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The Langlands Program for 3-Manifolds

三流形朗兰兹纲领

基本信息

批准号：
2202363
负责人：
Sam Gunningham
金额：
$ 28.04万
依托单位：
Montana State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-06-15 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2202363&HistoricalAwards=false
关键词：
Langlands Program Manifolds

项目摘要

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.

朗兰兹纲领是一个包含数论、代数、几何和物理学中的思想、定理和理论的庞大网络。程序的核心是对偶原理（物理学中电磁对偶的一种形式），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没有在数域或代数曲线的传统背景下研究这些现象，而是将朗兰兹计划设置在三维拓扑学领域-这是一个丰富而活跃的研究领域。在这种情况下，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结的不同方法来观察朗兰兹现象。这种方法打开了朗兰兹程序到一个更广阔的数学世界，并提供途径，让新一代的学生在这个基础研究领域。该项目为学生提供研究和培训机会。更详细地说，PI将定义（数学上）和研究状态空间相关的3流形在家庭的几何朗兰兹拓扑量子场论的卡普斯汀和维滕。拟议的研究是广泛的几何表示理论的领域，交叉的概念和工具，从规范理论，量子拓扑学（绞理论），并导出代数几何。研究的一些具体目标包括：建立一个关于skein模维数的对偶猜想，以及在一些重要情况下明确验证的工具;通过发展广义Springer理论的椭圆/量子变体，在一般水平上证明椭圆曲线的Betti和de Rham几何Langlands猜想;一个完善的建设的状态空间的B-模型使用工具从派生的代数几何;和计算的同源性的规范群的3流形方面的朗兰兹双谱惠特克理论。该项目由数学科学部的代数和数论项目、激励竞争性研究的既定项目（EPSCoR）和数学科学部的拓扑项目共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。