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Unipotent Representations and Associated Cycles

单能表示和相关循环

基本信息

批准号：
2000254
负责人：
Dan Barbasch
金额：
$ 18万
依托单位：
Cornell University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000254&HistoricalAwards=false
关键词：
Unipotent Representations Associated Cycles

项目摘要

Laws of nature, in particularly those of physics, satisfy basic symmetries. In mathematical language, these symmetries are most often expressed in terms of Lie groups and their representations (named after Sophus Lie, a 19th century mathematician who pioneered these concepts in his study of solutions of differential equations). This project is concerned with properties of unitary representations. The modern study of unitary representations stems from quantum physics. In addition, they play a crucial role in many other applications such as tomography, crystallography, and signal processing.In more technical terms, this project is focused on properties of unipotent representations, which are the building blocks of the unitary duals of real and p-adic reductive groups. The determination of the unitary dual is a major problem in the representation theory of such groups. The PI will formulate and sharpen conjectures on the signatures of hermitian forms for modules of the affine Hecke algebra. These are essential for the determination of the unitary duals of p-adic groups. Another focus of this project is to make explicit the role of unipotent representations in the description of the unitary dual. In addition to linear groups, the PI will study these notions for nonlinear and disconnected groups that arise in mathematical physics and in the study of automorphic forms.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.

自然规律，特别是物理规律，满足基本的对称性。在数学语言中，这些对称性通常用李群及其表示来表达（以Sophus Lie命名，Sophus Lie是一位世纪数学家，他在研究微分方程的解时开创了这些概念）。这个项目关注酉表示的性质。对幺正表示的现代研究起源于量子物理学。此外，它们在许多其他应用中发挥着至关重要的作用，如层析成像，晶体学和信号处理。在更专业的术语中，该项目专注于幂幺表示的性质，这是真实的和p-adic约化群的酉群的构建块。确定酉对偶是这类群的表示论中的一个主要问题。PI将制定和锐化仿射Hecke代数的模的埃尔米特形式的签名。这些对于确定p-adic群的酉群是必不可少的。这个项目的另一个重点是明确的作用，幂幺表示在描述的酉对偶。除了线性群之外，PI还将研究数学物理和自守形式研究中出现的非线性和断开群的概念。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。