ChCh, Tom Gate, Room 2

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.

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.

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.

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.

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

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.

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.

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.

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.