C5

Algebraically closed fields, and in general varieties are among the first examples
of Zariski Geometries.
I will consider specialisations of algebraically closed fields and varieties.
In the case of an algebraically closed field K, I will show that a specialisation
is essentially a residue map, res from K to a residue field k.  
In both cases I will show universality of the specialisation is controlled by the
transcendence degree of K over k.  

Given a field K and an ordered abelian group G, we can form the field K((G)) of generalised formal power series with coefficients in K and indices in G. When is this field decidable? In certain cases, decidability reduces to that of K and G. We survey some results in the area, particularly in the case char K > 0, where much is still unknown.

Given a field K and an ordered abelian group G, we can form the field K((G)) of generalised formal power series with coefficients in K and indices in G. When is this field decidable? In certain cases, decidability reduces to that of K and G. We survey some results in the area, particularly in the case char K > 0, where much is still unknown.

Inexact hardware trades reduced numerical precision against a reduction

in computational cost. A reduction of computational cost would allow

weather and climate simulations at higher resolution. In the first part

of this talk, I will introduce the concept of inexact hardware and

provide results that show the great potential for the use of inexact

hardware in weather and climate simulations. In the second part of this

talk, I will discuss how rounding errors can be assessed if the forecast

uncertainty and the chaotic behaviour of the atmosphere is acknowledged.

In the last part, I will argue that rounding errors do not necessarily

degrade numerical models, they can actually be beneficial. This

conclusion will be based on simulations with a model of the

one-dimensional Burgers' equation.

The common convention when dealing with hyperbolic groups is that such groups are finitely

generated and equipped with the word length metric relative to a finite symmetric generating

subset. Gromov's original work on hyperbolicity already contained ideas that extend beyond the

finitely generated setting. We study the class of locally compact hyperbolic groups and elaborate

on the similarities and differences between the discrete and non-discrete setting.

Since their introduction, Shimura varieties have proven to be important landmarks sitting right at the crossroads between algebraic geometry, number theory and representation theory. In this talk, starting from the yoga of motives and Hodge theory, we will try to motivate Deligne's construction of Shimura varieties, and briefly survey some of their zoology and basic properties. I may also say something about the links to automorphic forms, or their integral canonical models.

We state a conjecture about the size of the intersection between a bounded-rank progression and a sphere, and we prove the first interesting case, a result of Chang. Hopefully the full conjecture will be obvious to somebody present.