Rational Dynamics on Complex Surfaces

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

The project involves research on dynamical systems of several complex variables. Dynamical systems are processes that can be modeled mathematically by a set of states that evolve over time according to fixed rules. Dynamical systems are ubiquitous in science and engineering, arising in the context of weather prediction, macroeconomics, mechanics, and neural networks, to name just a small selection of application domains. Topics of interest in this area include making predictions about the future state of a system based on present conditions, understanding how sensitively future states depend on initial states, and characterizing the full array of eventual states. Dynamical systems given by polynomial or rational formulas constitute an important and interesting special class and are a focus of study in this project. Such systems are natural from a mathematical standpoint, and also arise in various scientific and mathematical applications including interior point methods in linear programming, thermodynamics, and root-finding algorithms. The study of rational dynamics in several complex variables draws on tools and techniques from a range of mathematical areas, and advances in multivariable complex dynamics in turn yield new insights in those areas. In addition to the expected research advances, the project contributes to the development of human resources via the support of doctoral students and STEM outreach activities at the Riverbend Community Math Center, a local non-profit organization devoted to working with K-12 mathematics students and teachers. The project involves the dynamics of rational mappings in the setting of several complex variables. One major thrust of the proposed work concerns the difficult problem of constructing measures of maximal entropy for rational maps on complex projective surfaces. Of particular interest are rational maps on complex surfaces that do not admit algebraically stable models. It is anticipated that any such map must preserve additional geometric structure, such as a fibration onto a curve or an invariant meromorphic canonical form. Progress towards this conjecture is an important goal of the project. In addition, the project aims to develop analytic tools tailored to such structures, with an eye towards producing invariant currents and measures, as were previously constructed in the setting of algebraically stable maps. 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.