Programme leaders: J.M. Aarts, J. van Mill
Central in this programme has been and will be the study of various topological objects ranging from the classical - Hilbert cube, plane continua arising in dynamical topological dynamics and the ech-Stone compactification - to the more recent - function spaces with the topology of pointwise convergence, two- and -points sets, and non-metric continua.
Techniques from many branches of mathematics are brought to bear on the study of these objects; first and foremost from the world of (general) topology itself but also from Functional Analysis, set Theory, Model Theory and Geometry.
Recent work on the structure of hereditarily indecomposable continua illustrates this: purely topological methods were combined with results from Model Theory in the construction of interesting examples of such spaces. In a totally different vein is the study of the structure of sets in the plane that meet every line in a specific number of points; the techniques here are very geometric in nature.
History and Philosophy of Mathematics
Since the last report the programme has been enriched by the addition of researchers in the history and philosophy of mathematics.
The historical research is embedded in the network of the Dutch Association for the History and Social Aspects of Mathematics. One focus is the history of geometry, in particular the history of kinematics and aspects of the history of topology. One Ph.D. thesis on the history of kinematics is being prepared. The philosophical research concerns the ideas of Imre Lakatos and investigatons with respect to some paradoxes.
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. 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.