Teichmuller Theory and Low-Dimensional Topology

Teichmüller theory originated from the extensive work of Teichmüller on quasi-conformal deformations of complex structures on Riemann surfaces in the 1930’s-40’s. Since then, it has developed enormously, making a great impact on both topology and complex analysis. In particular, Teichmüller theory helped the progress of low-dimensional topology in many ways. Let us give just a few examples. Recent development of three-dimensional topology owes much to Thurston’s far-reaching insights, in particular his work on uniformisation of Haken manfolds and his famous conjectures on Kleinian groups. Both his uniformisation of Haken manifolds and recent resolutions of his conjectures on Kleinian groups made full use of Teichmüller theory. Also, in recent studies on asymptotic geometry of mapping class groups, initiated by Gromov, Minsky, Bowditch and Bestvina among others, deep understanding of Teichmüller spaces with various metrics, such as the Teichmüller metric, the Weil-Peterson metric, and Thurston’s asymmetric metric, was indispensable. On the other hand, topologists have invented generalisations of Teichmüller theory in several ways. There are two which are very important among them. One is what is called quantum Teichmüller theory, which was started by Chekhov, Fock, Bonahan, Kashaev etc. It is expected to give a new direction in the study of quantum invariants of 3-manifolds making use of Teichmüller theory. The other is higher Teichmüller theory. This deals with the spaces of representations of surface groups into higher dimensional Lie groups, which turns out to be a very rich subject. These generalisations show that Teichmüller theory still has inexhaustible sources which continue to stimulate the further development of low-dimensional topology. We plan to devote this workshop to the interaction of Teichmüller theory and low-dimensional topology, not only subjects like hyperbolic manifolds or mapping class groups, but also knot theory, representation theory of fundamental groups of low-dimensional manifolds, four-dimensional topology, and so on. We believe that there should be more use of Teichmüller theory in low-dimensional topology than what has already been achieved. To promote this kind of research, we shall make a program focusing particularly on new applications of Teichmüller theory in low-dimensional topology. We plan to invite specialists in this field from all over the world. Although there already exists a strong group of low-dimensional topologists in China, we hope that the topics which this workshop deals with will give new impetus to their research.