Joseph Maher will investigate the large scale geometry of the mapping class group, with a view towards applications to 3-manifolds. Moduli spaces of Riemann surfaces, and their symmetries, the mapping class groups, are fundamental objects of interest in many branches of mathematics. Recently, techniques from coarse geometry have been applied to these spaces, and using these results Joseph Maher has shown that a random walk on the mapping class group gives rise to a pseudo-Anosov element with asymptotic probability one. This project involves extending this to obtain further information about the mapping class group, for example, investigating whether random walks on this group make linear progress in the complex of curves. Such results may have applications to the study of 3-manifolds, as every 3-manifold has a Heegaard splitting, determined by a gluing map, which is an element of the mapping class group. One aim of this project is to show that the volume of the resulting manifold grows linearly with the length of the random walk. Other goals of this project include attempting to extend the results for random walks to other counting problems, both in the mapping class group and in Teichmuller space, and to more general classes of weakly relatively hyperbolic groups.
The space of all possible shapes a surface may have, called the moduli space of the surface, is a fundamental object of interest in many parts of mathematics, and its structure is closely related to its symmetries, which are described by the mapping class group. These objects have applications in physics, for example in attempting to build models of quantum gravity in 2+1 dimensions, known as topological quantum field theories (TQFTs). The moduli spaces of surfaces also play an important role in certain conformal field theories and string theories. The study of surfaces and the mapping class group is also useful in the study of three- and four-dimensional spaces, which are important, as they are spaces which are locally modelled on the visible dimensions of the universe in which we live, which has three spatial dimensions, plus one time dimension.
