The Langlands Program for 3-Manifolds

朗兰兹计划是数论、代数、几何和物理中的思想、定理和猜想的巨大网络。该程序的核心是二元性原理(物理学中电磁二元性的一种形式)，它预测了先验无关的数学对象之间令人惊讶的联系。理解这种二元性的本质是数学和物理学中的一个基本问题。PI不是在传统的数域或代数曲线的背景下研究这些现象，而是将朗兰兹计划设置在三维拓扑领域--这是一个本身就是一个丰富而活跃的研究领域。在这种背景下，这些深刻的想法变得比以前更容易理解。例如，该项目的一个组成部分是通过比较在给定的三维空间中计算节点的不同方法来观察朗兰兹现象。这些方法使朗兰兹计划向更广泛的数学世界开放，并提供了让新一代学生参与这一基础研究领域的途径。这个项目为学生提供了研究和培训的机会。更详细地说，PI将定义和研究与Kapustin和Witten的几何朗兰兹拓扑量子场理论家族中的3-流形相关的态空间。所提出的研究广泛地在几何表示理论的领域，交叉了来自规范理论、量子拓扑(绞线理论)和衍生代数几何的概念和工具。这项研究的一些具体目标包括：建立关于Skein模的维度的对偶猜想，以及在一些重要情况下显式验证的工具；通过发展广义Springer理论的椭圆/量子变体，在一般水平上证明Betti和De Rham几何Langland猜想；利用导出的代数几何工具精化构造B-模型的状态空间；以及根据Langland对偶谱Whittaker理论计算3-流形的规范群的同调。该项目由数学科学部的代数和数论计划、激励竞争性研究的既定计划(EPSCoR)和数学科学部的拓扑学计划共同资助。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。

The Langlands program is a vast network of ideas, theorems, and conjectures in number theory, algebra, geometry, and physics. At the heart of the program is a duality principle (a form of electric-magnetic duality in physics), which predicts surprising connections between a priori unrelated mathematical objects. Understanding the nature of this duality is a fundamental problem in mathematics and physics. Rather than study these phenomena in the traditional contexts of number fields or algebraic curves, the PI sets the Langlands program in the realm of 3-dimensional topology - a rich and active area of research in its own right. In this setting, these deep ideas become much more accessible than was previously possible. For example, one component of the project is about observing Langlands phenomena by comparing different ways to count knots in a given 3-dimensional space. Such methods open up the Langlands program to a wider mathematical world and provide pathways to involve a new generation of students in this fundamental research area. This project provides research and training opportunities for students. In more detail, the PI will define (mathematically) and study the space of states associated to a 3-manifold in the family of geometric Langlands topological quantum field theories of Kapustin and Witten. The proposed research is broadly in the realm of geometric representation theory, intersecting with concepts and tools from gauge theory, quantum topology (skein theory), and derived algebraic geometry. Some particular goals of the research include: formulating a duality conjecture for the dimensions of skein modules, as well as tools to explicitly verify in a number of important cases; a proof of the Betti and de Rham geometric Langlands conjecture for elliptic curves at generic level by developing an elliptic/quantum variant of generalized Springer theory; a refined construction of the state space of the B-model using tools from derived algebraic geometry; and a computation of the homology of gauge groups of 3-manifolds in terms of the Langlands dual spectral Whittaker theory. This project is jointly funded by the Algebra and Number Theory program in the Division of Mathematical Sciences, the Established Program to Stimulate Competitive Research (EPSCoR), and the Topology program in the Division of Mathematical Sciences.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.