# Winter 2011

Jan. 6 2011

Circular orders on groups – Katie Mann

Abstract: a circular order on a group G is choice of orderings invariant under left multiplication that captures the same kind of idea as the natural “clockwise” order on the circle.  Why do I care?  Well, I won’t give away the punchline, but it has to do with which groups can and cannot act on the circle.  Or act on the line.  If you care about groups acting on manifolds, you might just care about this too.

I’ll define everything from scratch and give some surprising proofs showing that some of your favourite groups are in fact circular-orderable.  And if you really like my talk, I have a sequel planned in which you will see a cool theorem of Dave Witte that says that some of your other favourite groups are not circular-orderable, and explain why this is all part of the Zimmer program.

Jan 13, 2011

Circular orders part II – Katie Mann

In this talk I’ll give the other side of the story: groups that are not circular orderable.  The Zimmer program says that “big groups can’t act on small manifolds” and since the circle is a very small manifold, we should expect that “big” groups won’t be able to act on it.   The previous talk is not a prerequisite for most of this.

Jan 20, 2011

Statistical hyperbolicity in the curve complex – Spencer Dowdall

Abstract: The curve complex encodes the combinatorial relationships of simple closed curves on a surface, and it has proven to be a central object in the study of surface geometry and the Teichmuller space. As established in groundbreking work of Masur and Minsky, the curve complex turns out to be a hyperbolic metric space in the sense of
Gromov. In this talk, I will show that the curve complex is also “statistically hyperbolic” in the sense that most points on a metric sphere of radius $R$ have distance $2R$ from each other. Joint work with Moon Duchin and Howard Masur.

Jan 27, 2011

“Puncture homological stability” for the pure mapping class group – Rita Jimenez Rolland

Abstract:  A family of groups of interest is $\mathrm{PMod}_{g,r}^n$, the  pure mapping class group of an orientable surface of genus $g$, with $r$ boundary components and $n$ punctures. One basic question corresponds to understand how the ith homology group of $\mathrm{PMod}_{g,r}^n$ change as we vary the parameters $g$, $r$ and $n$, in particular when the parameters are very large with respect to $i$.  It is a classical result by Harer (1985) that $\mathrm{PMod}_{g,r}^n$ satisfies “genus homological stability”. By considering the parameter n one could ask whether “puncture homological stability” holds. Some particular cases indicate that it may not be case. In this talk, we will discuss how the question of “puncture homological stability” could be rephrased, in terms of the language of representation stability developed by Church and Farb, to give a meaningful answer.

Feb 10, 2011

Using distances to measures to infer homology with high confidence from noisy random samples. – Kate Turner

Abstract:
Suppose there is some reasonably nice compact set $A$ in $\mathbb{R}^d$ and we want to find out what it is. The problem is we only have the ability to take noisy samples at random (sampling from a probability measure on the set convolved with a Gaussian for noise). Since we are taking random samples our results can only ever be probabilistic in nature. We will construct (with high probability) a function on $\mathbb{R}^d$ which is $L_\infty$ close to the distance function from $A$ using these samples. We can infer the homology of $A$ via this function. This construction will use what is called a distance to a measure. There are some assumptions we need to make about the amount of noise in the Gaussian and the measure on $A$ and we will consider in what circumstances this construction works.

Feb 17, 2011

The Hero’s Journey: Bass-Serre Theory and Splittings over Submanifold Groups – Daniel Studenmund

Abstract: Bass-Serre theory is a fancy name for the study of actions of groups on trees.  But here’s a little-known fact: the theory was developed by Serre when he read Joseph Campbell’s “The Hero with a Thousand Faces”.  In this talk, we will start with the story of a brave closed geodesic on a hyperbolic surface who ventures into the big scary universal cover of hyperbolic space.  We then describe the general monomyth of Bass-Serre theory.  Time permitting, the conclusion will state the JSJ decomposition of 3-manifolds, and describe space-operatic spin-offs of this theory to general splittings of groups.

Feb 24, 2011

A Gentle Introduction to Spectral Sequences – Jenny Wilson

Abstract: We will discuss the basic structure of a spectral sequence, and look at some elementary examples of Leray-Serre spectral sequences, the spectral sequence associated to a fibration. Time permitting, I will outline the construction of the spectral sequence associated to a filtered complex.

Notes: The Leray-Serre Spectral Sequence

Monday, Feb 28, 3:00 p.m. in room E203

A gentle introduction to counting points over finite fields – Tom Church

Abstract:
I will give a gentle and accessible introduction to one of the most powerful and beautiful techniques in mathematics: if you have a collection of equations, you can use the Grothendieck–Lefschetz fixed-point formula to relate the *cohomology* of the solutions over the complex numbers with the *number* of solutions over a finite field. I will start by recalling the Lefschetz fixed point theorem and gradually move to the finite field case, where we will work through a number of examples. No knowledge of algebraic geometry will be necessary.

I will finish by discussing joint work with Jordan Ellenberg and Benson Farb on applying representation stability to certain combinatorial questions over finite fields. For example, the representation-stable cohomology of the pure braid group can be used to understand how polynomials factor in $\mathbb{F}_q[T]$ (number of linear factors, irreducible factors, etc.), while the representation-stable cohomology of flag varieties tells us about maximal tori in $\mathrm{GL}_n(\mathbb{F}_q)$.

Although my talk will be self-contained, I won’t have a lot of time to talk about representation stability. Fortuitously, Benson happens to be giving a colloquium this Friday on exactly this topic, if you’d like to know more about the theory these applications are based on. (This colloquium should also be a good overview of / advertisement for Geometric Literacy next quarter, if you’re wondering whether you’d be interested.)

http://www.math.uchicago.edu/research/colloquium_farb.shtml

Note unusual time and date.  This talk will take place in E203

March 3, 2011

Jon Chaika

Some results in ergodic theory.

I will present a few accessible results in ergodic theory.

1) An application of the Birkhoff Ergodic Theorem to inhomogeneous diophantine approximation.

2) Rokhlin’s Lemma which says that all measure preserving dynamical systems are close to being periodic from the standpoint of sets.

3) Kingman’s ergodic theorem which can often show that the norms of matrix products have definite growth.

March 10, 2011

Nonpositively curved triangles of groups – Tam Nguyen Phan

Abstract: I’ll define what a triangle of groups is and what it means for
it to be nonpositively curved. I’ll give some examples of manifolds whose
fundamental groups have this sort of structure. I’ll state and prove some
theorems.