William P. Thurston - US grants

9704135 Thurston This project is a multi-pronged investigation into low-dimensional geometry and topology with the central long-range goal of supporting and establishing the Geometrization Conjecture, namely, that every 3-dimensional manifold has a canonical decomposition into pieces with locally homogeneous Riemannian metrics. The first of these prongs is an investigation, partially aided by computer, of geometric Dehn filling spaces for 3-manifolds. Geometric Dehn fillings interpolate between manifolds with distinct topology. A suitable variation of this idea has the potential to connect all possible 3-manifolds and equip them with geometric decompositions. The second prong is an investigation of the geometry of taut foliations and essential laminations and their relationships to hyperbolic structures and other geometric structures. This approach has the promise to give a common generalization of the canonical decompositions of surface homeomorphisms developed by Thurston in 1976, and the geometrization of Haken manifolds developed by Thurston in the late 1970's and early 1980's. Additional prongs of this project involve contact structures, confoliations, and geometric group theory. In addition, the project will explore connections of low-dimensional geometry and topology to biology, particularly genetics. Low-dimensional geometry and topology is a beautiful area of mathematics that has undergone tremendous growth and transformation over the last two decades. Paralleling this internal development of the subject, lagging slightly behind, has been the external development of new connections and strengthened connections to other areas of mathematics and of science. Some of these connections are direct, like the cosmological issue of the shape of our universe or the more down-to-earth issue of the shapes of crystals. Other connections are indirect, to diverse topics such as databases, group theory, genetics, or computational chemistry. Any topic tha t can be formulated in a quantitative way can be studied with geometric methods (alongside the more prevalent symbolic, algebraic and analytic methods), so that in any particular case, the powerful discoveries and tools of low-dimensional geometry and topology have a reasonable chance to have a direct bearing. The aim of this project is further development of the tools of low-dimensional geometry and topology to make them more portable, more powerful and more universal. ***

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).

Thurston will investigate several areas of low-dimensional geometry and topology and their interconnections with other areas of mathematics and science. He will continue a collaboration with Allen Hatcher to analyze the topology of the space of branched polymers, extending the result that for configurations of more than or equal to 8 equal-size atoms the fifth homotopy group has high rank. Thurston will extend joint work with Hass and Thompson to investigate the global geometry of Heegaard splittings, bridge number of knots, generating sets for groups, as well as a continuous generalization, bridge measure for geodesic flows. Thurston will also investigate the geometry of the space of approximately finite k-generated groups: groups whose Cayley graphs are limits of Cayley graphs of finite groups. The space of approximately finite k-generated groups is compact, and important closed subsets are countable. This space gives insight into finite quotients of finitely-generated groups, and the dual property of residual finiteness.

Mathematician's understanding of 2-dimensional and 3-dimensional spatial phenomena has undergone a dramatic revolution culminating in Perelman's solution to Thurston's geometrization conjecture (which includes the famous Poincare conjecture), giving beautiful geometric answers to questions far beyond the wildest dreams of 40 years ago. These geometric insights and tools developed during this revolutionary change are understood mainly within a specialized community, but Thurston is interested in the strong potential for extending their explanatory power beyond three-dimensional topology into other domains, both inside and outside mathematics proper. One initiative is to analyze the space of branched polymers, an idealized theory that exhibits some interesting and unexpected topological phenomena. We hope the idealized topological theory will ultimately contribute to understanding real molecules. Another initiatives in this project involve the geometry of the space of all possible finite groups. Finite groups are pervasive throughout mathematics and science as descriptors of the symmetries both visible and hidden that shape our world. The predominant (and powerful) approach to finite groups is through tools of algebra and representation theory. We will investigate the geometry of finite groups to elucidate phenomena that are not readily seen from the algebraic point of view.