ChCh, Tom Gate, Room 2

This talk will summarize some of the problems and conjectures that I haven't managed to solve (although I have tried to) while spending my three years in this job. It will cover the areas of group theory, representation theory, both of general finite groups and of symmetric groups, and fusion systems.

The talk will start with the definition of amenable groups. I will discuss various properties and interesting facts about them. Those will be underlined with representative examples. Based on this I will give the definition and some basic properties of sofic groups, which only emerged quite recently. Those groups are particularly interesting as it is not know whether every group is sofic.

The long-open Tarski problem asked whether first-order logic can distinguish between free groups of different ranks. This was finally answered in the negative by the works of Sela and Kharlampovich-Myasnikov, which sparked renewed interest in the model theoretic properties of free groups. I will give a survey of known results and open questions on this topic.

Many problems in computer science can be modelled as metric spaces, whereas for mathematicians they are more likely to appear as the opening question of a second year examination. However, recent interesting results on the geometry of finite metric spaces have led to a rethink of this position. I will describe some of the work done and some (hopefully) interesting and difficult open questions in the area.

The representation theory of the symmetric groups is far more advanced than that of arbitrary finite groups. The blocks of symmetric groups with defect group of order pⁿ are classified, in the sense that there is a finite list of possible Morita equivalence types of blocks, and it is relatively straightforward to write down a representative from each class.

In this talk we will look at the case where n=2. Here the theory is fairly well understood. After introducing combinatorial wizardry such as cores, the abacus, and Scopes moves, we will see a new result, namely that the simple modules for any p-block of weight 2 "come from" (technically, have isomorphic sources to) simple modules for S_2p or the wreath product of S_p and C₂.

$\int_{0}^\infty$

This talk will be divided into three parts. In the first part we will recall basic notions and facts of differential geometry and the Ricci flow equation. In the second part we will talk about singularities for the Ricci flow and Ricci flow on homogeneous spaces. Finally, in the third part

of the talk, we will focus on the case of Ricci flow on compact homogeneous spaces with two isotropy summands.