Past

The evaluation of oscillatory integrals is often considered to be a computationally challenging problem. However, in many cases, the exact opposite is true. Oscillatory integrals are cheaper to evaluate than non-oscillatory ones, even more so in higher dimensions. The simplest strategy that illustrates the general approach is to truncate an asymptotic expansion, where available. We show that an optimal strategy leads to the construction of certain unconventional Gaussian quadrature rules, that converge at twice the rate of asymptotic expansions. We examine a range of one-dimensional and higher dimensional, singular and highly oscillatory integrals.

Introduction to the topological degree: Existence and Uniqueness of the Brouwer degree, Existence and Uniqueness of the Leray-Schauder degree.

We survey the classification of structures interpretable in o-minimal theories in terms of thorn-minimal types. We show that a necessary and sufficient condition for such a structure to interpret a real closed field is that it has a non-locally modular unstable type. We also show that assuming Zilber's Trichotomy for strongly minimal sets interpretable in o-minimal theories, such a structure interprets a pure algebraically closed field iff it has a global stable non-locally modular type. Finally, if time allows, we will discuss reasons to believe in Zilber's Trichotomy in the present context

A graph property is a first order property if it can be written as a logic sentence with variables ranging over the vertices of the graph.

A sequence of random graphs (G_n)_n satisfies the zero-one law if the probability that G_n satisfies P tends to either zero or one for every first order property P. This is for instance the case for G(n,p) if p is fixed. I will survey some of the most important results on the G(n,p)-model and then proceed to discuss some work in progress on other graph models.

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.

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

The Skorokhod reflection problem, originally introduced as a means for constructing solutions to stochastic differential equations in bounded regions, has found applications in many areas of Probability, for example in queueing-like stochastic dynamical systems; its uses range from methods for proving limit theorems to representations of local times of diffusions and control. In this talk, I will present several applications, e.g. to Levy stochastic networks and to queueing-like systems driven by local times of Levy processes, and give an order-theoretic approach to the problem by extending the domain of functions involved from the real line to a fairly arbitrary partially ordered set. I will also discuss how Palm probabilities can be used in connection with the Skorokhod problem to obtain information about stationary solutions of certain systems.

Recent joint work with Greg McShane has answered the following question: Which curves can be short in a given cell of the decomposition of Teichmueller space? The answer involves a new combinatorial structure called "screens on fatgraphs" as we shall describe. The techniques of proof involve Fock's path-ordered product expansion of holonomies, Ptolemy transformations, and the triangle inequalities. This is a main step in giving a combinatorial description of the Deligne-Mumford compactification of moduli space which we shall also discuss as time permits.

I consider the Metropolis Markov Chain based on the nearest neighbor random walk on the positive half of the Mean Field Ising Model, i.e., on those vectors from $\{−1, 1\}^N$ which contain more $1$ than $−1$. Using randomly-chosen paths I prove a lower bound for the Spectral Gap of this chain which is of order $N^-2$ and which does not depend on the inverse temperature $\beta$. In conjunction with decomposition results such as those in Jerrum, Son, Tetali and Vigoda (2004) this result may be useful for bounding the spectral gaps of more complex Markov chains on the Mean Field Ising Model which may be decomposed into Metropolis chains. As an example, I apply the result to two Multilevel Markov Chain Monte Carlo algorithms, Swapping and Simulated Tempering. Improving a result by Madras and Zheng (2002), I show that the spectral gaps of both algorithms on the (full) Mean Field Ising Model are bounded from below by the reciprocal of a polynomial in the lattice size $N$ and in the inverse temperature $\beta$.