Postgraduate research work supervised by Lorenz Halbeisen

I am happy to supervise students intending to work on a thesis at the Master's or PhD level in Combinatorial Set Theory as well as its applications to General Topology and Banach Space Theory. However, on the Master's level, I also supervise students interested in Finite Combinatorics, Graph Theory or Combinatorial Number Theory.

What goes on in modern Set Theory?
One of the most important tool in Set Theory is the so-called 'forcing technique'. Forcing was invented by Paul Cohen in the early 1960's to prove that the Axiom of Choice and the Continuum Hypothesis are independent from Zermelo-Fraenkel set theory. Forcing is a technique to extend models of Set Theory in such a way that a certain statement gets true in the extension (the statement is forced to get true). For example, with forcing one can build a model in which the real line can be covered by less than continuum many Lebesgue measure zero sets. On the other hand, it is also possible to force that this cannot happen (even if the continuum hypothesis fails). Further, forcing is also involved to produce well-founded models of Set Theory in which the Axiom of Choice fails. Such models can be very strange. For example it is possible to have a model in which there is a partition of the real line which has more parts than real numbers exist.

To get a first taste of forcing and what could be proved with forcing, consider the following:

Let N={0,1,2,...} be the set of natural numbers and let F be the set of all functions from N to N. For g,f in F we write g <* f if for all but finitely many n in N we have g(n) < f(n). A set X of size continuum is called a K-Lusin set, if for every f in F, the set of functions g in X for which g <* f is countable. Now, is it possible that K-Lusin sets exist?

In the joint-paper with Tomek Bartoszynski entitled "On a theorem of Banach and Kuratowski and K-Lusin sets" it is shown that the existence of K-Lusin sets is independent of ZFC, which means that one can neither prove nor disprove the existence of K-Lusin sets.

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