The principal investigator plans a research program for studying Teichmuller theory, a fundamentally important topic in geometry, by way of its connections and interactions with several other mathematical disciplines. Central to the fields of geometry and topology is the study of surfaces, for example the surface of a spherical solid like a planet, or the surface of a donut or of a more complicated three dimensional solid. A key objective in Teichmuller theory is to understand all possible geometric forms that a fixed type of surface can admit. For example, an astrophysicist might be interested in cataloging all possible topographies of a hypothetical planet. The possibilities are countless: there could be hilly or mountainous terrains, highlands, valleys of varying elevations, and so forth. Miraculously, Teichmuller theory gives a way to package the entire plethora of possible topographies into one geometric object which can then be studied. The practical applications of Teichmuller theory and of its mathematical relatives abound, from computer visualization and graphics design, to evolutionary biology and genetics, and to many other important disciplines in between.
In more technical detail, the principal investigator proposes a three part plan for studying Teichmuller and hyperbolic geometry, the mapping class group, and more general classes of finitely generated groups and metric spaces. First, the PI plans to develop dynamical and combinatorial tools for studying the Teichmuller space, the mapping class group, and hyperbolic 3-manifolds, and to use these tools to analyze fundamental questions at the intersection of combinatorics and dynamics, such as lattice point counting problems, and the study of hyperbolic 3-manifolds from a combinatorial perspective. For example, the PI plans to demonstrate relationships between the curve complex of a surface S and the geometry of a hyperbolic 3-manifold fibering over S, in such a way that is sensitive to the topology of the underlying surface S. Next, the PI will generalize and extend these tools to other groups and spaces, such as the outer automorphism group of the free group and the Outer space; for instance, the PI will initiate a study of metrics on moduli spaces of graphs that are inspired by the Weil-Petersson metric on Teichmuller space. Finally, the PI plans to pose analogs of counting and other types of dynamical problems in a wider class of groups and of spaces, and use the generalized tools to attack these questions in their respective contexts.
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.
