1.3. Topology
Programme leader: 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.
