### Overview

The conference will consist of 3 days of introductory talks for nonexperts (May 2-4) followed by two days of research talks (May 5-6).

### Registering for the conference

There is no formal registration for this conference. However, if you are interested in attending, please send an e-mail Nathan Broaddus. This will allow us to add you to the mailing list, and keep you informed of any activities during the week of the conference.

### Contacting the organizers

If you have any questions or concerns, please feel free to contact the organizers:

- Nathan Broaddus (Ohio State University), broaddus@math.ohio-state.edu

### Schedule

###### 4:00pm-7:00pm

### Barbecue at Nate's

63 Warren St., Columbus OH

###### 1:30pm-2:30pm in CH 240

### Tara Brendle, University of Glasgow

#### Mapping Class Group I

###### 3:00pm-4:00pm in CH 240

### Tara Brendle, University of Glasgow

#### Mapping Class Group II

###### 4:10pm-4:30pm in CH 240

### Dawid Kielak, University of Oxford

#### Linear and free representations of Out(F_{n})

##### Abstract

We will discuss low-dimensional linear representations of Out(`F _{n}`) and use
this knowledge to obtain a rigidity-like result about possible
homomorphisms between groups Out(

`F`) for varying parameter

_{n}`n`.

### 11am University Plaza Hotel Shuttle to OSU

###### 11:30am-12:30pm in CH 240

### Tom Church, University of Chicago

#### Stability I

##### Abstract

I will introduce homological stability and explain how it is proved in each of the major
cases where homological stability is known: braid groups `B _{n}`, special
linear groups SL

`, moduli spaces of Riemann surfaces`

_{n}(**Z**)`M`, symmetric groups

_{g}`S`, automorphism groups of free groups Aut(

_{n}`F`), and configuration spaces Conf

_{n}`.`

_{n}(M)###### 2:00pm-2:20pm in CH 240

### Catherine Pfaff, Rutgers University

#### Constructing and Classifying Fully Irreducible Outer Automorphisms of Free Groups

##### Abstract

The main goal of my research is to prove a fully irreducible outer automorphisms analogue
of the Howard Masur and John Smillie theorem which precisely records the lists of foliation
singularity indices which arise from pseudo-Anosov elements of Mapping Class Groups. The more
appropriate conjectured theorem in the Out`(F _{r})` setting actually records possibilities for an
even finer outer automorphism invariant, an ideal Whitehead graph. I have developed methods
for constructing and identifying fully irreducible outer automorphisms that have thus far been
used to completely determine which connected graphs with 5 vertices occur as ideal Whitehead
graphs for ageometric fully irreducible outer automorphisms of free groups.

###### 2:25pm-3:25pm in CH 240

### Tom Church, University of Chicago

#### Stability II

##### Abstract

*See Stability I above.*

###### 4:30pm-4:50pm in CH 240

### Funda Gultepe, University of Oklahoma

#### Normal tori in `#`^{n}(S^{2} × S^{1})

^{n}(S

^{2}× S

^{1})

##### Abstract

The fundamental group of `M = # ^{n}(S^{2} × S^{1})`
is

`F`, the free group with

_{n}`n`generators. There is a one-to-one correspondence between the

**Z**-splittings of

`F`and embedded essential tori in

_{n}`M`. We define and prove a local notion of minimal intersection for tori with respect to a maximal sphere system in

`M`which generalizes Hatcherâ€™s work [Hat95] on 2-spheres in the same manifold.

### 10am University Plaza Hotel Shuttle to OSU

###### 10:30am-11:15am in CH 240

### Martin Bridson, University of Oxford

#### Outer automorphisms of free groups

###### 11:30am-12:30am in CH 240

### Dan Margalit, Georgia Tech

#### Torelli Group I

###### 2:30pm-3:30pm in CH 240

### Dan Margalit, Georgia Tech

#### Torelli Group II

###### 3:40pm-4:00pm in CH 240

### Richard Wade, University of Oxford

###### 4:05pm-4:25pm in CH 240

### Masatoshi Sato, Osaka University

#### MMM classes

###### 4:30pm-5:30pm in CH 240

### Andrew Putman, Rice University

#### Teichmüller space

### 8am University Plaza Hotel Shuttle to OSU

###### 8:30am-9:20am in CH 240

### Tara Brendle, University of Glasgow

#### Hyperelliptic Birman Exact Sequences

##### Abstract

Motivated by a conjecture of Hain and Morifuji that the hyperelliptic Torelli group SI is generated by Dehn twists about symmetric separating curves, we will give Birman exact sequences for hyperelliptic mapping class groups. We will also describe various aspects of the structure of SI, including some relations in the Torelli group which allow us to write certain examples of elements of SI in terms of Hain's generators. (Joint work with Dan Margalit.)

###### 9:30am-10:20am in CH 240

### Mladen Bestvina, University of Utah

###### 10:30am-11:20am in CH 240

### Dan Margalit, Georgia Tech

#### The Torelli group is generated by bounding pair maps

##### Abstract

We give a new proof of that the Torelli group is generated by pair maps that is analogous to the proof that the mapping class group is generated by Dehn twists. This is joint work with Allen Hatcher.

###### 11:30am-12:20pm in CH 240

### Joan Birman, Columbia University

#### Invariants of pseudo-Anosov maps

##### Abstract

Let `F` be a pA map acting on a surface `S`. Extending the Bestvina-Handel
algorithm, Brinkmann, Kawamuro and I investigated the structure of the characteristic polynomial
det`(xI-T)` of a transition matrix `T` associated to `F`. We learned
that certain polynomial divisors of det`(xI-T)` are invariants of `F`. We
give examples of pA maps having the same dilatation which can be shown to be non-conjugate by
our invariants.

###### 2:30pm-3:20pm in CH 240

### Justin Malestein, Temple University

#### On genericity of pseudo-Anosovs in the Torelli group

##### Abstract

We will show that, for any (symmetric) finite generating set of the Torelli group of a closed surface, the probability that a random word is not pseudo-Anosov decays exponentially in the length of the word. As a consequence of our methods, we will prove the same statement for some other finitely generated subgroups of the mapping class group. (Joint work with Juan Souto)

###### 3:30pm-4:20pm in CH 240

### Benson Farb, University of Chicago

#### Representation Theory and Homological Stability

##### Abstract

In this talk I will explain some recent work with Tom Church. The story begins when we were working out some cohomology computations for a problem in geometric topology. Hints of a pattern emerged, but we struggled to find a way to describe it. We finally developed a language to do this, and called the phenomenon "representation stability".

As we began to look more broadly we started to see representation stability in many different areas of mathematics: from group cohomology to the topology of configuration spaces to classical representation theory to flag varieties to Lie algebras to algebraic combinatorics. We have been able to apply representation stability to prove theorems and make new predictions in these directions (some since proved by others). Many conjectures remain.

My goal in this talk will be to explain representation stability through examples and applications. I will also explain how Church, Jordan Ellenberg and I are applying this theory in order to compute and explain various combinatorial statistics in number theory. I will try to make this talk accessible to first-year graduate students.

###### 4:30pm-5:20pm in CH 240

### Masatoshi Sato, Osaka University

#### On the third cohomology group of the Torelli group

##### Abstract

This is a joint work with Nariya Kawazumi. We have been trying to determine whether the
even Morita-Mumford-Miller classes are trivial or not in the Torelli group. If a third
cohomology class `m _{2,1}` called the

`(2,1)`-twisted MMM class in the Torelli group is trivial, the second MMM class

`e`is also trivial. In this talk, I will talk about some results about the class

_{2}`m`.

_{2,1}### 8am University Plaza Hotel Shuttle to OSU

###### 8:30am-9:20am in CH 240

### Martin Bridson, University of Oxford

#### Abelian covers of graphs and maps between outer automorphism groups of free groups

##### Abstract

This is joint work with Karen Vogtmann. It is part of a programme whereby I'd like to
understand all maps between outer automorphism groups of free groups. The main result here
is that if `m=r ^{n}(n-1)+1`, where

`r`is coprime to

`(n-1)`, then there is an embedding of Out

`(F`into Out

_{n})`(F`. This is proved by determining for which abelian covers of finite graphs one can lift the action of the group of homotopy equivalences of the base to the cover.

_{m})###### 9:30am-10:20am in CH 240

### Andrew Putman, Rice University

#### Small generating sets for the Torelli group

##### Abstract

Proving a conjecture of Johnson, we show that for `g` at least 3, the genus
`g` Torelli group has a finite generating set `S _{g}` with

`|S`. The main tool is a new space (the "handle graph") on which the Torelli group acts cocompactly.

_{g}| = O(g^{3})###### 11:00am-11:50am in CH 240

### Matthew Day, Caltech

#### Generalized Johnson homomorphisms and non-finitely presentable subgroups of
Aut(`F`_{n})

_{n}

##### Abstract

We investigate a subgroup `K` of Aut(`F _{n}`) that is an analogue
of the kernel of the Birman exact sequence in mapping class groups. We produce homomorphisms
on

`K`and its subgroups that are restrictions of crossed homomorphisms, in a way that is formally parallel to the classic higher Johnson homomorphisms in mapping class groups. Our homomorphisms suffice to compute the abelianization of

`K`, and are a key ingredient in building infinitely many independent classes in the second rational cohomology groups of

`K`. This is joint with Andrew Putman.

###### 12:00pm-12:50pm in CH 240

### Piotr Przytycki

#### The ending lamination space for the five-punctured sphere

##### Abstract

Joint with Sebastian Hensel. We prove that the ending lamination
space (i.e. the boundary of the curve complex) for the 5-punctured sphere
is the so called Noebeling curve. This is the space obtained from **R**^{3} by
removing all points with at least 2 rational coordinates.