Oxford

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.

Abstract: For a large class of compactifications of interest in string phenomenology, the task of finding vacua of the four dimensional effective theories can be rewritten as a simple problem in algebraic geometry. Using recent developments in computer algebra, the task can then be rapidly dealt with in a completely algorithmic fashion. I shall review the main points of hep-th/0606122 and hep-th/0703249 in which this approach to finding vacua was set out, before moving on to a description of the Mathematica package STRINGVACUA (as described in arXiv:0801.1508 [hep-th]). This package uses the power of the computer algebra system Singular and provides a user-friendly implementation of our methods, intended for use by physicists, within the comfortable working environment of Mathematica.

Abstract: The moduli space of Calabi-Yau manifolds have a natural geometrical structure that has come to be known as special geometry. This geometry will be reviewed in the complex context and it will be shown that much of the structure persists for p-adic Calabi-Yau manifolds.