Statement of research interests:
I am especially active in
that part of topology which is involved with
set-theoretic consistency and independence of some very basic
topological concepts. This is part of what I call
the legacy of Kurt Goedel, who first showed that there are
mathematical statements whose truth or falsity cannot be decided
on the basis of the axioms on which all of mathematics up
to Goedel's day was based. Although over sixty years have
passed since then, no one has come up with any
new axioms that are generally seen to be true.
However, in the meantime,
a multitude of statements in mathematics, including
some in almost every one of the main areas of pure
mathematics, have been shown to be undecidable on the basis
of the generally accepted axioms. Many of them are easy
to state, in fact easier to state and more fundamental to some
branches of mathematics, than most of the true statements that are being
proven in these branches today.
Topology is one branch
that has been completely revolutionized in the past three decades
as a result. Ever since 1977, I have been a leading researcher
in the part of topology that deals with these consistency and
independence results. To take just one example: in 1948, M. Katetov
proved that a compact space is metrizable if, and only if, every subspace of
its cube is normal; he then asked whether ``cube'' could be
replaced by ``square''. In 1977, I showed that it is consistent
that the answer is negative, while in 2001, P. Larson and
S. Todorcevic showed that it is consistent that the
answer is affirmative.
An expository article of mine on independence results has been recently
in Topology Atlas.