Outer Space is an important object in Geometric Group Theory and can be described from two viewpoints: as a space of marked graphs and a space of actions on trees. The latter viewpoint can be used to prove that Outer Space is contractible; and this fact together with some arguments using the first viewpoint enables us to say something about the Outer Automorphism group of a free group - I will sketch both these proofs.

# Past Kinderseminar

By way of shameless advertising for a TCC course I hope to give next term on the theory of totally disconnected locally compact groups, I will present two interesting and illuminating examples of such groups: the full automorphism group of a regular tree, and Neretin's group of spheromorphisms

Classifying line arrangements on the plane is a problem that has been around for a long time. There has been a lot of work from the perspective of incidence geometry, but after a paper of Hirzebruch in in 80's, it has also attracted the attention of algebraic geometers for the applications that it has on classifying complex algebraic surfaces of general type. In this talk I will present various results around this problem, I will show some specific questions that are still open, and I will explain how it relates to complex surfaces of general type.

Modular forms holomorphic functions on the upper half of the complex plane, H, invariant under certain matrix transformations of H. The have a very rich structure - they form a graded algebra over C and come equipped with a family of linear operators called Hecke operators. We can also view them as functions on a Riemann surface, which we refer to as a modular curve. It transpires that the integral homology of this curve is equipped with such a rich structure that we can use it to compute modular forms in an algorithmic way. The modular symbols are a finite presentation for this homology, and we will explore this a little and their connection to modular symbols.

This talk will hopefully highlight the general framework in which Penrose tilings are proved to be aperiodic and in fact a tiling.

In the game 'Set', players compete to pick out groups of three cards sharing common attributes. But how many cards must be dealt before such a group must appear?

This is an example of a "cap set problem", a problem in Ramsey theory: how big can a set of objects get before some form of order appears? We will translate the cap set problem into a problem of geometry over finite fields, discussing the current best upper bounds and running through an elementary proof. We will also (very) briefly discuss one or two implications of the cap set problem over F_3 to other questions in Ramsey theory and computational complexity

In my talk I will give a basic introduction to the finiteness properties of groups and their relation to subgroups of direct products of groups. I will explain the relation between such subgroups and fibre products of groups, and then proceed with a discussion of the n-(n+1)-(n+2)-Conjecture and the Virtual Surjections Conjecture. While both conjectures are still open in general, they are known to hold in special cases. I will explain how these results can be applied to prove that there are groups with arbitrary (non-)finiteness properties.

There are many natural questions one can ask about presentations of finite groups- for instance, given two presentations of the same group with the same number of generators, must the number of relations also be equal? This question, and closely related ones, are unsolved. However if one asks the same question in the category of profinite groups, surprisingly strong properties hold- including a positive answer to the above question. I will make this statement precise and give the proof of this and similar results due to Alex Lubotzky.

For every $\epsilon>0$ does there exist some $n\in\mathbb{N}$ and a bijection $f:\mathbb{Z}_n\to\mathbb{Z}_n$ such that $f(x+1)=2f(x)$ for at least $(1-\epsilon)n$ elements of $\mathbb{Z}_n$ and $f(f(f(f(x))))=(x)$ for all $x\in\mathbb{Z}_n$? I will discuss this question and its relation to an important open problem in the theory of countable discrete groups.