1.3. TopologyProgramme leaders: J. van Mill
Description of the programme.
The programme focuses on the study of the topology of classical objects, like the Hilbert cube, continua in the plane which arise in topological dynamics, various topological groups and the CechStone compactifications of the reals and the integers, as well as more recent objects like certain compact Lspaces, spaces of chaotic maps and "nice" twopoint sets in the plane. Most of these objects are studied using methods from several branches of general and geometric topology, as well as nontopological methods, most notably from functional analysis, measure theory, set theory and geometry. The topological methods can be as diverse as infinitedimensional topology, descriptive set theory, continuum theory and the theory of ultrafilters. A good illustration of the kind of interaction one gets is provided by the study of socalled colourings of maps, related to the question whether a fixedpointfree selfmap f on X extends to a fixedpointfree selfmap on the CechStone compactification of X.
Here one meets classical results from algebraic and geometric topology, like the LjusternikSchnirelman theorem and the BorsukUlam theorem, nontrivial results in topological groups, methods from dimension theory and topological dynamics, as well as settheoretic methods to construct counterexamples.
This theory also gives rise to applications in the form of fixed point theorems, for instance.
Of course similar things can be said for other specific research interests within this programme.
Status of the programme.
The Dutch topology groups enjoy an excellent international reputation, and collaborate extensively with renowned researchers in Canada, the United States, Poland and the Czech Republic for instance. They play an important role in the international organization of topological research.
Within the Netherlands there are links to analysis, dynamical systems theory and probability theory, which sometimes provide sources of problems for topological research.
Examples are: Lyapounov's convexity theorem and Cantor sets in the plane with positive logarithmic capacity.
Especially the links with dynamical systems theory could provide insights in the asymptotical behaviour of physical systems.
This requires a deep understanding of the topology of the underlying spaces, and a solid knowledge of dimension theory and other branches of topology.
Research staff (situation at January 1,2007)
 Permanent staff
 Prof.dr. J.M. Aarts (TUD)
 Dr. J.J. Dijkstra (VUA)
 Dr. R.J. Fokkink (TUD)
 Dr. K.P. Hart (TUD)
 Dr.ir. T. Koetsier (VUA)
 Prof.dr. J. van Mill (VUA)
 Dr. M.L.J. van der Vel (VUA)
 Ph.D. students
 M. Abry Msc. (VUA)
 D. Basile (VUA)
 Drs. W. Rekers (VUA)
 Drs. G.J.F. Ridderbos (VUA)
 Drs. K.I.S. Valkenburg (VUA)
 D. Visser (VUA)
