Complex Dynamics in Higher Dimensions

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

A dynamical system, at its most general, is any set of circumstances or physical object—e.g., the economy, a viral epidemic, or a solar system—that evolves from one moment to the next according to definite mathematical rules. It is often important but quite difficult to use such moment-to-moment rules to predict the state of the system in the relatively distant future. Will a change in regulation lead to a speculative bubble? Will present interventions suffice to stop an outbreak? Will the solar system fly apart? In a broad mathematical sense, all of these questions reduce to understanding whether some aspect of the dynamical system in question is 'stable' or 'unstable'. That is, does it vary slowly and predictably as the system evolves, or is it prone to change rapidly and chaotically with a small variation in the system. The research in this project aims at better understanding the mathematics of dynamical systems, and in particular at determining and describing those parts of a system which are most unstable. Funding for this project will, among other things, support the principal investigator's graduate student as well as several undergraduates who are helping to run math circles for local K-12 students. It's broader impact will be further felt through the principal investigator's involvement with the Riverbend Math Center, which is a local independent non-profit organization dedicated to promoting math education at all levels. This project concerns problems in the intersection between analysis, complex algebraic geometry and dynamical systems. The problems stem from a very general program for constructing and analyzing measures of maximal entropy for rational self-maps of projective space. The work will prominently feature the particular case of rational maps preserving a meromorphic two form. Specific issues to be investigated include `algebraic stability' for degree growth of maps, the manufacture and intersection of dynamically natural closed currents to produce invariant measures, and combinatorial `train track' models for the dynamics of real automorphism on rational surfaces. 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.