Winter 2012

January 12, 2012

Postnikov towers — Fedya Manin

Abstract: The Postnikov tower of a space is a way to construct the space from its homotopy groups, similarly to the way the cell decomposition is constructed from homology groups.  The role of the attaching map is played by the $k$-invariant of a fibration, which relates to obstruction theory and the Serre spectral sequence.  In this talk we’ll construct the Postnikov tower of a simply connected space, going over the other stuff as we go along.  If we have time we’ll also discuss what happens when $\pi_1$ enters the picture.

January 19, 2012 *** Special time: 3:00 PM in E308***

Perfectness of Diffeomorphism Groups — Katie Mann

January 26, 2012

Margulis’ Normal Subgroups Theorem — Wouter van Limbeek

Suppose you have a lattice in a semisimple Lie group. Then what are its normal subgroups? In this talk I will prove Margulis’ amazing answer: If the Lie group is higher rank and the lattice is irreducible, then every normal subgroup is finite or finite index.

February 2, 2012

“Hyperbolization Procedures” — Tam Nguyen Phan

1. Hyperbolization is a machine such that for each

Input = a triangulable closed manifold $M$,

Output = an locally CAT(0) (hence aspherical) closed manifold $h(M)$ such that a few nice things hold and there is a degree $1$ map $f: h(M) \to M$.

2. Cor: Any close manifold is cobordant to a closed aspherical manifold.

3. Strict hyperbolization: this does the same thing as in (1) but with CAT(0) replaced by CAT(-1).

4. and some other interesting if time permits.

February 9, 2012

Birrational geometry of (some) moduli spaces — Cesar Lozano Huerta

We will analyze a very specific type of deformation (birrational models) of some moduli spaces (like $M_g$, Hilbert Schemes, Stable maps). Despite the fact that such deformations are badly behaved from the topological point of view, these moduli spaces under such transformations remain moduli spaces. There were no reasons, a priori, for this fact to be true in general (for any moduli space), but in this talk we will give evidence suggesting that this phenomenon warrants further exploration.

February 16, 2012

Tautological Class of $\mathcal{M}_{g,n}$ — Rita Jiminez Rolland

Little is known about the cohomology rings of $\mathcal{M}_{g,n}$ (the moduli space of $n$-pointed Riemann surfaces of genus $g$) and of $\overline{\mathcal{M}_{g,n}}$ (its Deligne-Mumford compactification). Instead of trying to understand the whole $H^*(\mathcal{M}_{g,n};\mathbb{Q})$ we will focus on a subring generated by those classes that come naturally from geometric constructions (e.g. as Chern classes of some vector bundles).
In this talk I will define the tautological rings of $\mathcal{M}_{g,n}$ and $\overline{\mathcal{M}_{g,n}}$ and discuss some of their known and unknown properties. If time permits I will say a word about the existence of non-tautological classes.

February 23, 2012

Minimal but not uniquely ergodic interval exchanges — Jon Chaika

I will present a construction of a minimal but not uniquely ergodic interval exchange. Some time will be spent on explaining why one cares about this and why it is surprising. Time permitting I will present a second construction.

March 1, 2012

A Theorem of Borel — Grigori Avramidi

In this talk, I will discuss the following theorem of Borel:

“A closed aspherical manifold with centerless fundamental group  has no homotopically trivial isometries.”

I will describe extensions of this to some noncompact aspherical manifolds (in particular, locally symmetric spaces).

March 8, 2012

An Introduction to Coxeter groups — Jenny Wilson

This talk will give an overview of the basic theory of Coxeter groups: the structure of Coxeter groups, Cartan matrices, root systems, the length function, parabolic subgroups, Coxeter graphs, and the classification of finite Coxeter groups. It should lay the groundwork for future talk(s) on applications to coinvariant algebras and the cohomology of flag varieties, or braid groups and hyperplane complements.