Past

Abstract: I will talk about developments in my ongoing project to understand algebraic modules for finite groups, in particular for V_4 blocks, and their relation with the Puig finiteness conjecture. I will discuss a new (as in 5th of November) theorem of mine that generalizes results of Alperin and myself.

We give different criteria for transience of branching Markov chains. These conditions enable us to give a classification of branching random walks in random environment (BRWRE) on Cayley graphs in recurrence and transience. This classification is stated explicitly for BRWRE on $\Z^d.$ Furthermore, we emphasize the interplay between branching Markov chains, the spectral radius, and some generating functions.

Apologies - this seminar is CANCELLED

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.

Defined in terms of $\zeta(\frac{1}{2} +it)$ are the Riemann-Siegel functions, $\theta(t)$ and $Z(t)$. A zero of $\zeta(s)$ on the critical line corresponds to a sign change in $Z(t)$, since $Z$ is a real function. Points where $\theta(t) = n\pi$ are called Gram points, and the so called Gram's Law states between each Gram point there is a zero of $Z(t)$, and hence of $\zeta(\frac{1}{2} +it)$. This is known to be false in general and work will be presented to attempt to quantify how frequently this fails.

We study randomized (i.e. Monte Carlo) algorithms to compute expectations of Lipschitz functionals w.r.t. measures on infinite-dimensional spaces, e.g., Gaussian measures or distribution of diffusion processes. We determine the order of minimal errors and corresponding almost optimal algorithms for three different sampling regimes: fixed-subspace-sampling, variable-subspace-sampling, and full-space sampling. It turns out that these minimal errors are closely related to quantization numbers and Kolmogorov widths for the underlying measure. For variable-subspace-sampling suitable multi-level Monte Carlo methods, which have recently been introduced by Giles, turn out to be almost optimal.

Joint work with Jakob Creutzig (Darmstadt), Steffen Dereich (Bath), Thomas Müller-Gronbach (Magdeburg)

In ordinary percolation, sites of a lattice are open with a given probability and one investigates the existence of infinite clusters (percolation). In dynamical percolation, the sites randomly flip between the states open and closed and one investigates the existence of "atypical" times at which the percolation structure is different from that of a fixed time.

1. I will quickly present some of the original results for dynamical percolation (joint work with Olle Haggstrom and Yuval Peres) including no exceptional times in critical percolation in high dimensions.

2. I will go into some details concerning a recent result that, for the 2 dimensional triangular lattice, there are exceptional times for critical percolation (joint work with Oded Schramm). This involves an interesting connection with the harmonic analysis of Boolean functions and randomized algorithms and relies on the recent computation of critical exponents by Lawler, Schramm, Smirnov, and Werner.

3. If there is time, I will mention some very recent results of Garban, Pete, and Schramm on the Fourier spectrum of critical percolation.

The Snaer program calculates the posterior mean and variance of variables on some of which we have data (with precisions), on some we have prior information (with precisions), and on some prior indicator ratios (with precisions) are available. The variables must satisfy a number of exact restrictions. The system is both large and sparse. Two aspects of the statistical and computational development are a practical procedure for solving a linear integer system, and a stable linearization routine for ratios. We test our numerical method for solving large sparse linear least-squares estimation problems, and find that it performs well, even when the $n \times k$ design matrix is large ( $nk = O (10^{8})$ ).