LEAPS-MPS: Limits in Mating and in Several Complex Variables

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).Mathematics often uses simpler functions (such as polynomials) to approximate the behavior of more complicated functions. The area of dynamical systems studies the behavior of iterated functions, in which a function such as a polynomial is applied repeatedly. One basic dichotomy is the distinction between the set where these iterates get larger and larger and the set where they stay relatively small. For a polynomial, the latter is called the filled Julia set. In some cases, a technique called mating can be used to glue together the filled Julia sets for two polynomials. One aspect of this project will look at limits of matings to see if it is possible to construct a similar gluing for exponentials. A second aspect of this project will look at approximations in higher-dimensional spaces to extend one-dimensional approximation results to that context. In this project, the PI will recruit undergraduate and graduate students from underrepresented minorities for research opportunities and travel to demographically relevant conferences, which will provide community support and information on opportunities for future education or employment. Additionally, the project will provide remedial and enrichment activities for middle- and high-school students through a branch of the Math Corps program, which has a long history of creating a community around mathematical excellence among underserved students. The program will recruit students from local middle schools that contain large populations of students from underrepresented groups, with high levels of economic need and low levels of performance in mathematics. This project provides two new directions to extend work on polynomial approximations of transcendental functions. The first direction is to extend into the area of matings. Mating provides a method to combine the dynamics of two polynomial functions together to obtain a rational function with characteristics of both polynomials. Most examples of matings come from quadratic or cubic polynomials, due to computational difficulties. The project will use a new criterion and the special structure of the polynomials in the family to give significant new examples of matings of arbitrarily large degree. The examination of the limits of these matings will use recently developed tools for transcendental functions to identify the limit, if it exists. The eventual goal is to develop a theory of mating for transcendental functions. The second direction is to generalize examples and results for limits of functions of one complex variable to limits of functions of several complex variables. In this new context, there are three different types of degree (algebraic, topological, and dynamical). The examples will illustrate how the limits interact with the different types of degree for maps of several complex variables. Eventually, an improved understanding of different examples could help answer open questions, such as which values can be attained as a first dynamical degree. Additionally, our examples will lead us to prove more general theorems, for example, showing when the convergence of the functions implies the convergence of the Julia sets. By looking for families with constrained critical behavior, it may be possible to classify the periodic and preperiodic Fatou components that appear in the families or in the limits, as a first step toward completing such a classification for transcendental functions of several complex variables.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.