There is no conformal map taking a square to a non-square rectangle. H.Gr\"{o}tzsch posed a problem in the 1920s, asking for the map of this kind that is ``closest" to being conformal. This question led to the discovery of quasiconformal maps, which now play an essential role in several branches of analysis, and in Teichm\"{u}ller Theory in particular.

The fundemental objects in Teichmuller Space are closed Riemann Surfaces. I will define Reimann Surfaces and give several equivallent definitions on Teichmuller Space. With a focus on examples, we will see the Teichmuller space of genus zero and one. Time permitting I will state Teichmuller's Theorem which gives the structure of Teichmuller Space of larger genus surfaces.

We continue where last week's lecture left off with further explanation of Teichmüller's theorem. We then show the parametrization of Teichmüller space by harmonic maps due to Wolf. We define the representation space of Riemann surfaces, of which Teichmüller space is one component, and discuss a possible extension of Wolf's parametrization to other components of representation space.

A fibre of marked complex projective structures over a point in Teichmuller space has both an analytic and geometric parameterization. The former is via bounded holomorphic quadratic differentials, and the later is via grafting. An interesting question to pursue is the interplay between the two parameterizations. This last Teichmuller Theory seminar will focus on defining complex projective structures and giving the analytic parameterization, with an introduction to the geometric parameterization if time allows.

James Cooper: Homology is a tool that we use to find the higher dimensional holes that the fundamental group fails to find. This provides us another topological invariant we may use to distinguish between spaces. I define a cellular complex and n-chains on that complex. With a boundary operator, we form a chain of abelian groups of these chains. We then use this sequence of groups to define homology. I will end with examples of computing homology and, time permitting, the statement of the Mayer–Vietoris sequence.
Derek Goto: The correspondence between covering spaces and deck groups is basically Galois theory in another category. I will talk about this and go over some relevant examples. If there is time, I will say a bit about Cayley complexes.

Abstract

Contact Homology, one specialized version of Floer Homology, is currently a very active area of research in Topology. In this talk, I will be following notes of a talk given by John Etnyre at the 2009 GITC as well as discussing some of what I learned at the MSRI Workshop in Symplectic and Contact Geometry and Topology. I will give a general discussion Contact Homology which studies the properties of M, a compact, oriented, 2n-1 dimensional manifold endowed with a contact structure. I will specialize this to the discussion of Legendrian Contact Homology which gives an invariant of Legendrian submanifolds of M, which is useful in the study of the isotopy classes of Legendrian knots in R^3 given the standard contact structure. If time permits, we will look at relationships between Contact Homology and other Floer Homology Theories.

Khovanov homology is a categorification of the Jones polynomial, a polynomial knot invariant discovered in the mid 80s. I will discuss what this statement means, in the context of constructing the Khovanov chain complex. I will also talk about how we measure the amount of new information Khovanov homology can bring to the table, and how similar features show up in other categorifications of knot polynomials.

Abstract

The main purpose of my talk will be to develop some of the elementary theory around sheaves and varieties in Algebraic Geometry. I will conclude my talk with a demonstration of how to view projective space as a variety.

Abstract

Abstract

Abstract

I will give an introduction to handles, including handle decompositions, handle cancellation, and obtaining homology from handles. This will be useful in proving the h-cobordism theorem later in the series.

I will give some background needed to state and understand the h-Cobordism theorem and outline its proof.

I will discuss the more technical details of the proof of the h-cobordism theorem, specifically the "Whitney Trick". The Whitney trick will demonstrate why this theorem does not work for dimension n=4. I will also give a similar theorem for when our manifolds are not simply connected.

Abstract

Abstract

Abstract

This talk will introduce the notion of uniform distribution. Weyl's criterion and Van der Corput's difference theorem will be stated. Examples and non-examples will be presented. If time permits variants of uniform distribution will be discussed.

Cancelled

Abstract

Abstract

Abstract

Abstract

Abstract