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Sheaves, Representations, and Dualities

滑轮、表示和对偶性

基本信息

批准号：
2101507
负责人：
Roman Bezrukavnikov
金额：
$ 67.5万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-01 至 2026-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101507&HistoricalAwards=false
关键词：
Sheaves Representations Dualities

项目摘要

A duality in mathematics or theoretical physics is a correspondence between two theories which allows one to relate key phenomena in one theory to those in the other, despite their somewhat different, in a sense opposite, nature. A rough analogy is to the relation between a visual image and its reflection in a mirror. This project is in representation theory, that is, the study of algebraic structure of symmetries. Fundamental dualities have been playing an increasingly central role in the subject. One such duality is geometric Langlands duality, an outgrowth of reciprocity laws in number theory. Another is mirror symmetry, a phenomenon in algebraic geometry with origins in physics. The present project will derive further consequences of these fundamental ideas to representation theory. An additional goal is to study the common features of different dualities that appear in representation theory in order to find a unified approach resulting in a better understanding of the nature of the observed phenomena. The Principal Investigator will involve graduate and undergraduate students in projects thus promoting mathematical education and research at various levels. The project has several aims. One is to advance rapidly developing topological methods in the theory of modular representations. Another is to further develop etale sheaf methods in harmonic analysis on p-adic groups and finite groups of Lie type. Here we plan to work on uncovering the geometric phenomena underlying the theory of endoscopy in harmonic analysis on p-adic groups. Finally, the PI will continue to work on the theory of quantized symplectic resolutions, an exciting recent chapter in representation theory with surprising connections to physics and algebraic geometry.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.

数学或理论物理学中的二元性是两种理论之间的对应关系，它允许一种理论中的关键现象与另一种理论中的关键现象联系起来，尽管它们的性质有些不同，在某种意义上是相反的。一个粗略的类比是视觉图像与其在镜子中的反射之间的关系。该项目属于表示论，即对称性代数结构的研究。基本的二元性在该学科中发挥着越来越重要的作用。其中一种对偶性是几何朗兰兹对偶性，它是数论中互易定律的产物。另一个是镜像对称，这是代数几何中的一种现象，起源于物理学。本项目将进一步推导出这些基本思想对表示理论的影响。另一个目标是研究表示论中出现的不同对偶性的共同特征，以便找到统一的方法，从而更好地理解观察到的现象的本质。首席研究员将让研究生和本科生参与项目，从而促进各个层面的数学教育和研究。该项目有几个目标。一是推进模表示理论中快速发展的拓扑方法。另一个是进一步发展etale层方法在p进群和李型有限群的调和分析中。在这里，我们计划致力于揭示 p 进群调和分析中内窥镜理论背后的几何现象。最后，PI 将继续致力于量子化辛分辨率理论的研究，这是表示论中令人兴奋的最新章节，与物理学和代数几何有着惊人的联系。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。