Josef Slapal (Brno), Martin Schneider (Dresden)
Josef Slapal (Brno University of Technology): Digital topology
Digital topology is a term that has arisen for the study of geometric and topological properties of digital images. Because of our incresingly digital world, such a study plays an important role in a wide range of diverse applications. Digital topology provides theoretical foundations for important image processing operations such as object counting, thinning, boundary detection and contour filling having so important applications in computer graphics, computer tomography, pattern analysis, robotic design, etc. A basic problem of digital topology is to provide the digital plane ZxZ with a convenient structure, meaning by this that the structure should satisfy analogues of some basic geometric properties of the Euclidean plane RxR. Of these analogues, the validity a digital version of the Jordan curve theorem plays a crucial role. Despite the name, the traditional approach to digital topology has been based on using graph-theoretic tools rather than topological ones. It was only in late 90's that an alternative, topological approach was proposed which uses topology explicitly for structuring the digital plane. In the lecture, we will develop and generalize the topological approach by using pretopologies (i.e., structures more general than topologies) instead of topologies for structuring the digital plane.
Martin Schneider (TU Dresden): Amenable topological groups
Having its roots in the beginnings of modern measure theory, the concept of amenability has become of central importance to infinite group theory and topological dynamics. The study of amenable groups began in 1929 with von Neumann, who introduced them in order to explain why the Banach-Tarski paradox occurs only for dimension greater than two. Since then amenable groups have proved advantageous in many respects. In fact, the class of amenable groups is vast and allows defining dynamical invariants by means of a well-behaved averaging process. Among the most important tools for studying amenable groups ranges Folner's amenability criterion, as it offers a method of approximating an amenable group in terms of almost invariant finite subsets. Folner's theorem has facilitated a number of interesting applications within several areas of mathematics, including operator theory and ergodic theory.
Recent developments in Ramsey theory have revealed the necessity of a better understanding of amenability for general (in particular non-locally compact) topological groups. In the talk I will give an introduction to amenability for discrete and topological groups, and present a generalization of Folner's theorem for topological groups by means of topological matchings, which originates from a joint work with Andreas Thom.