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Brown’s Spectral Measure: New Computational Methods from Stochastics, Partial Differential Equations, and Operator Theory

布朗谱测量：来自随机学、偏微分方程和算子理论的新计算方法

基本信息

批准号：
2055340
负责人：
Todd Kemp
金额：
$ 32.42万
依托单位：
University of California-San Diego
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：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055340&HistoricalAwards=false
关键词：
Brown Spectral Measure New Computational

项目摘要

The modern world is built on data, which is usually represented in rectangular arrays of numbers. Mathematicians call those tables matrices, and try to find hidden structure within them. One such hidden signature is a list of numbers called eigenvalues that describes some of the properties of the larger matrix of numbers. Eigenvalues not only summarize the information in a large array; they can reveal information that was hidden from plain view, through ubiquitous tools like principal component analysis that statisticians have used to great effect for decades. The main purpose of this proposal is to explore promising new computational tools to understand the behavior of eigenvalues of very large matrices lacking any overall symmetry. The PI and his team have discovered new and surprising connections between several different mathematical fields of study that can be used to compute the large-scale behavior of the eigenvalues of such symmetry-free matrices, opening the door to solve problems that have remained inaccessible for decades. As a new methodology, there is much to explore including low-hanging fruit that is perfectly suited to research by Ph.D. students and postdoctoral researchers. Funds will be used to develop these computational tools in collaboration with colleagues and postdoctoral researchers, and will support the research of a diverse body of Ph.D. students – both developing their research skills to prepare them for academic or technical careers, and furthering the research goals of the project.This project will address questions relating operator theory, stochastic differential equations, diffusion on Lie groups, and random matrix theory. The central theme will be to deploy a promising new set of computational tools, based on stochastic and partial differential equations, to calculate and prove regularity of spectral measures of non-normal operators in von Neumann algebras. These arise naturally as the high-dimension limits of random matrix models that appear in wireless communication and information theory, as well as throughout geometry and analysis in pure mathematics. The project concerns eleven research directions, which yield connections between these topics and applications to others. The intended research, upon completion, will settle several interesting open questions and present a major contribution to the theory, as well as provide ample opportunity for the mentoring of Ph.D. students and postdoctoral researchers.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.

现代世界是建立在数据之上的，数据通常用矩形数字数组来表示。数学家将这些表格称为矩阵，并试图找到其中隐藏的结构。一个这样的隐藏签名是一个称为特征值的数字列表，它描述了更大的数字矩阵的一些属性。本征值不仅将信息汇总在一个大数组中；它们还可以通过主成分分析等无处不在的工具揭示隐藏在普通视图中的信息，几十年来，主成分分析一直发挥着巨大的作用。这一建议的主要目的是探索有前景的新计算工具，以了解缺乏任何整体对称性的非常大的矩阵的特征值的行为。PI和他的团队在几个不同的数学研究领域之间发现了令人惊讶的新联系，可以用来计算这种对称自由矩阵的特征值的大规模行为，为解决几十年来一直无法解决的问题打开了大门。作为一种新的方法，有很多需要探索的地方，包括非常适合博士生和博士后研究人员研究的容易摘到的果实。资金将用于与同事和博士后研究人员合作开发这些计算工具，并将支持不同博士生的研究--既发展他们的研究技能，为他们的学术或技术职业做准备，又促进项目的研究目标。本项目将解决与算子理论、随机微分方程、李群上的扩散和随机矩阵理论相关的问题。中心主题将是部署一套有前途的新的计算工具，基于随机和偏微分方程，来计算和证明von Neumann代数中非正规算子的谱测度的正则性。这些问题自然而然地出现在无线通信和信息理论中出现的随机矩阵模型的高维极限，以及整个几何和纯数学分析中。该项目涉及11个研究方向，这些方向产生了这些主题与其他应用之间的联系。这项预期的研究完成后，将解决几个有趣的开放问题，并对该理论做出重大贡献，并为指导博士生和博士后研究人员提供充足的机会。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。