[Topology and Geometry Seminar] "Curves, surfaces, and symmetries" by Dr Robert Tang
Abstract: Surfaces are some of the most familiar examples of topological spaces one encounters both in mathematics and in the real world. These talks will focus on symmetries of surfaces - the ways of mapping a surface back onto itself. I will begin with the case of the torus, a surface otherwise known as a donut. Symmetries of the torus are well understood, and can be easily represented using matrices. For higher genus surfaces (those with more 'holes'), the situation is more complicated. In particular, it is still an open question whether matrix representations for their symmetries exist in general. Thus, more sophisticated techniques are needed to deal with higher genus surfaces.
I will discuss methods for studying surface symmetries using ideas from geometric group theory. The key tool is the curve complex - a mathematical object which organises all the curves on a surface into a single geometric entity. These objects have been used to solve problems not only in surface topology, but also in 3-manifold geometry and Teichmueller Theory. These talks will conclude with a description of my current research projects relating curve complexes to geometric and dynamical properties of flat surfaces.