Past Kinderseminar

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.

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.

I will give a short introduction to non-standard analysis using Nelson's Internal Set Theory, and attempt to give some interesting examples of what can be done in NSA. If time permits I will look at building models for IST inside the usual ZFC set theory using ultrapowers.

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.

I will give an introduction to Kashiwara's theory of crystal bases. Crystals are combinatorial objects associated to integrable modules for quantum groups that, together with the related notion of crystal bases, capture several combinatorial aspects of their representation theory.

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.

Given a block, b, of a finite group, Alperin's weight conjecture predicts a miraculous equality between the number of isomorphism classes of simple b-modules and the number of G-orbits of b-weights. Radha Kessar showed that the latter can be written in terms of the fusion system of the block and Markus Linckelmann has computed it as an Euler characteristic of a certain space (provided certain conditions hold). We discuss these reformulations and give some examples.

I am going to introduce Thompson's groups F, T and V. They can be seen in two ways: as functions on [0,1] or as isomorphisms acting on trees.

We will investigate what one can detect about a discrete group from its profinite completion, with an emphasis on considering geometric properties.

This talk will be an introduction to property (T). It was originally introduced by Kazhdan as a method of showing that certain discrete subgroups of Lie groups are finitely generated, but has expanded to become a widely used tool in group theory. We will take a short tour of some of its uses.

From a categorical point of view, the standard Zermelo-Frankel set theoretic approach to the foundations of mathematics is fundamentally deficient: it is based on the notion of equality of objects in a set. Equalities between objects are not preserved by equivalences of categories, and thus the notion of equality is 'incorrect' in category theory. It should be replaced by the notion of 'isomorphism'.

Moving higher up the categorical ladder, the notion of isomorphism between objects is 'incorrect' from the point of view of 2-category, and should be replaced by the notion of 'equivalence'...

Recently, people have started to take seriously the idea that one should be less dogmatic about working with set-theoretic axiomatisiations of mathematics, and adopt the more fluid point of view that different foundations of mathematics might be better suited to different areas of mathematics. In particular, there are currently serious attempts to develop foundations for mathematics built on homotopy types, or, in another language, ∞-groupoids.

An (∞,1)-topos should admit an internal 'homotopical logic', just as an ordinary (1-)topos admits an internal logic modelling set theory.

It turns out that formalising such a logic is rather closely related to the problem of finding good foundations for 'intensional dependent type theory' in theoretical computer science/logic. This is sometimes referred to as the attempt to construct a 'homotopy lambda calculus'.

It is expected that a homotopy theoretic formalisation of the foundations of mathematics would be of genuine practical significance to the average mathematician!

In this talk we will give an introduction to these ideas, and to the recent work of Vladimir Voevodsky and others in this area.

In this talk we will survey some aspects of social choice theory: in particular, various impossibility theorems about voting systems and strategies. We begin with the famous Arrow's impossibility theorem -- proving the non-existence of a 'fair' voting system -- before moving on to later developments, such as the Gibbard–Satterthwaite theorem, which states that all 'reasonable' voting systems are subject to tactical voting.

Given time, we will study extensions of impossibility theorems to micro-economic situations, and common strategies in game theory given the non-existence of optimal solutions.

Not only does the definition of an (abstract) saturated fusion system provide us with an interesting way to think about finite groups, it also permits the construction of exotic examples, i.e. objects that are non-realisable by any finite group. After recalling the relevant definitions of fusion systems and saturation, we construct an exotic fusion system at the prime 3 as the fusion system of the completion of a tree of finite groups. We then sketch a proof that it is indeed exotic by appealing to The Classification of Finite Simple Groups.

We will introduce both the classical Hanna Neumann Conjecture and its strengthened version, discuss Stallings' reformulation in terms of immersions of graphs, and look at some partial results. If time allows we shall also look at the new approach of Joel Friedmann.

I'll start with the definition of the first Grigorchuk group as an automorphism group on a binary tree. After that I give a short overview about what growth means, and what kinds of growth we know. On this occasion I will mention a few groups that have each kind of growth and also outline what the 'Gap Problem' was. Having explained this I will prove - or depending on the time sketch - why this Grigorchuk group has intermediate growth. Depending on the time I will maybe also mention one or two open problems concerning growth.

There are two competing notions for a normal subsystem of a (saturated) fusion system. A recent theorem of mine shows how the two notions are related. In this talk I will discuss normal subsystems and their properties, and give some ideas on why this might be useful or interesting.

The first part of Galois' Second Mémoire, less than three pages of manuscript written in 1830, is devoted to an amazing insight, far ahead of its time. Translated into modern mathematical language (and out of French), it is the theorem that a primitive soluble finite permutation group has prime-power degree. This, and Galois' ideas, and counterexamples to some of

them, will be my theme.