SR1

In euclidean space there is a well-known parallelogram law relating the

length of vectors a, b, a+b and a-b. In the talk I give a similar formula

for translation lengths of isometries of CAT(0)-spaces. Given an action of

the automorphism group of a free product on a CAT(0)-space, I show that

certain elements can only act by zero translation length. In comparison to

other well-known actions this leads to restrictions about homomorphisms of

these groups to other groups, e.g. mapping class groups.

... for Torelli groups of surfaces.

We will attempt to introduce fusion systems in a way comprehensible to a Geometric Group Theorist. We will show how Bass--Serre thoery allows us to realise fusion systems inside infinite groups. If time allows we will discuss a link between the above and $\mathrm{Out}(F_n)$.

First of all, we are going to recall some basic facts and definitions about homogeneous Riemannian manifolds. Then we are going to talk about existence and non-existence of invariant Einstein metrics on compact homogeneous manifolds. In this context, we have that it is possible to associate to every homogeneous space a graph. Then, the graph theorem of Bohm, Wang and Ziller gives an existence result of invariant Einstein metrics on a compact homogeneous space, based on properties of its graph. We are going to discuss this theorem and sketch its proof.

We begin by showing the underlying ideas Bourgain used to prove that the Cayley graph of the free group of finite rank can be embedded into a Hilbert space with logarithmic distortion. Equipped with these ideas we then tackle the same problem for other metric spaces. Time permitting these will be: amalgamated products and HNN extensions over finite groups, uniformly discrete hyperbolic spaces with bounded geometry and Cayley graphs of cyclic extensions of small cancellation groups.

We'll discuss 2 ways to decompose a 3-manifold, namely the Heegaard

splitting and the celebrated geometric decomposition. We'll then see

that being hyperbolic, and more in general having (relatively)

hyperbolic fundamental group, is a very common feature for a 3-manifold.

I will describe why e^{\pi\sqrt{163}} is almost an integer and how this is related to Q(\sqrt{-163}) having class number one and why n^2-n+41 is prime for n=0,...,39. Bits and pieces about Gauss's class number problem, Heegner numbers, the j-invariant and complex multiplication on elliptic curves will be discussed along the way.