This project aims to study problems involving surfaces and three-manifolds, primarily from the perspective of group theory and geometry, but also drawing on ideas and techniques from complex analysis and dynamics. This includes (1) an ongoing project with R.P. Kent on convex cocompactness in the mapping class group, various parts of which are also joint work with M. Bestvina, J. Brock, K. Bromberg, and S. Dowdall; (2) ongoing work with D. Margalit on dynamics of pseduo-Anosov homeomorphisms and their relation to three-manifolds, parts of which are also joint work with B. Farb; (3) various projects on length functions and geometric structures on surfaces which involves work with M. Duchin and K. Rafi, as well as work done by the PI's graduate students, A. Bankovic, S.-W. Fu, R. Maungchang, and C. Uyanik.
Surfaces - two-dimensional spaces 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, shaping the way people solve problems. Three-dimensional topology - the study of three-dimensional spaces like our physical universe - has been profoundly impacted by developments in the study of surfaces. This may not seem surprising since a surface is a two-dimensional space, and one might expect a three-dimensional space to enjoy some of the same properties. While it is true that direct analogies in the situations have produced some interesting results, it is the use of surfaces as building blocks for understanding three-manifolds (the three-dimensional spaces) which has most significantly shaped the field. One important aspect of this project is the strategy of "reversing the flow of information" by (1) applying new technology developed to study three-manifolds into the theory of surfaces, rather than vice-versa, and (2) exploring the subtle effects the geometry of three-manifolds imposes on the geometry of the surfaces used to construct them.
