L3

The Rado Graph R is a graph on a countably infinite number of vertices that can be characterized by the following property: for every pair of finite, disjoint subsets of vertices X and Y, there exists a vertex that is adjacent to every vertex in X and none in Y. It is often called the Random Graph for the following reason: for any 0

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.

Let $K \subset {\mathbb R}^d$ be a convex set. Choose $n$ random points in $K$, and denote by $P_n$ their convex hull. We call $P_n$ a random polytope. Investigations concerning the expected value of functionals of $P_n$, like volume, surface area, and number of vertices, started in 1864 with a problem raised by Sylvester and now are a classical part of stochastic and convex geometry. The last years have seen several new developments about distributional aspects of functionals of random polytopes. In this talk we concentrate on these recent results such as central limit theorems and tail inequalities, as the number of random points tends to infinity.

A central task in conservation biology is measuring, predicting, and preserving biological diversity as species face extinction. Dating back to 1992, phylogenetic diversity is a prominent notion for measuring the biodiversity of a collection of species. This talk gives a flavour of some the combinatorial and algorithmic problems and recent solutions associated with computing this measure. This is joint work with Magnus Bordewich (Durham University, UK) and Andreas Spillner (University of East Anglia, UK).

We show that the diameter of G(n,p) is concentrated on one of three values provided the average degree p(n-1) goes to infity with n. This is joint work with N. Fountoulakis even though he refuses to admit it.

On parabolic and elliptic equations with VMO coefficients.

Abstract: An L_p-theory of divergence and non-divergence form elliptic and parabolic equations is presented.

The main coefficients are supposed to belong to the class VMO_x, which, in particular, contains all functions independent of x.

Weak uniqueness of the martingale problem associated with such equations is obtained