In the early 70's I started doing research in category theory,
topology and analysis.
General problem:
Let G be an arbitrary topological group and let
TOPG be the category of
G-spaces and equivariant continuous mappings. Which
results about the category TOP{e}
(i.e., which theorems from ordinary general topology) can be
generalized to the category TOPG for
non-trivial G? For a general overview, see [2]; for a more
specific result, see [1] and [3].
Publications [4] and [5] deal with an equivariant version of
Glickberg's Theorem.
Related with this, from a categorical point of view, is [6].
Gradually, I became interested in abstract topological dynamics: the theory of dynamical systems phrased in terms of G-spaces (where G plays the role of the space of time-values; in ordinary dynamical systems theory G = R or Z). This eventually led to the book Elements of Topological Dynamics. Some gems not contained in the book are in [7] and [8]. An overview of the research area of (abstract) topological dynamics is given in [9].
Other interests: almost periodic functions on groups and semigroups (see [10]), representation theory of topological and Lie groups (see [11] and [12]), and the structure of semisimple Lie groups (see [13]).
After my retirement I re-worked and translated my lecture notes on topological dynamics (discrete systems on intervals, shift systems and an introduction into chaos) - in Dutch - into the introductory book Topological Dynamical Systems.
| [1a] | Topological transformation groups, a
categorical approach,
Mathematical Centre Tracts 65, Mathematisch Centrum, Amsterdam, 1975. |
| [1b] | Linearization of actions of locally
compact groups, Proc. Steklov Inst. Math., 1984, Issue 4, pp. 57-74. |
| [2] | Problems and results in the category of topological
transformation groups, CWI Quarterly 1 (1988), 29-35. |
| [3a] | Equivariant embeddings of G-spaces, in: J.Novak (ed.), General Topology and its Relations to Modern Analysis and Algebra IV, Part B (Proc. 4th Prague Top. Sym., 1976), Prague, 1977, pp. 485-49. |
| [3b] | On the existence of G- compactifications, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 275-280. |
| [4a] | On the G- compactification of products, Pacific J. Math. 110 (1984), 447-470. |
| [4b] | A note on the G- space version of Glicksberg's
theorem, Pacific J. Math. 122 (1986), 493-495. |
| [5] | G- spaces: compactifications
and pseudocompactness, in: A.Czaszar (ed.), Topology, theory and Applications (Proc. Conference at Eger, 1983), Colloquia Mathematica Societatis Janos Bolyai 41, North-Holland, Amsterdam, 1985, 655-666. |
| [6a] | (with M.Husek) Preservation of products by functors close
to reflectors, Topology and Appl. 27, 171-189. |
| [6b] | (with M.Husek) A note on compactifications of products of
semigroups, Semigroup Forum 38 (1989), 85-89. |
| [7] | Two applications of topological dynamics in combinatorial
number theory, CWI Newsletter 2, 1984, pp. 2-17. |
| [8] | The Furstenberg structure theorem in topological
dynamics, CWI Quarterly 4 (1991), 27-44. |
| [9] | Abstract topological dynamics, in: M.Husek and J.van Mill (eds.), Recent progress in General Topology, North-Holland, Amsterdam, 1992, pp. 641-672. |
| [10] | Equivalence of almost periodic compactifications, Compositio Mathematica 22 (1970), 453-456. |
| [11] | Pseudocompactness and the Stone-Cech compactification for
topological groups, Nieuw Archief voor Wiskunde (3) 23 (1975), 35-48. |
| [12] | Integration on locally compact groups, in: T.H.Koornwinder (ed.), Representations of locally compact groups, MC Syllabus 38, Mathematisch Centrum, Amsterdam, 1979. |
| [13] | The Furstenberg boundary of a semisimple Lie group, in: T.H.Koornwinder (ed.), The structure of semisimple Lie groups, MC Syllabus 49, Mathematisch Centrum, Amsterdam, 1982, pp. 79-112. |