Principal Investigator: Christopher J. Leininger
This project aims to study the geometry of various structures on surfaces, actions of the mapping class group on these spaces, and topological/dynamical aspects of surface homeomorphisms. This includes (1) an ongoing project with R.P. Kent on convex cocompactness in the mapping class group, with a focus on free groups and surface groups; (2) a topological investigations of algebraic relations in the mapping class group and the closely related braid groups with D. Margalit; (3) a study, via geodesic length functions, of certain singular euclidean geometric structures on surfaces and their degenerations in a joint project with M. Duchin and K. Rafi; (4) a continuing project with B. Farb and D. Margalit to provide a topological model for "minimal complexity" surface homeomorphisms.
Surfaces---like the surface of a ball or a doughnut---have been studied for hundreds of years, and are fundamental and beautiful objects in mathematics. The study of surfaces is intrinsically interesting, but is also responsible for the creation of entire fields of mathematics, as well as the development of techniques in many others. As such, the theory of surfaces and their geometries lies at the juncture of several fields of mathematics including complex analysis, differential geometry, low-dimensional topology, geometric group theory and dynamics. Many geometric objects can be described using surfaces as the basic building blocks. To study these objects one naturally encounters the notion of a "homeomorphism" of a surface: this is a kind of "symmetry" which preserves only the most basic properties of the surface. The set of all homeomorphisms is somewhat unwieldy, but the most important features can be distilled into a more manageable structure called the "mapping class group." This project proposes the study of several related problems about surfaces, their homeomorphisms and mapping class groups, and the implications of these studies to related geometric objects.
