Mapping Class Groups and Polynomials

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The main goal of this project is to study surfaces and their symmetries. A surface is a two-dimensional space, in other words, a two-dimensional version of the world we live in. Surfaces come in many shapes (for instance the surface of a ball is different from the surface of a doughnut) and they arise in many varied contexts, from physics to robotics to data analysis to quantum field theory. The symmetries of a surface form a beautiful and rich theory that has been the focus of intense study over the past century. The close connection between those symmetries and hyperbolic geometry facilitates the investigation of some questions that are not obviously geometric, such as algorithms for working with these groups.The mapping class group of a surface is the group of homotopy classes of orientation-preserving homeomorphisms of the surface. Among other things, the mapping class group encodes the outer automorphism group of the surface fundamental group, the (orbifold) fundamental group of the moduli space of the surface, and the isomorphism types of surface bundles over arbitrary spaces. The mapping class group also has connections to many, many areas of mathematics, including dynamics, group theory, number theory, quantum field theory, representation theory, and algebraic geometry, just to name a few. Goals of this research program include these: (1) Establish a quadratic-time algorithm for the conjugacy problem in the mapping class group. Work done under prior NSF support developed a quadratic-time algorithm for the Nielsen-Thurston type of a mapping class, and an extension of that theory is expected to give similar speed for the conjugacy problem. (2) Understand polynomials from the point of view of mapping class groups, including a new approach to the question of which branched covers of surfaces come from polynomials. (3) Describe the structure of an arbitrary normal subgroup of the mapping class group. For instance, descriptions are available of the normal closures of many types of elements, and new examples have been found of finitely generated right-angled Artin subgroups.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.