# Past Kinderseminar

In 1924, James W. Alexander constructed a 2-sphere in R3 that is not ambiently homeomorphic to the standard 2-sphere, which demonstrated the failure of the Schoenflies theorem in higher dimensions. I will describe the construction of the Alexander horned sphere and the Antoine necklace and describe some of their properties.

Since the symmetric group is a finite group it’s representation theory is not too complex, however in this special case we can realise these representations in a particular nice combinatorial way using young tableaux and young symmetrizers. I will introduce these ideas and use them to describe the representation theory of Sn over the complex numbers.

I will talk about a remarkable theorem by Paulin, which says

that if a one-ended hyperbolic group has infinite outer automorphism

group, then it splits over a two-ended subgroup. In particular, this

gives a condition which ensures a hyperbolic group doesn't have property

(T).

The representation dimension of an algebra was introduced in the early 70's by M. Auslander, with the goal of measuring how far an algebra is from having finite number of finitely generated indecomposable modules (up to isomorphism). This invariant is not well understood. For instance, it was not until 2002 that O. Iyama proved that every algebra has finite representation dimension. This was done by constructing special quasihereditary algebras. In this talk I will give an introduction to this topic and I shall briefly explain Iyama's construction.

I will give a historical overview of some of the theorems proved by the

Ancient Greeks, which are now taken for granted but were, and are,

landmarks in the history of mathematics. Particular attention will be

given to the calculation of areas, including theorems of Hippocrates,

Euclid and Archimedes.

Many people talk about properties that you would expect of a group. When they say this they are considering random groups, I will define what it means to pick a random group in one of many models and will give some properties that these groups will have with overwhelming probability. I will look at the proof of some of these results although the talk will mainly avoid proving things rigorously.