Grant: Discovery Launch Supplement

The long term vision for my research program is to understand the geometry of the mapping class group, Map(S), the group of symmetries of a surface where a surface is a 2-dimensional shape. An example of a surface is the glaze of a ring donut, where the hole of the donut is called the genus of the surface. Over the next five years, I will focus on three key objectives that will contribute to my overarching vision. My first objective is to provide a presentation of Map(S) using involutions, which are elements of order two meaning that applying the element twice results in the identity. A generating set is a subset of the group where any element of the group can be expressed as a finite combination of the elements in the generating set. A presentation is a generating set together with a set of relations. This work will connect two different subfields of mathematics, namely, it will help determine a relationship between Map(S) and the Coxeter group. My second objective is to understand whether pseudo-Anosov elements are the typical elements in the case of the mapping class group of a punctured disk where each puncture stays fixed point-wise. The pseudo-Anosov mapping classes mix the underlying surface in a complicated way. For example, a taffy pulling machine can be mathematically visualized as a pseudo-Anosov map on disk with three punctures. By considering the word metric, which is the minimal number of generators required to write a group element, a long-standing open problem asks whether pseudo-Anosov mapping classes are generic in Map(S). Generic means that the proportion of pseudo-Anosov elements in a ball of radius R tends to 0 as R tends to infinity. As we can visualize each puncture as a rod and the disk as the surface of a fluid, this work can be applied to questions of how to efficiently mix a fluid. Hyperbolic groups are fundamental objects in the field of geometric group theory. Surfaces can either be finite type, such as surfaces with finitely many genus, or infinite type, such as surfaces with infinitely many genus. A group is hyperbolic if triangles in the group are delta-thin, meaning that a point located on any edge of the triangle is a maximum distance of delta away from the other two edges. For finite type surfaces, the mapping class group is never hyperbolic if the genus is greater than one. My final objective will provide an explicit classification of which mapping class groups of infinite type surfaces are hyperbolic, providing new examples of hyperbolic groups with an uncountably infinite generating set. This work is significant since it aims to connect various areas of mathematics, solve long standing open problems, and provide a classification theorem for MCG(S) where S is of infinite type, each allowing us to further study the underlying geometry of the group.