Topology & Geometry Seminar: Spring 2011 Abstracts

This will be the first of two mostly expository talks about arithmetic groups and hyperbolic geometry.

This will be the second of two mostly expository talks about arithmetic groups and hyperbolic geometry.

I will discuss the structure of subgroups of Mod(S) generated by three Dehn twists along simple closed curves with "small" geometric intersections.

We consider k-step recurrences of the form $z_{n+k} = A(z)/B(z)$, where A and B are linear functions of $z_n, z_{n+1}, ..., z_{n+k-1}$, which we call k-step linear fractional recurrences. In case $k=2,3$ we determine all the possible periodicities within this family or equivalently the induced birational maps of finite order.

I will talk about 3-manifold of Category 3, especially the classification of 3-manifold covered by 3 pieces of S^1 contractible sets.

I will give an introduction to simplicial sets and illustrate how homotopy theory can be done in the realm of simplicial sets. In particular I will explain how simplicial sets and topological spaces give rise to equivalent homotopy theories.

Curves that are very different when considered in their entirety can still share significant features. For example, if an object in a 2-D image is cropped or partially hidden by another object, then the object's boundary curve will only partially match a model curve corresponding to an unoccluded object of the same type. We describe an algorithm for finding partial similarities between curves in the square-root velocity (SRV) framework.

The stable/unstable manifold theorem has proven to be integral to the theory of differentiable and topological dynamics. The Local Unstable Manifold Theorem for a Point will be demonstrated for Lipschitz maps.

Penner gave a class of examples to give an upper bound for the minimal dilatation of pseudo-Anosov mapping classes on closed surfaces. We will show that these examples are all realized by fibrations of a single 3-manifold. Then we will discuss the Teichmuller polynomial and compute it for this example comparing the Teichmuller polynomial to the characteristic polynomial of the train track maps.