Olaf Klinke
Research
Papers can be found here.Thesis working title:
A theory of locally compact bitopological spaces.My work goes into the fields of denotational semantics of functional programming languages and the theory of infinite data types. The main tools employed are topology and category theory. When giving a denotational semantics of a functional programming language it has been proven useful to consider data types as ordered sets and programs as functions between these ordered sets which preserve the order. The order can be understood as a measure of information. Order preservation then is just the reasonable assumption that the output of a program contains more useful information that its input.
When doing this systematically, one quickly is led to take notions of continuity and approximation into account. The first one to do this in a rigorous way was Dana Scott. These notions have their historical origin in geometry, whence it is quite remarkable that the very same definitions work so beautifully in this setting.
The ordered sets, which are of algebraic nature, and the topological spaces are just two sides of the same medal. The language of translation is provided by the celebrated family of Stone dualities, which transform an algebraic structure into a topological strucure and vice versa. This translation has a logical reading: Algebraically one can consider the set of programs of a given type and look at all the observable properties they have. This yields a topological structure on all the programs. Conversely, starting off with a bunch of observable properties one can try to give a specification of a program by stating all the observable properties it should have. Of course the existence and uniqueness of a program matching the specification is not trivial but has to be proved. My supervisor Achim Jung together with Andrew Moshier observed that a number of these dualities are all special cases of a duality employing two topological structures at the same time, hence bitopological spaces.
The most convenient classes of structures one has considered so far have a property named compactness which can be seen as a generalised form of being finite. One of the data types one would like to extend the theory to is the real numbers, which unfortunately do not have this compactness property. I shall try to push the theory beyond the borders of compactness to the next weaker property which is local compactness.
Visit my papers section to view my thesis proposal, intermediate results and other work I have done.
In order to explain the basic ideas of the research described above, I set up these pages. A german version is also available here.