Mapping Class Groups, Branched Covers, and Rational Maps

多项式不仅仅是方程，它们是描述运动和变化的复数的函数。 通过重复应用多项式，可以创建动态系统，可以模拟天气模式，经济或生态变化。 但是，还有许多描述动力系统的函数不能用简单的公式给出，因此研究起来要困难得多。 通过调整用于研究表面对称性的技术，主要研究员将研究这些函数的动力学特性。 首席研究员还将继续她的推广工作与小学和中学生，本科生的研究，并在研究生的专业发展计划的领导。这些项目的重点是映射类组，这是一组同胚的表面到同伦。 随着使用技术从覆盖空间理论和几何群论的统一主题，这个议程有两个广泛的方向，一个动力学和一个代数。 在动力学方向，计划中的研究将致力于开发一种算法，以确定何时球体的分支覆盖等同于有理函数。 首席研究员和她的合作者已经开发了一个类似的算法多项式使用技术的研究映射类组，这些调查将适应现有的算法，以某些家庭的分支覆盖的领域。 在代数方向，计划的研究旨在了解代数结构和表示的映射类组的子群规范化有限的组action.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

Polynomials are more than equations, they are functions of complex numbers that describe motion and change. By repeatedly applying polynomials, one creates dynamical systems, which may model weather patterns, economics, or ecological changes. But there are many other functions that describe dynamical systems that cannot be given by simple formulas, and are therefore much more difficult to study. By adapting techniques used to study the symmetries of surfaces, the principal investigator will study the dynamical properties of such functions. The principal investigator will also continue her outreach efforts with elementary and middle-school students, undergraduate research, and leadership in professional development programs for graduate students.These projects focus on the mapping class group, which is the group of homeomorphisms of a surface up to homotopy. With a unifying theme of using techniques from covering space theory and geometric group theory, this agenda has two broad directions, one dynamical and one algebraic. In the dynamical direction, the planned research will work towards developing an algorithm to identify when a branched cover of the sphere is equivalent to a rational function. The principal investigator and her collaborators have developed an analogous algorithm for polynomials using techniques from the study of mapping class groups, and these investigations will adapt the existing algorithm to certain families of branched covers of the sphere. In the algebraic direction, the planned research seeks to understand the algebraic structure and representations of subgroups of the mapping class group that normalize a finite group action.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.