A Topological Approach to Questions in Dynamical Systems

University Of Alabama At Birmingham, Birmingham AL

Investigators

Abstract

Mathematical models are widely used to understand the world around us and to predict how changes in the parameters will affect the future. Understanding the long-term behavior of models is an important scientific endeavor. In dynamical systems, a research sub-area in mathematics, one studies such mathematical models and attempts to understand their long-term behavior as well as consequences of changes in parameters. A standard approach to studying complicated mathematical models is to perturb them to simpler models and to show that certain conclusions for the simpler models are also valid for the more complicated models. Research funded by this award will contribute to the solution of three main problems that address basic aspects of dynamical systems and offer insights for the creation of new models. Many mathematical models are based on homogeneous spaces. In recent work, the investigator and his collaborators provided a complete classification of homogeneous plane continua, solving a problem that dates back to the 1920s. In this project these techniques will be exploited to address classification of all one-dimensional homogeneous continua. Parameter spaces are in general extremely complicated and very little is known about them. Thurston showed that the parameter space of complex polynomials of degree two can be modeled by the space of degree two invariant laminations. This combinatorial model is simpler than the Mandelbrot set and is completely described. Every complex polynomial of degree two corresponds to a single point in this combinatorial model and its location in the combinatorial model determines the topological dynamics of the polynomial. In another direction, the investigator will focus on understanding the topology and geometry of the spaces of invariant laminations of higher degrees, which is the first step in extending Thurston's work. 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.