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Local Geometric Langlands Conjecture in Quasi-Classical Setting

准古典背景下的局域几何朗兰兹猜想

基本信息

批准号：
1903391
负责人：
Dmytro Arinkin
金额：
$ 39万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1903391&HistoricalAwards=false
关键词：
Local Geometric Langlands Conjecture Quasi

项目摘要

The PI will conduct research in two directions in the area of the geometric Langlands program. The Langlands program started as a series of conjectures proposed by Robert Langlands beginning in 1967. From its origins in number theory and representation theory, it developed into a framework connecting and influencing many areas of mathematics. In particular, the geometric Langlands program is formulated in terms of algebraic geometry and can be interpreted in physical terms as a duality in quantum field theory. This project introduces new geometric ideas into the area. It is likely that the techniques developed in this project will find applications in other areas of mathematics. This award provides training of graduate students through research.In more detail, the PI will work on the classical limit of the local geometric Langlands conjecture and on the duality for compactified Jacobians of formal curves. In earlier work, the PI has studied the classical limit of the global geometric conjecture; while progress in the special case of the group GL(n) was made, it became apparent that the general case will require new tools. One goal of this project is to develop such tools using local methods. At the same time, the research focuses on an essentially unexplored direction in the geometric Langlands program: while the general shape of conjectures is known (and stated in the proposal), the details are unclear at this point. The project aims to clarify the situation by using the easier case of GL(n) to test various conjectures before attempting to prove them in general.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.

PI将在几何朗兰兹计划领域的两个方向进行研究。朗兰兹纲领最初是罗伯特·朗兰兹从 1967 年开始提出的一系列猜想。它起源于数论和表示论，发展成为连接和影响数学许多领域的框架。特别是，几何朗兰兹纲领是用代数几何来表述的，并且可以用物理术语解释为量子场论中的对偶性。该项目将新的几何理念引入该地区。该项目中开发的技术很可能会在其他数学领域得到应用。该奖项通过研究为研究生提供培训。更详细地说，PI 将研究局部几何朗兰兹猜想的经典极限和形式曲线的紧致雅可比行列式的对偶性。在早期的工作中，PI研究了全局几何猜想的经典极限；虽然 GL(n) 组的特殊情况取得了进展，但很明显，一般情况将需要新的工具。该项目的目标之一是使用本地方法开发此类工具。与此同时，该研究的重点是几何朗兰兹纲领中一个本质上尚未探索的方向：虽然猜想的一般形式是已知的（并在提案中陈述过），但目前细节尚不清楚。该项目旨在通过使用更简单的 GL(n) 案例来测试各种猜想，然后再尝试普遍证明它们，从而澄清情况。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。