Oxford

Packings and coverings in graphs are related to two main problems of

graph theory, respectively error correcting codes and domination.

Given a set of words, an error correcting code is a subset such that

any two words in the subset are rather far apart, and can be

identified even if some errors occured during transmission. Error

correcting codes have been well studied already, and a famous example

of perfect error correcting codes are Hamming codes.

Domination is also a very old problem, initiated by some Chess problem

in the 1860's, yet Berge proposed the corresponding problem on graphs

only in the 1960's. In a graph, a subset of vertices dominates all the

graph if every vertex of the graph is neighbour of a vertex of the

subset. The domination number of a graph is the minimum number of

vertices in a dominating set. Many variants of domination have been

proposed since, leading to a very large literature.

During this talk, we will see how these two problems are related and

get into few results on these topics.

Taking the intersection form of a 4n-manifold defines a functor from a category of cobordisms to a symmetric monoidal category of quadratic forms. I will present the theory of the Maslov index and some higher-categorical constructions as variations on this theme.

In these (three) lectures, I will discuss the following topics:

1. The theorems of Ax on the elementary theory of finite and pseudo-finite fields, including decidability and quantifier-elimination, variants due to Kiefe, and connection to Diophantine problems.

2. The theorems on Chatzidakis-van den Dries-Macintyre on definable sets over finite and pseudo-finite fields, including their estimate for the number of points of definable set over a finite field which generalizes the Lang-Weil estimates for the case of a variety.

3. Motivic and p-adic aspects.

In these (three) lectures, I will discuss the following topics:

1. The theorems of Ax on the elementary theory of finite and pseudo-finite fields, including decidability and quantifier-elimination, variants due to Kiefe, and connection to Diophantine problems.

2. The theorems on Chatzidakis-van den Dries-Macintyre on definable sets over finite and pseudo-finite fields, including their estimate for the number of points of definable set over a finite field which generalizes the Lang-Weil estimates for the case of a variety.

3. Motivic and p-adic aspects.

Abstract: The alternative action for Yang-Mills theory, which Lionel Mason formulated in twistor space, explains some of the simplicities of gluon scattering amplitudes. We will review the derivation of the familiar CSW rules concerning tree-level scattering, show that the `missing' three-point amplitude can be correctly recovered and elucidate the connection with the canonical Lagrangian approach of Mansfied, Morris, et. al.