CAREER: Algebraic structures in complex dynamics

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

The simplest yet non-trivial complex dynamical systems are algebraic: iteration of noninvertible polynomials in one complex variable and, more generally, rational self-maps of a complex algebraic variety. Certain analytic quantities associated to these holomorphic systems have surprising connections to algebraic constructions. Examples from this project that illustrate the rich algebraic structures in a complex dynamical setting include: (1) the study of one-dimensional polynomials by their associated trees and the connection with valuations on the field of regular functions on the moduli space; (2) an analysis of regular self-maps on toric varieties, the resultant of such mappings, and connections with pluri-potential theory; (3) the structure of the moduli spaces of regular self-maps (e.g. of projective spaces) and their "dynamical" compactifications (describing how such systems degenerate) as algebraic varieties. One of the most familiar examples of such a dynamical system is Newton's method, an iterative algorithm for finding roots of polynomials, first introduced in calculus courses. Even for this famous and seemingly simple example, we have only recently begun to understand its structure, and its failure in general, using modern mathematical techniques. With Newton's method (its history, its applications, and recent studies) as a guide, I will pursue a collection of educational projects for students: (1) undergraduate research projects, with an emphasis on computer exploration of examples; (2) the development of an undergraduate course in dynamics, to present both the theory and recent applications; (3) two workshops for graduate students; and (4) regular student seminars. The research described above is designed to explore the interplay between chaotic dynamical systems (such as Newton's method and its generalizations) and any underlying algebraic structures. Such structures add an extra element of symmetry or regularity to a system which might otherwise seem intractable. The experts in these aspects of complex dynamical systems are concentrated in France, England, Japan, and the US (with smaller groups in Chile, Spain, Poland, and Germany, to name a few); with this project, the Principal Investigator will bring together some of these experts to work with students and clarify these connections between algebra and dynamics.