L3

H. Jerome Keisler suggested to associate to each classical structure M a family of "random" structures consisting of random variables with values in M . Viewing the random structures as structures in continuous logic one is able to prove preservation results of various "good" model theoretic properties e.g., stability and dependence, from the original structure to its randomisation. On the other hand, simplicity is not preserved by this construction. The work discussed is mostly due to H.

Jerome Keisler and myself (given enough time I might discuss some applications obtains in joint work with Alex Usvyatsov).

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: We consider deformations of torsion-free $ G_2 $ structures, defined by the $ G_2 $-invariant 3-form $ \phi $ and compute the expansion of the Hodge star of $ \phi $ to fourth order in the deformations of $ \phi $. By considering M-theory compactified on a $ G_2 $ manifold, the $ G_2 $ moduli space is naturally complexified, and we get a Kahler metric on it. Using the expansion of the Hodge star of $ \phi $ we work out the full curvature of this metric and relate it to the Yukawa coupling.