Complex Analysis and Random Geometry

The project explores probabilistic and deterministic aspects of self-similar geometry. Self-similar sets are characterized by the property that they look the same at different scales. Such sets arise in the study of dynamical systems, for instance, in complex dynamics and the study of the Mandelbrot set. On the other hand, in probability theory and statistical physics one often encounters stochastically self-similar sets. Such objects only have the same statistical properties at different scales. There are surprising analogies between the probabilistic theory and its deterministic counterpart. The research supported by this award explores these analogies and addresses foundational questions regarding self-similar objects, using methods from complex analysis. The project also provides opportunities for the training and mentoring of junior researchers, including graduate students and postdoctoral researchers. The PI will contribute to the dissemination of mathematical knowledge through the organization of conferences, workshops, and summer schools. Research to be conducted under this award involves the geometry of conformally self-similar structures, both in stochastic and deterministic settings. Julia sets for the iteration of complex mappings illustrate the latter setting, while the former includes topics such as Schramm-Loewner evolution. The project aims to answer fundamental regularity questions for conformally self-similar objects, including Jordan curves of finite Loewner energy. A new parametrization of the Teichmueller spaces of punctured spheres will also be studied. Additional motivation for the project arises from the interaction between the deterministic and stochastic frameworks, notably, the transfer of methods and results between these two areas. For instance, the concept of conformal mating of polynomials in complex dynamics bears close similarity to Sheffield's mating of trees construction for random spheres. The PI’s research uses methods developed in complex dynamics to provide analytic constructions for random structures. Conversely, insights from the probabilistic theory translate to new research avenues in complex dynamics. Conformal welding is a tool of central importance in both theories, and the proposal aims to resolve several fundamental questions regarding Weil-Petersson curves, welding of Liouville Quantum Gravity discs, and Werner's conformal restriction measure on Jordan curves. 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.