The recent surge of results in the geometry and topology of 3-manifolds has provided many new tools for understanding 3-manifolds combinatorially and geometrically. In particular, the question of how to gain an explicit understanding of the internal geometry of a closed hyperbolic 3-manifold can be addressed with new techniques developed by the P.I. together with Dick Canary and Yair Minsky that apply to the infinite volume case. Such an understanding would be a kind of effective version of Mostow rigidity, wherein one not only knows the uniqueness of the hyperbolic structure but additionally an explicit decription of its geometry. The P.I., in joint work with Juan Souto, seeks to develop this kind of picture for closed hyperbolic 3-manifolds admitting a Heegaard splitting, given in terms of the Heegaard surface. Additionally, with Howard Masur and Yair Minsky the P.I. will relate the internal geometry of hyperbolic 3-manifolds homotopy equivalent to a surface to the geometry of surfaces along a Weil-Petersson geodesic G in Teichmueller space.
A recent trend in the study of geometry and topology is to develop combinatorial models for geometric spaces. This kind of description of a space or shape sacrifices a certain degree of precision in the interest of capturing more of the large-scale structure, and often general theorems guarantee that knowing this large-scale structure is sufficient to completely determine the space. In a recent result of the P.I. with R. Canary and Y. Minsky, such models were used to classify all `constantly negatively curved,' or `hyperbolic' 3-dimensional spaces of infinite volume that are nevertheless tame in a certain sense. This result solved a long-standing conjecture of William thurston, where in a certain piece of data (akin to a kind of DNA-sequence for the space) completely determines its structure. We have developed a similar setup in the finite-volume case, and hope to prove a similar classification theorem for such spaces. Information of this large-scale type is more useful than existence results for geometric structures, in that it gives one a more complete picture of how such spaces behave. Such large-scale data arise and describe phenomena in many contexts, whether it is the spaces themselves, or their parameter spaces. The P.I., together with his collaborators, will continue to develop a complete description of the large-scale geometry of all hyperbolic 3-dimensional manifolds, as well as for related spaces that parameterize them.
