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

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。数学通常使用更简单的函数（如多项式）来近似更复杂函数的行为。动力系统领域研究迭代函数的行为，其中一个函数如多项式被反复应用。一个基本的二分法就是迭代值越来越大的集合和迭代值保持相对较小的集合之间的区别。对于多项式，后者称为填充Julia集合。在某些情况下，可以使用一种称为配对的技术将两个多项式的填充Julia集合粘合在一起。这个项目的一个方面将着眼于配对的极限，看看是否有可能为指数构建类似的粘合。本项目的第二个方面将着眼于高维空间中的近似，将一维近似结果扩展到该上下文中。在这个项目中，PI将从代表性不足的少数民族中招收本科生和研究生，以获得研究机会，并前往参加与人口统计学有关的会议，这些会议将提供社区支持和关于未来教育或就业机会的信息。此外，该项目还将通过数学队项目的一个分支为初高中学生提供补习和充实活动，该项目在为缺课学生创建数学优秀社区方面有着悠久的历史。该项目将从当地中学招收学生，这些中学中有大量来自弱势群体的学生，他们的经济需求很高，数学成绩很差。本课题为超越函数的多项式近似的研究提供了两个新的扩展方向。第一个方向是延伸到交配领域。配合提供了一种将两个多项式函数的动力学性质结合在一起，从而得到一个同时具有两个多项式特征的有理函数的方法。由于计算困难，大多数匹配的例子来自二次多项式或三次多项式。该项目将使用一个新的准则和族中多项式的特殊结构来给出任意大程度配对的重要新例子。检查这些匹配的极限将使用最近开发的超越函数工具来识别极限，如果它存在的话。最终的目标是发展一种超越函数的配对理论。第二个方向是将单复变函数的极限的例子和结果推广到多复变函数的极限。在这个新的背景下，有三种不同类型的度（代数、拓扑和动态）。示例将说明极限如何与几个复杂变量映射的不同类型的度相互作用。最终，对不同例子的更好理解将有助于回答一些悬而未决的问题，例如哪些值可以作为一级动力学度获得。此外，我们的例子将引导我们证明更一般的定理，例如，说明函数的收敛何时意味着Julia集合的收敛。通过寻找具有约束临界行为的族，有可能对出现在族或极限中的周期和前周期法图分量进行分类，这是完成对多个复杂变量的超越函数进行分类的第一步。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.