# Spring 2012

March 29, 2012

Solvable Lie Groups: A Menagerie — Daniel Studenmund

Every connected, simply connected real Lie group decomposes as the semidirect product of its solvable radical with a semisimple subgroup.  We hear about semisimple groups all the time, but why don’t we hear about solvable groups?  A superficial answer to that question is that they are, to loosely quote a professor at this university, “a total mess”.  I will provide numerous examples illustrating nonalgebraicity of solvable groups, nonrigidity of lattices, nondensity of lattices, and even nonarithmeticity of lattices.  After devastating you with negative examples, I will give some indication of the methods used to prove, and statements of, positive results.  Expect few proofs, many heuristics.

April 5, 2012

NOTE: Special Time: The talk will be 3:00 — 4:00 in Eckhart 308

Counting curves and Quantum cohomology: there and back again — Simion Filip

Around 1994, Kontsevich generalized the following theorem of Euclid – through two points in the plane there is a unique line. A little bit more seriously, he gave an answer to how many degree d rational plane curves pass through $3d-1$ points in general position (he only used the $d=1$ case). I’ll try to explain how to find such numbers by hand (using combinatorics and a lot of patience) and how to find them scientifically (using quantum cohomology). I will start from scratch and no previous knowledge of any of these subjects will be assumed.

April 12, 2012

Flat tori in the homology of $\mathrm{SO}(3)\backslash \mathrm{SL}(3,\mathbb{R})/G$ — Grigori Avramidi

Let $M$ be the locally symmetric space $\mathrm{SO}(3)\backslash \mathrm{SL}(3,\mathbb{R})/G$, for a torsion free finite index subgroup $G$ of $\mathrm{SL}(3,\mathbb{Z})$. I will explain how to find some two-dimensional flat tori which are nontrivial in the homology (with rational coefficients) of the locally symmetric space. This is joint work with Tam Nguyen Phan (who would like to add that this is not joint work with Church and Farb.)

April 19, 2012

Gromov’s Polynomial Growth Theorem — Wouter van Limbeek

Given a finitely generated group $G$, one would like to understand the relationship between the algebraic properties of $G$ and the geometry of its Cayley graph. In pizza seminar this week I proved / am proving / will prove the following results of Milnor and Wolf:

1. A finitely generated virtually nilpotent group has polynomial growth,
2. A finitely generated solvable group has exponential growth unless it is virtually nilpotent,

thus classifying the virtually nilpotent groups among all solvable groups via the geometry of their Cayley graphs. In this talk, I will sketch the proof of Gromov’s amazing extension of this result, recognizing virtually nilpotent groups among all finitely generated groups via their geometry:

Gromov’s Polynomial Growth Theorem: A finitely generated group $G$ is virtually nilpotent if and only if it has polynomial growth.

April 26, 2012

Homomorphisms between diffeomorphism groups — Katie Mann

1. Question: Can you detect how big a manifold is by looking at how big its group of diffeomorphisms is?
2. Exercise: How about this – show that $\mathbb{R}^2$ is bigger than $\mathbb{R}^1$ by demonstrating that the group of compactly supported diffeomorphisms of $\mathbb{R}^2$ is not isomorphic to any subgroup of the group of compactly suppored diffeos of $\mathbb{R}^1$. Now do the same for $2$-manifold/$1$-manifold. Now do the same for $n$-manifold/$m$-manifold, where $n \geq m$.

In my talk, I’ll explain everything I know about diffeomorphism groups of $1$-manfiolds and answer the first two parts of the exercise. The third part is still open – it’s a question of Ghys from 1991.

May 3, 2012

Homotopy types of diffeomorphism groups — Bena Tshishiku

I will prove that the group of diffeomorphisms of the $2$-sphere is homotopy equivalent to the orthogonal group $O(3)$ by explicit construction of a deformation retract. I will mention how this relates to characteristic classes and mention some surrounding results.

May 10, 2012

What is an FI–module? — Jenny Wilson

Last month, Church–Ellenberg–Farb posted their preprint “FI–modules: a new approach to stability for $S_n-$representations”. I will give the definition of an FI–module, and discuss how this new concept allows us to strengthen the theory of “representation stability” for a sequence of representations of the symmetric groups. There will be many examples, and, time permitting, applications to configuration spaces and coinvariant algebras.

May 17, 2012

Frame flow and hyperbolic rank rigidity — Dave Constantine

I will introduce some geometric rank rigidity problems and then show how to use the dynamics of the frame flow to prove hyperbolic rank rigidity for certain non-positively curved manifolds.

May 24, 2012

Exotic aspherical/locally CAT(0) manifolds and invariants at infinity — Tam Nguyen Phan

I will talk about how Cartan-Hadamard theorem fails in the CAT(0) set up. I will sketch two constructions of locally CAT(0) manifolds that do not admit any Riemannian nonpositively curved metric, and explain why by using, for example, the fundamental group at infinity (which I will define), etc.

May 31, 2012

Finite volume, nonpositively curved manifolds — Tam Nguyen Phan

I will talk about a theorem of Eberlein that classifies (compact or noncompact) finite volume, nonpositively curved manifolds with continuous isometry group of the universal cover. I will talk about questions and conjectures on finite volume, nonpositively curved manifolds of geometric rank one (I will define the geometric rank).

June 7, 2012

An introduction to smooth dynamics in dimension two — Alex Wright

What can you say about the dynamics of the “generic” diffeomorphism of a surface? We will discuss this question, while introducing key ideas in smooth dynamics.