C5

We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}} )$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction (joint work with Ilya Kapovich).

This will be an informal discussion developing the details of the Amplituhedron for the loop integrand.

The 'pyjama stripe' is the subset of the plane consisting of a vertical

strip of width epsilon about every integer x-coordinate. The 'pyjama

problem' asks whether finitely many rotations of the pyjama stripe about

the origin can cover the plane.

I'll attempt to outline a solution to this problem. Although not a lot

of this is particularly representative of techniques frequently used in

additive combinatorics, I'll try to flag up whenever this happens -- in

particular ideas about 'limit objects'.

TQFTs have received widespread attention in recent years. In mathematics

for example due to Lurie's proof of the cobordism hypothesis. In physics

they are used as toy models to understand structure, especially

boundaries and defects.

I will present a lattice construction of 2d Spin TFT. This mostly

motivated as both a toy model and stepping stone for a mathematical

construction of rational conformal field theories with fermions.

I will first describe a combinatorial model for spin surfaces that

consists of a triangulation and a finte set of extra data. This model is

then used to construct TFT correlators as morphisms in a symmetric

monoidal category, given a Frobenius algebra as input. The result is

shown to be independent of the triangulation used, and one obtains thus

a 2dTFT.

All results and constructions can be generalised to framed surfaces in a

relatively straightforward way.

This is joint work with Angus Macintyre. I will discuss new developments in 
our work on the model theory of adeles concerning model theoretic 
properties of adeles and related issues on adelic geometry and number theory.

One of the most classical questions of modern algebra is whether the group algebra of a torsion-free group can be embedded into a skew field. I will give a short survey about embeddability of group algebras into skew fields, matrix rings and, in general, continuous rings.