Geometry, groups, and dynamics

William Marsh Rice University, Houston TX

Investigators

Abstract

The shape of an object can influence a wide variety of data observed for that object. An important mathematical problem is to determine the extent to which what is observed determines the shape of an object. More generally - what mathematical features of a shape can one determine from the observables? The principal investigator will study both concrete and abstract problems of this nature. For example, suppose a particle is enclosed in a polygon and is traveling in a straight line, bouncing off any side it encounters. The shape of the polygon could be inferred by observing the sequences of sides encountered by the particle. The principal investigator will study the extent to which the shape of the region is determined by these observables. More abstract objects studied in these projects involve algebraic systems probed using "cross-sectional images" analogous to those recorded by an MRI. These cross-sectional images can be fit together to completely reconstruct an object. The principal investigator will develop techniques for predicting certain properties without carrying out the reconstruction. Using these approaches will also allow one to develop problem solving techniques and methods that could find applications in concrete settings. The principal investigator will investigate a variety of problems, each of which is connected, either directly or indirectly, to surfaces through geometry, group theory, and dynamics. The research activities are organized into four themes. (1) Studying the large-scale geometry of surface bundles. The goal is to determine hyperbolicity properties of surface bundles from the geometric data of their associated monodromy representation to the mapping class group. (2) Probing geometric and dynamical properties of free-by-cyclic groups from the cross section of flows on 2-complexes, and pursuing a fruitful analogy with fundamental groups of surface bundles over the circle. (3) Analyzing singular Euclidean metrics on surfaces and understanding the extent to which the limited data encoded by the support of its associated Liouville current determines the geometry of the metric. This is directly related to studying the extent to which the symbolic dynamics of billiards in polygons determines the shape of the polygon. (4) Studying the geometry at infinity of spaces of structures on a surface, specifically the Teichmueller space, the curve complex, and variants of these spaces. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.