Exploring the Topology and Geometry of Dynamical Subvarieties

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Dynamical systems are all around us: they govern the motion of the planets, the weather, the stock market, the ecosystems in which we live. These systems depend on a variety of parameters, and as these parameters change, the corresponding system is affected. Understanding how dynamical systems change with different parameters is a very complicated and delicate question that is not even completely understood in the simplest of mathematical models. However, there is one shining success story in this regard: that is, the parameter space, or "moduli space", of complex quadratic polynomials. This space contains the famous Mandelbrot Set, which has been thoroughly studied over the last 40 years. The research outlined in this proposal explores different parameter spaces associated to particular dynamical systems, with a view toward understanding them to the same extent that the mathematical community understands the space where the Mandelbrot set lives. This proposal also contains a significant outreach component to support the Math Corps at U(M), a free math Summer Camp for middle school students and high school mentors. A major goal in the field of complex dynamics is to understand dynamical moduli spaces. The most successful endeavor in this regard has been the study of the moduli space of quadratic polynomials where the Mandelbrot Set lives, a fundamental object in the subject. Ultimately, we strive to understand the moduli space of rational maps of arbitrary degree to the same extent that we understand the moduli space of quadratic polynomials. Many tools from complex analysis that pave the way for key breakthroughs in the one-dimensional setting do not carry over to higher dimensions. So instead of considering the whole moduli space, PI follows an approach initiated by William Thurston and investigate sub-varieties of moduli space. The most natural sub-varieties to study are those that come from dynamical conditions, like imposing combinatorial constraints on the forward orbits of critical points. One may view Thurston’s Topological Characterization of Rational Maps as a first step. It provides a way to understand zero-dimensional dynamical sub-varieties; that is, those that consist of postcritically finite parameters. Following Epstein, the PI will adapt Thurston’s ideas and constructions and shall develop a setting in which to study higher-dimensional dynamical sub-varieties of moduli spaces. PI will explore this theory by working with one-dimensional dynamical sub-varieties in the moduli space of quadratic rational maps, and one-dimensional dynamical sub-varieties in the moduli space of cubic polynomials, where there are already very challenging and fundamental problems concerning their topology. 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.