Mapping Class Groups, Branched Covers, and Rational Maps

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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.