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

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.