January 24, 2013
Note: FFSS this week will start at 3pm in E308. |

Extension Problems for Groups of Diffeomorphisms — Katie Mann |

If is a manifold with boundary , restricting an (isotopically trivial) diffeomorphism of to its action of the boundary gives a surjective homomorphism from to . Does this map ever admit a section? The answer is usually no, and the reasons are usually interesting. In this talk, we describe obstructions to sections and in a number of cases obstructions to lifting finitely generated subgroups of to groups of diffeomorphisms of . Our methods range from the dynamical to the cohomological, and will be illustrated as much as possible with concrete examples. |

Notes in pdf: Extension Problems for Groups of Diffeomorphisms |

January 31, 2013 |

PSL(2,R) Representations of a Rank 2 Free Group — Alden Walker |

The character variety of representations of a free group is subset of . I’ll describe a classification (not made up by me) of the representations as realizations of the free group as the fundamental group of a hyperbolic pair of pants or once-punctured torus. This will be all about playing with matrices. |

Reference: Section 4 of Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces |

February 12, 2013
Note: There are FFSS talks on Tuesday and Thursday this week. |

Surfaces in 4–Space — Joshua Batson |

I will talk about surfaces embedded and immersed in –space, how to visualize them and their normal bundles, and how to build branched covers along them. When you cut a –manifold containing a surface, you get a –manifold with a knot in it — the study of the (knot, surface) pair goes back to Wirtinger (1905). For orientable surfaces bounding torus knots, this is the relative version of the Thom conjecture, resolved in the 90’s using Floer homology. My research concerns nonorientable surfaces, where all the old theorems fail, and I’ll talk about some new conjectures and techniques. |

Preprint: Nonorientable four-ball genus can be arbitrarily large |

February 14, 2013 |

Polynomial Invariants of Finite Reflection Groups — Jenny Wilson |

It is well-known that the algebra of symmetric polynomials in variables is generated by (for example) the elementary symmetric polynomials, algebraically independent homogeneous generators. A classical theorem of Chevalley states that this phenomenon holds in general for the algebra of invariant polynomials associated to a finite reflection group — and the degrees of the homogeneous generators have some unexpected combinatorial properties. In this talk, I will briefly review the basics of Coxeter groups, and describe their polynomial invariants. I will define coinvariant algebras, and their relation to the cohomology of flag varieties. Time permitting, I will outline a proof of Chevalley’s theorem. |

Notes in pdf: Polynomial Invariants of Finite Reflection Groups |

February 21, 2013 |

Slim Unicorns and Hyperbolicity of Curve Graphs — Sebastian Hensel |

The curve graph is a graph which encodes the intersection pattern of simple closed curves on a surface. This graph has proven extremely useful in the study of Teichmüller theory, –manifolds and mapping class groups. I will define the curve graph and show some of its applications. Finally, I will describe joint work with Piotr Przytycki and Richard Webb giving a new short proof of hyperbolicity of curve and arc graphs using unicorns. |

Notes in pdf: Slim Unicorns and Hyperbolicity of Curve Graphs |

February 28, 2013 |

Surface Bundles Over Surfaces with Multiple Fiberings — Nick Salter |

The theory of the Thurston norm shows that it is common for a given three-manifold to fiber in infinitely many ways as a surface bundle over the circle. Unfortunately, this ultimately seems to be a story about manifolds that fiber over the circle, rather than about surface bundles. In four dimensions, the situation is much more rigid, and it is not even known if there are examples of surface bundles (of nonzero Euler characteristic) that fiber in three ways! In this talk, I will survey what little we do know about surface bundles over surfaces with multiple fiberings. This will lead into a quest to find surface subgroups of mapping class groups, and I will describe a few constructions. |

Notes in pdf: Surface Bundles Over Surfaces with Multiple Fiberings |