October 8th, Kevin Wildrick, Quasisymmetry and 2-dimensional spaces
Quasisymmetric mappings are homeomorphisms between metric spaces which almost preserve relative distance. The surprisingly subtle definition allows quasisymmetric maps to have strong properties which one often associates with conformal maps while still providing a large supply of such maps. We will discuss the basic definitions, some key results, and report on current research exploring a 2-dimensional quasisymmetric "uniformization" theorem.

October 15th, Dennis Clark, Automatic Groups
Let G be a finitely presented group. In general, it is impossible to determine algorithmically whether a word in the generators of G is the identity; this is called the "word problem" in G. It is a natural question to try to classify the language corresponding to the word problem. While regularity of this language implies G is finite, a slight generalization (due to Epstein et al.) yields the class of automatic groups, which is broad enough to contain many known classes of finitely generated groups and yet narrow enough to permit Gromov-hyperbolic-style geometric arguments growing out of a purely language-theoretic definition. We will present some of the background and implications of this theory.

November 5th, Alan Stapledon, Geodesics in SO(n)
Geodesics can be defined on any Riemannian manifold and lead to the defintion of an exponential map. Why is it called an exponential map? Were the founders of differential geometry out of their minds? It turns out that the motivation for this choice of words is the example of SO(n). We'll compute the geodesics of SO(n), sketch them, and determine the exponential map in this case.

November 12th, Cagatay Kutluhan, Dehn Surgery
There are many techniques used to understand the topology of low dimensional manifolds, i.e. 3 and 4 dimensional manifolds. One of the most natural ones is performing a sugery on a link embedded in a 3-manifold, so called the knot or the Dehn surgery. A surgery along a knot is mainly cutting out a tubular neigbourhood of a knot in a 3-manifold (which is a solid torus) and attaching a solid torus along its boundary to the manifold by an orientation reversing homeomorphism of the torus T^2. Before introducing knot surgery on the three sphere S^3, I will talk about some basic facts concerning Heegard decompositions and links. Then we will see how we can obtain any closed orientible 3-manifold via knot surgery on S^3.

November 19th, Dave Anderson, Sheaves in Geometry: an Introduction
A sheaf on a topological space is a way of keeping track of local information, and as such, the notion appears frequently in geometric mathematics. The sheaf of functions on a manifold provides a typical example; others include vector bundles, differential forms, and orientations. In this talk, I'll introduce sheaves and try to motivate their study through examples; little or no prior experience with sheaves will be assumed. I'll also touch on sheaf cohomology, and time permitting, say a few words about how the concept of a sheaf leads to generalized topologies on a category.

December 3rd, Jose Gomez-Guerra, Classifying Space for Real Vector Bundles

December 10th, Aaron Magid, Hyperbolic Knot Complements
For a knot K embedded in the 3-sphere, the complement, S^3 - K, sometimes admits a hyperbolic structure. A geometric invariant for a hyperbolic 3-manifold is also a topological invariant, so we are interested in how geometric properties of knot complements can be analyzed. For instance, the volumes of these manifolds and the existence of certain types of surfaces in the complements can distinguish these manifolds. We will present some basic results about hyperbolic knot complements with some examples, and then show the existence of a totally geodesic Seifert surface by construction as an example of a particularly rigid geometric property.