**Section:**001

*Teichmuller theory-higher and lower*

**Term:**FA 2017

**Subject:**Mathematics (MATH)

**Department:**LSA Mathematics

: If S is a closed orientable surface of genus at least 2, the Teichmuller T(S) of S is the space of isometry classes of marked hyperbolic surfaces homeomorphic to S. Equivalently, we may think of T(S) as the space of conformal classes of Riemann surfaces which are homeomorphic to S. The study of Teichmuller space arises naturally in many fields. For example, one may view Teichmuller space as the ``universal cover'' of Moduli space, since T(S) is homeomorphic to an open ball and the mapping class group acts properly discontinuously on T(S) so that we may identify its quotient with Moduli space.

We will assume only material from the alpha courses, so we will begin with a basic introduction
to hyperbolic geometry. We will produce parameterizations of Teichmuller space from both
geometric and complex analytic viewpoints. We will also discuss dynamics on hyperbolic surfaces, *e.g.* the ergodicity of the geodesic flow.

We will also, as time permits, study two natural generalizations of Teichmuller space. First, we will study spaces of marked hyperbolic 3-manifolds homotopy equivalent to S, e.g. quasifuchsian space. Second, we will study the Hitchin component of the space of representations of the fundamental group of S into PSL(n,R). The exact topics covered will depend on the interests of the students. Moreover, we will give a topological viewpoint on the Deligne-Mumford compactification of Moduli space

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