The PI plans to investigate aspects of the interaction of topology and geometry in 2 and 3 dimensions, focusing in particular on the relations between topological and geometric complexity. A primary feature in this investigation is a certain localization of complexity that occurs in maps of surfaces, and has deep relations to the theory of 3-manifolds and their hyperbolic structure, as well as to the geometry of the Teichmuller and moduli space of conformal structures on a surface. The structure of the complex of curves on a surface plays a role in all these settings, which the PI plans to continue investigating and refining. Particular goals include genus-independent versions of existing theorems on the complex of curves, or versions where genus dependence is explicit; quantitative control of Thurston's skinning map; and better understanding of other settings where gluing maps determine the structure of hyperbolic 3-manifolds. Particular attention will be paid to the compressible-boundary case. In Teichmuller theory, the PI will investigate the behavior of Weil-Petersson geodesic flow and its relation to combinatorial invariants arising from the complex of curves.
Geometry and topology play a fundamental role in mathematics. Geometric spaces arise in many settings, either as explicit objects we visualize directly, or via abstractions such as configuration spaces of systems or parameter spaces. The topology of a space is the way it is combinatorially put together without regard to geometry, but often these topological descriptions suffice to determine the geometry uniquely. This phenomenon is called rigidity and plays a central role. In low dimensions there is a confluence of many different aspects of mathematics which interact with rigidity, including analysis, complex analysis and dynamics. Moreover, the visualizability and accessibility of the low-dimensional setting brings many subtle phenomena into focus and serves as a testing ground for new mathematical ideas. Within this setting we propose to investigate a number of quantitative aspects of rigidity, particularly ways in which complexity in the topological description translates to features of the geometry.
