1.3. Topology
Programme 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
Cech-Stone compactifications of the reals and the integers, as well as more
recent objects like certain compact L-spaces, spaces of chaotic maps and "nice"
two-point sets in the plane. Most of these objects are studied using methods from
several branches of general and geometric topology, as well as non-topological
methods, most notably from functional analysis, measure theory, set theory and
geometry. The topological methods can be as diverse as infinite-dimensional
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 so-called colourings of maps, related to the question whether a
fixed-point-free selfmap f on X extends to a fixed-point-free
selfmap on the Cech-Stone compactification of X.
Here one meets
classical results from algebraic and geometric topology, like the
Ljusternik-Schnirelman theorem and the Borsuk-Ulam theorem, non-trivial results
in topological groups, methods from dimension theory and topological dynamics, as
well as set-theoretic 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)
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