Some topics in Analysis and Probability in Metric Measure Spaces, Random Matrices, and Diffusions

度量测度空间、随机矩阵和扩散中的分析和概率中的一些主题

• 批准号: 2247117
• 负责人: Vasileios Chousionis
• 金额: $ 49.22万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247117&HistoricalAwards=false
• 关键词: Some; topics; Analysis; Probability; Metric

This project lies at the intersection of several areas of mathematics: analysis, geometry, and probability. A primary focus of the research resides in the development of these theories in settings which lack traditional notions of smoothness or regularity, for instance, in fractal spaces. Some of the research topics under consideration are motivated by questions in physics, engineering, or mathematical finance. A potential benefit of success in this project lies in the possibility to bring tools from one mathematical field to bear on other fields, thereby increasing the interactions between areas of mathematics. The project also provides opportunities for collaboration and for the mentoring and training of graduate students.The project focuses on three subjects within the broad field of nonsmooth analysis, geometry, and probability. First, spaces of Sobolev functions and functions of bounded variation will be considered on general metric measure space. The theory of Sobolev spaces on abstract metric measure spaces has attracted substantial attention over the past few decades. In this context the upper gradient approach has proved to be one of the most successful approaches. However, due to the lack of sufficient connectivity, the approach via upper gradients fails to be effective in many fractal spaces. This project will explore an alternative approach to Sobolev spaces, building on prior work of Korevaar and Schoen, which is more effective in fractal settings. A second direction of research involves fractional Gaussian fields and the parabolic and hyperbolic Anderson models on Dirichlet metric measure spaces. A key goal here is to develop a general theory of fractional Gaussian fields and Anderson models on general Dirichlet spaces, including fractals. Motivation arises from mathematical physics, and challenging properties such as intermittency and localization will be investigated. Finally, the project takes up the study of random matrices and symmetric spaces, exploring how Riemannian fibrations of symmetric spaces enable the construction of integrable random matrix functionals. Integrable here is understood in the sense that the Laplace transforms of such functionals can explicitly be expressed using special functions. These explicit formulas will be employed to obtain suitable limit theorems.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.

该项目位于多个数学领域的交叉点：分析、几何和概率。研究的主要焦点在于在缺乏平滑或规则性传统概念的环境中（例如分形空间）发展这些理论。正在考虑的一些研究主题是由物理、工程或数学金融问题引发的。该项目成功的一个潜在好处在于可以将一个数学领域的工具应用于其他领域，从而增加数学领域之间的相互作用。该项目还提供了合作以及指导和培训研究生的机会。该项目重点关注非光滑分析、几何和概率等广泛领域内的三个主题。首先，将在一般度量测度空间上考虑Sobolev函数和有界变差函数的空间。在过去的几十年里，关于抽象度量测度空间的索博列夫空间理论引起了广泛的关注。在这种情况下，上梯度方法已被证明是最成功的方法之一。然而，由于缺乏足够的连通性，通过上部梯度的方法在许多分形空间中并不有效。该项目将在 Korevaar 和 Schoen 之前的工作基础上探索 Sobolev 空间的替代方法，该方法在分形环境中更有效。第二个研究方向涉及分数高斯场以及狄利克雷度量测度空间上的抛物线和双曲安德森模型。这里的一个关键目标是发展分数高斯场的一般理论和一般狄利克雷空间（包括分形）的安德森模型。动机源于数学物理学，并且将研究间歇性和定位等具有挑战性的特性。最后，该项目开始研究随机矩阵和对称空间，探索对称空间的黎曼纤维如何构建可积随机矩阵泛函。这里的可积被理解为这样的泛函的拉普拉斯变换可以使用特殊函数显式地表达。这些明确的公式将用于获得合适的极限定理。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。