Past

We consider a linearly edge-reinforced random walk

on a class of two-dimensional graphs with constant

initial weights. The graphs are obtained

from Z^2 by replacing every edge by a sufficiently large, but fixed

number of edges in series.

We prove that a linearly edge-reinforced random walk on these graphs

is recurrent. Furthermore, we derive bounds for the probability that

the edge-reinforced random walk hits the boundary of a large box

before returning to its starting point.

Part I will also include an overview on the history of the model.

In part II, some more details about the proofs will be explained.

We shall present work in progress in collaboration with E. Hrushovski on the geometry of spaces of stably dominated types in connection with non archimedean geometry \`a la Berkovich

When liquidating large portfolios of securities one faces a trade off between adverse market impact of sell orders and the impatience to generate proceeds. We present a Black-Scholes model with an impact factor describing the market's distress arising from previous transactions and show how to solve the ensuing optimization problem via classical calculus of variations. (Joint work with Dirk Becherer, Humboldt Universität zu

Berlin)

Meshfree methods become more and more important for the numerical simulation of complex real-world processes. Compared to classical, mesh-based methods they have the advantage of being more flexible, in particular for higher dimensional problems and for problems, where the underlying geometry is changing. However, often, they are also combined with classical methods to form hybrid methods.

In this talk, I will discuss meshfree, kernel based methods. After a short introduction along the lines of optimal recovery, I will concentrate on results concerning convergence orders and stability. After that I will address efficient numerical algorithms. Finally, I will present some examples, including one from fluid-structure-interaction, which will demonstrate why these methods are currently becoming Airbus's preferred solution in Aeroelasticity.

The partition function of the random cluster model on a graph $G$ is also known as its Potts model partition function. (Only the points at which it is evaluated differ in the two models.) This is a multivariate generalization of the Tutte polynomial of $G$, and encodes a wealth of enumerative information about spanning trees and forests, connected spanning subgraphs, electrical properties, and so on.

An elementary property of electrical networks translates into the statement that any two distinct edges are negatively correlated if one picks a spanning tree uniformly at random. Grimmett and Winkler have conjectured the analogous correlation inequalities for random forests or random connected spanning subgraphs. I'll survey some recent related work, partial results, and more specific conjectures, without going into all the gory details.

We outline a method to solve the stationary Einstein equations with source a body in rigid rotation consisting of elastic matter.

This is work in progress by R.B., B.G.Schmidt, and L.Andersson