Approximation Theory and Complex Dynamics

Data management plan for the grant, "Approximation Theory and Complex Dynamics." This project involves the study of approximation theory in the setting of complex functions, with applications to complex dynamics. Approximation theory seeks to understand the extent to which the behavior of a general function can be effectively modeled by that of functions drawn from a more restricted class. Efficient approximation of functions is of relevance for numerical calculation. Since the only calculations that can be carried out numerically are the elementary operations of addition, subtraction, multiplication, and division, in practical terms it is of importance to understand when the values of general functions are well approximated by the values of either polynomial or rational functions. In many situations, the values of the approximant resemble those of the general function only for a sampling of input values. What can be said about values of the approximant for other choices of input? This is the main question studied in this project, with the following application in mind: when a general function is iterated to produce a dynamical system, to what extent does the dynamical behavior of an approximant resemble the dynamical behavior of the original function? The project will also contribute to the development of human resources through educational outreach at the high school level as well as mentoring and training at the undergraduate and graduate levels, and will facilitate the interaction of different fields of mathematics through the organization of conferences and seminars