Complex Dynamics and Moduli Spaces

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A major goal in dynamics is to understand moduli spaces. The most successful endeavor in this regard has been the study of the moduli space of quadratic polynomials, which contains the Mandelbrot Set, a fundamental object in the subject. One hopes to understand other moduli spaces to a similar extent, like the moduli space of rational maps of a given degree. Understanding the analytic and algebraic structure of these spaces is quite challenging; one reason is that few of the one-dimensional tools carry over to higher dimensions. The projects outlined in this proposal incorporate topology, algebraic geometry, complex analysis, Teichmueller theory, and the nondynamical moduli spaces of curves (and various compactifications thereof) to better understand complex dynamical systems (in one and several variables) and their associated dynamical moduli spaces from both analytic and algebraic points of view. The projects are organized into three main topics, and each topic is related in some way to Thurston's Topological Characterization of Rational Maps, a central theorem in the field of complex dynamics. The research program outlined in this proposal weaves Thurston's theorem into these research topics in a variety of different ways. Dynamical systems are all around us: 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 which is not even completely understood in the simplest of mathematical models. The research outlined in this proposal forges new connections between different parameter spaces (or moduli spaces) associated to certain dynamical systems, which will be exploited to further understand the spaces in question. One dynamical system that arises across different scientific fields is Newton's Method, an essential tool for solving equations that is employed by scientists in every field. There are still many fundamental questions surrounding this dynamical system (in one and several variables) that have yet to be understood. Progress on the research outlined in the proposal has implications for this dynamical system in certain cases.