Past

In the Examination Schools

Reflected Brownian motion (RBM) in a two-dimensional wedge is a well-known stochastic process. With an appropriate drift, it is positive recurrent and has a stationary distribution, and the invariant measure is absolutely continuous with respect to Lebesgue measure. I will give necessary and sufficient conditions for the stationary density to be written as a finite sum of exponentials with linear exponents. Such densities are a natural generalisation of the stationary density of one-dimensional RBM. Using geometric ideas reminiscent of the reflection principle, I will give an explicit formula for the density in such cases, which can be written as a determinant. Joint work with Ton Dieker.

The symmetries of a dynamical system have a big effect on its typical behaviour. The most obvious effect is pattern formation - the dynamics itself may be symmetric, though often the symmetry of the system is 'broken', and the state has less symmetry than the system. The resulting phenomena are fairly well understood for steady and time-periodic states, and quite a bit can be said for chaotic dynamics. More recently, the concept of 'symmetry' has been generalised to address applications to physical and biological networks. One consequence is a new approach to patterns of synchrony and phase relations. The lecture will describe some of the high points of the emerging theories, including applications to evolution, locomotion, human balance and fluid dynamics.

Adaptive discretizations and iterative multilevel solvers are nowadays well established techniques for the numerical solution of PDEs.

The development of efficient multilevel techniques in the context of PDE-constrained optimization methods is an active research area that offers the potential of reducing the computational costs of the optimization process to an equivalent of only a few PDE solves.

We present a general class of inexact adaptive multilevel SQP-methods for PDE-constrained optimization problems. The algorithm starts with a coarse discretization of the underlying optimization problem and provides

1. implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and

2. implementable accuracy requirements for iterative solvers of the PDE and adjoint PDE on the current grid

such that global convergence to the solution of the infinite-dimensional problem is ensured.

We illustrate how the adaptive refinement strategy of the multilevel SQP-method can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equation. Moreover, we discuss the efficient handling of control constraints and describe how efficent multilevel preconditioners can be constructed for the solution of the arising linear systems.

Numerical results are presented that illustrate the potential of the approach.

This is joint work with Jan Carsten Ziems.

I will discuss how one can construct nice cellular

algebras using the cohomology of Springer fibres associated with two

block nilpotent matrices (and the convolution product). Their

quasi-hereditary covers can be described via categories of highest

weight modules for the Lie algebra sl(n). The combinatorics of torus

fixed points in the Springer fibre describes decomposition

multiplicities for the corresponding highest weight categories. As a

result one gets a natural subcategory of coherent sheaves on a

resolution of the slice to the corresponding nilpotent orbit.

Will discuss several constructions of the Grothendieck group in different contexts together with Wall's solution of the problem of determining homotopy types of finite CW complexes as a motivating application.

I will speak about the Zilber trichotomy for weakly minimal difference varieties, and the definable structure on them.

A difference field is a field with a distinguished automorphism $\sigma$. Solution sets of systems of polynomial difference equations like

$3 x \sigma(x) +4x +\sigma^2(x) +17 =0$ are the quantifier-free definable subsets of difference fields. These \emph{difference varieties} are similar to varieties in algebraic geometry, except uglier, both from an algebraic and from a model-theoretic point of view.

ACFA, the model-companion of the theory of difference fields, is a supersimple theory whose minimal (i.e. U-rank $1$) types satisfy the Zilber's Trichotomy Conjecture that any non-trivial definable structure on the set of realizations of a minimal type $p$ must come from a definable one-based group or from a definable field. Every minimal type $p$ in ACFA contains a (weakly) minimal quantifier-free formula $\phi_p$, and often the difference variety defined by $\phi_p$ determines which case of the Zilber Trichotomy $p$ belongs to.

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 ( $nk = O (10^{8})$ ).

There are many triangulated categories that arise in the study

of group cohomology: the derived, stable or homotopy categories, for

example. In this talk I shall describe the relative cohomological

versions and the relationship with ordinary cohomology. I will explain

what we know (and what we would like to know) about these categories, and

how the representation type of certain subgroups makes a fundamental

difference.

Given an algebraic variety $X$ over the finite field ${\bf F}_{q}$, it is known that the zeta function of $X$,

$$ Z(X,T):=\mbox{exp}\left( \sum_{k=1}^{\infty} \frac{#X({\bf F}_{q^{k}})T^{k}}{k} \right) $$

is a rational function of $T$. It is an ongoing topic of research to efficiently compute $Z(X,T)$ given the defining equation of $X$.

I will summarize how we can use Berthelot's rigid cohomology (sparing you the actual construction) to compute $Z(X,T)$, first done for hyperelliptic curves by Kedlaya. I will go on to describe Lauder's deformation algorithm, and the promising fibration algorithm, outlining the present drawbacks.