The Weil-Petersson Geodesics on the universal Teichmuller space

After a quick review of the classical theory of the universal Teichmuller space, we shall discuss the problems it poses: the lack of being a topological group, the divergence of the formula for the Weil-Petersson metric introduced in 1990 by Nag and Verjovsky. Then, the solution to these problems due to Takhtajan and Teo will be presented. This then naturally leads to the the study of the Weil-Petersson geodesics and their fundamental properties will be presented. The regularity of the elements in the universal.Teichnmuller space will also be discussed since it raises interesting questions about the product and composition of sobolev class maps below critical index. Time permitting, an application to computer vision will be presented where the long time existence of geodesics is needed to find a representative in a homotopy class of maps sending one quasi-circle to another.