Fall 2015

November 12, 2015
Kevin Casto: The Iwahori-Hecke algebra and braid group representations
The Iwahori-Hecke algebra is a certain deformation of the group ring of a Coxeter group, which was originally introduced in the study of algebraic groups over finite fields. However, as V. Jones first showed, it also has deep connections to topology (in his case, knot theory), since it is a quotient of the group ring of the braid group. Here, we discuss how interesting representations of the braid group factor through the Iwahori-Hecke algebra, and following S. Bigelow, how we can construct such representations from the action of the braid group on the homology of configuration spaces.
November 12, 2015
Nick Salter: Configurations of points on surfaces and the Johnson filtration
The Johnson filtration is a family of subgroups of the mapping class group related to the induced action of a mapping class as an automorphism of a nilpotent quotient of the fundamental group of the surface. Since D. Johnson (Dennis, not Dwayne “The Rock”) , the relationship between the Johnson filtration and the structure of the cohomological structure of surface bundles has been reasonably well-understood; S. Morita has also related the Johnson filtration to the theory of homology 3-spheres and the Casson invariant. The purpose of my talk is to explain some of the ideas of a remarkable 2008 paper of T. Moriyama giving a completely new perspective on the Johnson filtration defined in terms of the action of the mapping class group on the cohomology of the configuration space of points on surfaces.
November 5, 2015
Clark Butler: The classification problem for partially hyperbolic diffeomorphisms in dimension 3
We survey one of the most exciting and ambitious current programs in smooth dynamics, which is the classification of partially hyperbolic diffeomorphisms in dimension 3 up to leaf conjugacy. The principal tool in this classification is Novikov’s theorem from the theory of codimension one foliations. Highlights include:
  • The Brin-Burago-Ivanov proof that S^3 admits no partially hyperbolic diffeomorphisms
  • The complete classification of partially hyperbolic diffeomorphisms on 3-manifolds with solvable fundamental group by Hammerlindl and Potrie
  • The Franks-Williams surgery construction of Anosov flows on all sorts of 3-manifolds
  • The new Bonatti-Gogolev-Potrie construction of partially hyperbolic diffeomorphisms of the unit tangent bundle of a surface of genus > 1 in the isotopy class of any element of the mapping class group.
No dynamics knowledge will be assumed from the audience in this talk so even if what I wrote above makes absolutely no sense to you, you will still learn something awesome!
October 29, 2015
Lei Chen: Extension of Johnson homomorphism and MMM classes
Review of Morita’s papers
In this talk, I want to first extend the Johnson homomorphism to the whole mapping class group. Then We give a representation of MCG using this. Secondly, I will talk about the representation of Sp(2g,Z), how that relates to trivalent graphs. Finally I will talk about how MMM classes all coming from this representation.
October 22, 2015
Ian Frankel: Train Tracks, Measured Foliations, Quadratic Differentials, and
Teichmüller Geodesics
This is a more grown-up version of Nick Salter’s recent pizza seminar talk. We will describe how the combinatorics of train track splitting sequences give us a mechanism for proving properties that we see in hyperbolic space. For example, we will see why it is typical for two long geodesics in Teichmüller space that start and end distance 1 apart to get very close in the middle.
October 15, 2015
Benjamin Peters [Karlsruhe Institute of Technology]: Translation Surfaces and Teichmüller Curves
We will define Teichmüller curves. There will be constructions with translation surfaces (which, if not known from last week’s talk, will be defined briefly) leading to those special curves. Furthermore, we will see some examples demonstrating that normally life is much more messy in the translation surface world.
October 8, 2015
Sven Caspart [Karlsruhe Institute of Technology]: Translation Surfaces
Translation surface are (among others things) a special kind of flat surfaces (i.e. changes of coordinates are translations). They exhibit an interesting action of GL(2,R) and, surprisingly, have very interesting dynamics with respect to their geodesic flow.
In this talk I will give a small introduction into this subject and discuss some of the arising phenomena on concrete examples.
October 1, 2015
Julia Heller [Karlsruhe Institute of Technology]: Intro to Buildings
In this talk, I will give a short introduction to buildings as geometric objects with the focus on spherical buildings. In an important example I will explain the basic properties of buildings and demonstrate the connection between geometry and group theory.