover, the richness of these Diophantine problems has profound connections to other branches of mathematics. One can fruitfully study the set of solutions to these equations through its geometry and dynamics, repackaging them as a Galois representation. In the 1970s, Robert Langlands made far-reaching predictions that connect Galois representations to mathematical physics and harmonic analysis. A p-adic version of Langlands' conjectures, which incorporates divisibility by powers of a prime, has led to exciting progress on Langlands' original conjectures and, consequently, a range of classical number-theoretic questions like Fermat's Last Theorem. Further analogues of Langlands' conjectures in other contexts have spurred activity in representation theory, algebraic geometry, and modern physics. More recently, a number of mathematicians have proposed a unifying categorical framework for these different versions of Langlands' conjectures. This project aims to study the p-adic version of Langlands' conjectures through the categorical framework by modelling the space of Galois representations. The geometry of the space of Galois representations is a new and fertile research direction that provides opportunities to train younger researchers and to connect communities in number theory and geometric representation theory. The PI also plans to recruit and train researchers through outreach, by disseminating accessible background material on number theory aimed at beginning graduate students, and by organizing local and regional seminars and conferences.The goal of this project is to investigate the hypothetical mod p and p-adic Langlands correspondences that relate Galois representations and modular and integral representations of p-adic reductive groups. This correspondence is conjecturally mediated by sheaves on stacks of Galois representations. The Principal Investigator will construct local models for stacks of Galois representations to analyze these conjectural sheaves and answer questions around modularity lifting, the Breuil-Mezard conjecture, the weight part of Serre's conjecture, and the mod p cohomology of locally symmetric spaces using the Taylor-Wiles method. These investigations will allow for applications to and from modular and geometric representation theory. This project gives undergraduate and graduate students the opportunity to probe local models via combinatorial methods and computer algebra packages.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.