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

这个项目是几个数学领域的交叉：分析、几何和概率论。研究的主要焦点在于这些理论在缺乏传统平滑或规则概念的情况下的发展，例如，在分形空间中。一些正在考虑的研究课题是由物理、工程或数学金融问题所激发的。这个项目成功的潜在好处在于有可能将一个数学领域的工具应用到其他领域，从而增加数学领域之间的相互作用。该项目还为研究生的合作、指导和培训提供了机会。该项目侧重于非光滑分析、几何和概率这三个广泛领域内的三个主题。首先，在一般度量度量空间上考虑Sobolev函数和有界变分函数的空间。抽象度量测度空间上的Sobolev空间理论在过去几十年中引起了广泛的关注。在这种情况下，上梯度法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空间的另一种方法，这种方法在分形设置中更有效。第二个研究方向涉及分数高斯场和Dirichlet度量度量空间上的抛物型和双曲型Anderson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上的安德森模型，包括分形。动机源于数学物理，并将研究间歇性和局部性等具有挑战性的特性。最后，该项目研究了随机矩阵和对称空间，探索对称空间的黎曼振动如何使可积随机矩阵泛函的构造成为可能。这里的可积性理解为这样的泛函的拉普拉斯变换可以用特殊函数显式地表示。这些显式公式将被用来得到合适的极限定理。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.