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

这个项目位于数学的几个领域的交叉点：分析，几何和概率。研究的一个主要焦点在于这些理论在缺乏光滑性或规则性的传统概念的环境中的发展，例如，在分形空间中。正在考虑的一些研究课题是由物理学，工程学或数学金融学的问题所激发的。这个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的机会，并为指导和研究生的培训。该项目侧重于非光滑分析，几何和概率的广泛领域内的三个科目。首先，在一般的度量测度空间上考虑Sobolev函数空间和有界变差函数空间。抽象度量测度空间上的Sobolev空间理论在过去的几十年里引起了人们的广泛关注。在这方面，上梯度方法已证明是最成功的方法之一。然而，由于缺乏足够的连通性，通过上梯度的方法不能有效地在许多分形空间。该项目将探索Sobolev空间的另一种方法，建立在Korevaar和Schoen之前的工作基础上，这在分形设置中更有效。研究的第二个方向涉及分数高斯场和抛物型和双曲型安德森模型的Dirichlet度量测度空间。这里的一个关键目标是发展一个一般理论的分数高斯场和安德森模型一般狄利克雷空间，包括分形。动机来自数学物理，并将研究具有挑战性的性质，如不连续性和本地化。最后，该项目承担了随机矩阵和对称空间的研究，探索如何对称空间的黎曼纤维化使可积随机矩阵泛函的建设。这里的可积是指这样的泛函的拉普拉斯变换可以用特殊函数显式表示。这个奖项反映了NSF的法定使命，并被认为是值得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的支持。

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.