Quasiconformal geometry of random and deterministic spaces

Many classical problems in mathematics ask when a complicated shape can be represented by a simpler or more standard one without losing its essential geometric structure. This project investigates the geometry of spaces that arise naturally in complex analysis, geometric function theory, probability, and low-dimensional topology. Such questions are central to understanding conformal and quasiconformal mappings, which are transformations that preserve angles or controlled geometric distortion. The work advances mathematical knowledge in geometric analysis and helps build bridges between analysis, probability, geometry, and topology. The project also supports the training of students and early-career researchers through research mentoring, seminars, and dissemination of results. The project studies both deterministic spaces, such as surfaces and domains with complicated boundary structure, and random spaces, such as those generated by stochastic processes. The investigator studies quasiconformal geometry of random and deterministic spaces, with emphasis on uniformization, rigidity, removability, conformal dimension, and the geometry of Riemann surfaces and metric spaces. The project develops analytic and geometric tools for understanding when spaces admit conformal or quasiconformal parametrizations by standard models, and when such parametrizations are unique or rigid. A central theme is the interaction between curve families, modulus methods, boundary structure, and large-scale geometry. The project also investigates random fractal spaces and their quasiconformal invariants, including questions related to conformal dimension and Brownian motion. The expected contributions include new mapping and rigidity theorems, new examples and counterexamples in quasiconformal geometry, and a deeper understanding of the role of modulus and metric geometry in classical and modern uniformization problems. 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.