Oxford

By classical results of Tarski and Artin-Schreier, the elementary theory of the field of real numbers can be axiomatized in purely Galois-theoretic terms by describing the absolute Galois group of the field. Using work of Ax-Kochen/Ershov and a p-adic analogue of the Artin-Schreier theory the same can be proved for the field $\mathbb{Q}_p$ of p-adic numbers and for very few other fields.

Replacing, however, the absolute Galois group of a field K by that of the rational function field $K(t)$ over $K$, one obtains a Galois-theoretic axiomatiozation of almost arbitrary perfect fields. This gives rise to a new approach to longstanding decidability questions for fields like

$F_p((t))$ or $C(t)$.

Hypergraph Transversals have been studied in Mathematics for a long time (e.g. by Berge) . Generating minimal transversals of a hypergraph is an important problem which has many applications in Computer Science, especially in database Theory, Logic, and AI. We give a survey of various applications and review some recent results on the complexity of computing all minimal transversals of a given hypergraph.

One of the main tools in studying sparse random graphs with independence between different edges is local comparison with branching processes. Recently, this method has been used to determine the asymptotic behaviour of the diameter (largest graph distance between two points that are in the same component) of various sparse random graph models, giving results for $G(n,c/n)$ as special cases. Nick Wormald and I have applied this method to $G(n,c/n)$ itself, obtaining a much stronger result, with a best-possible error term. We also obtain results as $c$ varies with $n$, including results almost all the way down to the phase transition.

Random planar curves arise in a natural way in statistical mechanics, for example as the boundaries of clusters in critical percolation or the Ising model. There has been a great deal of mathematical activity in recent years in understanding the measure on these curves in the scaling limit, under the name of Schramm-Loewner Evolution (SLE) and its extensions. On the other hand, the scaling limit of these lattice models is also believed to be described, in a certain sense, by conformal field theory (CFT). In this talk, after an introduction to these two sets of ideas, I will give a theoretical physicist's viewpoint on possible direct connections between them.

John Cardy studied Mathematics at Cambridge. After some time at CERN, Geneva he joined the physics faculty at Santa Barbara. He moved to Oxford in 1993 where he is a Senior Research Fellow at All Souls College and a Professor of Physics. From 2002-2003 and 2004-2005 he was a member of the IAS, Princeton. Among other work on the applications of quantum field theory, in the 1980s he helped develop the methods of conformal field theory. Professor Cardy is a Fellow of the Royal Society, a recipient of the 2000 Paul Dirac Medal and Prize of the Institute of Physics, and of the 2004 Lars Onsager Prize of the American Physical Society "for his profound and original applications of conformal invariance to the bulk and boundary properties of two-dimensional statistical systems."