The Zakharov-Kuznetsov equation (ZK) is a higher dimensional analogue ot the famous Korteweg-de Vries equation (KdV). For three space dimensions this equation was derived in 1974 by Zakharov and Kuznetsov as a model for the propagation of sound waves in a magnetized plasma. We consider the Cauchy Problem in R^n, n=2,3, for a generalisation of (ZK) to higher integer powers. The data are assumed to belong to the classical Sobolev spaces H^s or, more generally, to the corresponding Besov spaces. For data of critical regularity local well-posedness and small data global well-posedness can be obtained. The proof relies substantially on the Kato-smoothing effect and a maximal function estimate for solutions of the homogeneous linear equation.

The Muskat problem is a classical model for the motion of two immiscible fluid layers in a porous medium. The model was first proposed in 1934 to describe the intrusion of water into an oil reservoir and it is connected to the oil extraction process. An important aspect of the mathematical analysis of the Muskat problem is the fact that the equations of motion can be formulated as an evolution problem for the free boundary, separating the fluids, only.

In this talk we discuss two different regimes of the Muskat problem: a confined regime where the layers have bounded positive thicknesses, and secondly the unconfined Muskat problem where the fluid layers are considered unbounded. We establish the well-posedness of the problem and other qualitative aspects of the fluid motion for both settings.

Given a set of polynomial equations with real coefficients, we would like to study the set of solutions. If this one is infinite, we can define the dimension, and try to understand what is the topology of such locus. In particular, can we get any possible surface? And can we do this with simple equations?

I will give a survey on the topic, supposed to be understandable by a large audience.

Self similar fractals are often used in modeling porous media. Hence, defining a Laplacian and a Brownian motion on such sets describes transport through such materials. However, the assumption of strict self similarity could be too restricting. So, we present several models of random fractals which could be used instead. After recalling the classical approaches of random homogenous and recursive random fractals, we show how to interpolate between these two model classes with the help of so called V-variable fractals. This concept (developed by Barnsley, Hutchinson & Stenflo) allows the definition of new families of random fractals, hereby the parameter V describes the degree of ‘variability’ of the realizations. We discuss how the degree of variability influences the geometric, analytic and stochastic properties of these sets.
- These results have been obtained with Ben Hambly (University of Oxford) and John Hutchinson (ANU Canberra).

Symmetry of a manifold allows to obtain information on its topology, as is demonstrated for example by the classical theorem of Poincaré-Hopf. The main topic of this talk is GKM theory, which associates to certain kinds of torus actions on manifolds a graph from which one can compute the equivariant cohomology of the action, as well as the de Rham cohomology of the manifold that is acted on. At the end we will explain a result about the GKM theory of isometric actions on positively curved manifolds.

We discuss function spaces that are defined with the help of nonlocal quadratic forms. One aim of the talk is to investigate conditions under which such a form is comparable with the quadratic form that defines the seminorm of a fractional Sobolev space. We provide several examples including those with singular measures and one example related to the study of the Boltzmann equation. We outline the proof of comparability in the latter case, which involves delicate chaining and renormalization arguments. A second aim of the talk is to explain the significance of the aforementioned comparability results with regard to Dirichlet forms and regularity of solutions to integrodifferential equations. Last, we mention recent results on function spaces that appear in the study of nonlocal Dirichlet problems. The talk is based on joint works with Bartek Dyda resp. with Kai-Uwe Bux and Tim Schulze.

We give an overview over the classification of 3-dimensional manifolds and we report on the recent breakthroughs by Ian Agol and Dani Wise.

A module - a linear representation of a ring (or other object) – usually is defined to be an abelian group (or vector space) with a structured collection of actions. But under Morita, or more generally tilting, equivalence the underlying group is replaced and the collection of actions is replaced, yet we have essentially the same module. From the point of view of model theory this shift entails choosing a different home sort in the associated category of sorts. Algebraically, the process can be seen as choosing a different generator for an abelian category canonically associated to the module. Through this we are led to an alternative view of what a module is, which I will illustrate with some examples and applications.