Topology and Dynamics of Geometric Structures

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This project concerns the classification of geometric structures on manifolds. A "geometric structure" means a set of coordinate systems where the coordinates live in a space with a "classical geometry:" examples include Euclidean geometry, non-Euclidean geometry, projective geometry, affine geometry or conformal geometry. Since the mid 19th century, considerations of symmetry emphasize the viewpoint that a classical geometry is just the study of properties which are unchanged under the transitive action of a Lie group. Closely related is the importance of symmetry in modern physics, and many of objects in this investigation have physical motivations. Classification of geometric structures on a fixed topology means putting the more rigid geometric measurements on a loosely organized collection of points, which is the topology. A simple example is the fact that the sphere cannot support Euclidean geometry. This is just the mathematical abstraction of the fact that there is no metrically accurate world atlas. At least one page of the world atlas will distort distance. On the other hand the torus (the surface of a doughnut, bagel or inner tube) does admit Euclidean structures. In fact the different ways of putting Euclidean geometry on a torus has a rich geometry of its own. Technology plays a key tool both in this investigation and its dissemination to the public. Many mathematicians are involved in this project, including colleagues, postdocs, graduate students, undergraduate students and even high school students. The objects of study are often two- and three-dimensional and can be visualized with the use of computers. All the software developed in the project is publicly available and invites further education and experimentation in this research project.While the traditional methods of classification of geometric structures involve a quotient space, frequently the moduli spaces are pathological. Simple examples lead to quotients of nice spaces by chaotic group actions. The PI adopts the viewpoint that classification of geometric structures is really a dynamical system. Many well known interesting dynamical systems arise from this context. In particular the extension of the Teichmuller geodesic flow to the universal character variety over the Riemann moduli space has rich dynamics, impacting low-dimensional geometry, topology and mathematical physics. The PI will study this dynamical system as the fusion of the well-developed subject of Teichmuller dynamics and his previous work on mapping class group dynamics on character varieties and moduli spaces of geometric structures.