Principal Investigator: Yair Minsky
The PI proposes to investigate a number of different aspects of hyperbolic geometry in 2 and 3 dimensions, the structure of the representation spaces that parametrize these geometric systems, and the dynamics of the groups that act on them. The SL(2,C)-character variety, X(F), of a group F parametrizes conjugacy classes of representations of F into SL(2,C), the isometry group of hyperbolic 3-space. Discrete faithful elements of X(F) correspond to hyperbolic 3-manifolds, and the PI will work to understand better the interaction between topological and geometric features of these manifolds, as well as the structure of the discrete-faithful locus itself. In addition, he will pursue a study of X(F) as a whole, considered as a dynamical system under the action of the outer automorphism group of F (particularly when F is a free group). A new dynamical decomposition of X(F) was recently discovered, whose structure seems to be rich and relatively unexplored. In addition the PI will study geometric aspects of Teichmuller spaces, which parametrize hyperbolic structures in two dimensions, and of Mapping Class Groups, their natural automorphism groups.
The interactions between geometry, topology and dynamics play a entral role in mathematics as well as its applications. A geometric space, such as our own universe or the configuration space of some system, may admit dynamical phenomena such as flows, iterations or group actions. The behavior of these phenomena, as well as the geometry of the space, can be strongly influenced by its topological structure, namely the underlying connective tissue on which the geometry is overlaid. Furthermore, we often find that geometry and dynamics persist at higher levels of abstraction: the collection of all geometric structures on a given space can itself be organized into a new "higher" space, with its own geometry and its own inherent symmetries which give rise to dynamical structure. The interaction between these phenomena at different levels can enrich our insight about the original systems. The PI's own research focuses on particular instances of this general template, namely the geometry and topology of 2- and 3- dimensional spaces, and the corresponding dynamics for their higher parameter spaces. This low-dimensional setting is particularly amenable to our intuition and is partly motivated by direct visual analogies with our physical world, but it also happens, for a variety of reasons, to be a meeting place for a number of different areas of mathematics as well as physics, so that a fuller understanding in this domain can enrich, by analogy as well as direct mathematical connection, our approaches to other parts of mathematics. This award is to be funded jointly by the programs in Topology, Geometric Analysis, and Analysis.
The interactions between geometry, topology and dynamics play a central role in mathematics as well as its applications. A geometric space, such as our own universe or the configuration space of some system, may admit dynamical phenomena such as flows, iterations or group actions. The behavior of these phenomena, as well as the geometry of the space, can be strongly influenced by its topological structure, namely the underlying connective tissue on which the geometry is overlaid. This project focussed on phenomena like these in the context of 2- and 3-dimensional spaces and their higher dimensional parameter spaces. Some outcomes of this project include an improvement of our understanding of deformation spaces of hyperbolic 3-manifolds, including a convergence theorem that governs the process in which an infinite sequence of structures approaches a limiting structure. Another result was a study of Thurston's "skinning map", which is the basis of an iterative process used by William Thurston to construct and glue together hyperbolic structures on 3-manifolds. Another outcome of the project was a gluing theorem that provides information on the relations between geometry and topology for new classes of hyperbolic 3-manifolds (including for example some manifolds with infinitely generated fundamental group, a class for which some of the previous theory does not apply). One of the PI's students studied parameter spaces of structures on 2-dimensional manifolds (surfaces), and the delicate flow (called the Weil-Petersson geodesic flow) that arises as the structures are deformed in certain ways. Another worked on refining our understanding of the combinatorial structure of configurations of curves on surfaces, and how these connect to the study of topology in 3 dimensions. The field of low-dimensional topology and geometry is undergoing a sort of pivot, in which many of the central problems originating with Thurston in the late 1970's have been solved, but where the solutions have produced new connections and new techniques. This is a period in which refinements to our current techniques, and extensions to broader settings, are becoming possibilities, and our focus is shifting in new directions. The work in this award contributes to some of this shift. The visual nature of work in low dimensions lends itself to educational efforts, and in view of this the award was also used to support a public lecture series at Yale for local families and school-age children. Topology and geometry played a major (but not exclusive) role in these lectures and contributed to Yale's outreach efforts bringing mathematics and science to the local community.
