CAREER: Coarse geometry and quasimorphisms

University Of Oklahoma Norman Campus, Norman OK

Investigators

Abstract

This project represents a continuing effort of the PI to expand our knowledge of surface theory. It also involves the training of graduate students to become research mathematicians. A surface is a two-dimensional space, like the surface of a ball or a saddle, or more abstractly, a surface is the evolution space of a string moving in space-time. The study of surfaces is a classical but still vibrant area of research, in mathematics and in physics. A surface can take on many geometric shapes. Teichmuller theory is the study of all the variable shapes a surface can have. The PI is particularly interested in studying how the shapes can change by deforming certain one-dimensional curves on the surface. She is also interested in investigating how a surface can sit inside a space of higher dimension. The tools she will employ come from various areas of mathematics, such as hyperbolic geometry, dynamics, and topology. The educational component involves organize a series of intense workshops, departmental seminars, a yearly public symposium in mathematics and a literacy course in geometry and topology. The PI will continue her research in Teichmuller theory from the perspective of the Thurston metric. This is an asymmetric Finsler metric defined on Teichmuller spaces, using the hyperbolic lengths of geodesic laminations on a surface and Lipschitz maps between surfaces, as opposed to using measured foliations and quasiconformal maps which give rise to the Teichmuller metric. This metric was introduced by Thurston over thirty years ago but it has not been studied extensively until recently. It has a distinctive and rich structure that is already apparent in two-dimensional Teichmuller space. In this case, the PI and her collaborators have developed a clear picture of the infinitesimal and coarse geometry of this metric. The PI plans to extend these results to higher dimensional Teichmuller spaces as well as explore dynamics of the Thurston metric. The PI will also study stable commutator lengths via quasimorphisms on right-angled Artin groups, right-angled Coxeter groups, and more generally, virtually special groups. Plans to organize graduate student workshops dedicated to these topics and related topics are also included.