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函数。通过反复应用多项式，可以创建动态系统，它可以模拟天气模式、经济或生态变化。但是，还有许多其他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。通过采用用于研究表面对称性的技术，首席研究员将研究这些函数的动力学性质。首席研究员还将继续在中小学生、本科生研究和研究生专业发展项目中发挥领导作用。这些项目的重点是映射类群，映射类群是一个曲面到同伦的同胚的群。该议程以空间理论和几何群论的技术为统一主题，有两个广泛的方向，一个是动态的，一个是代数的。在动力学方向上，计划的研究将致力于开发一种算法来识别球体的分支覆盖何时等同于有理函数。首席研究员和她的合作者利用映射类群研究的技术开发了一种多项式的类似算法，这些研究将使现有算法适应于球体分支覆盖的某些族。在代数方向上，计划的研究试图理解正则化有限群行为的映射类群的代数结构和子群的表示。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。

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.