Uniformization of non-uniform geometries

Cuny City College, New York NY

Investigators

Abstract

Recent decades have witnessed increased interest in the understanding of fractal and random objects that exhibit non-uniform geometries. Fractals are spaces that possess either a degree of self-similarity, or roughness, or both, and randomness refers to uncertainty in these patterns. Non-uniform geometry here means that the geometric features of the given objects do not adhere to the same quantitative standards across all locations and scales. Within mathematics, substantial interest in the study of fractals comes from their applications in the theory of dynamical systems. Outside of mathematics, fractals have numerous applications, for instance, in antenna design, terrain analysis, and target recognition. This project revolves around questions and problems regarding surface uniformization and simultaneous conformal welding, in particular, in the presence of randomness. The principal investigator will adapt and extend conformal uniformization and welding techniques to new non-uniform and fractal dynamical settings. The investigator also intends to mentor students at various levels, from high school to Ph.D., and foster research in the actively evolving field of fractal geometry and dynamics. The geometrically non-uniform objects considered in this project are metric curves and spaces that are not quasisymmetrically equivalent to circles, equilateral triangulations, etc. Examples of such spaces are abundant in the dynamics of quadratic polynomials, in conformal welding problems, and in random uniformization. One specific goal of the project is to introduce a new class of random surfaces spread over the sphere and to consider the type problem for surfaces of this class. In the case when such surfaces are (almost surely) parabolic, the value distribution properties of associated functions, particularly the order of growth of such functions, will be investigated. Another goal is to develop simultaneous conformal welding tools and techniques that in turn will extend recent results on merging reflection groups with critically fixed anti-rational maps. The methods to be employed are complex analytic in nature. Success in this project will result in new contributions to the growing literature on random surface uniformization and conformal welding, will raise new questions in random value distribution, and will develop new tools which go beyond those employed in current methodology. 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.