Randall Dougherty
rld@math.ohio-state.edu

Title: Congruences between open sets

Abstract:
A famous result of Hausdorff states that a sphere with countably many
points removed can be partitioned into three pieces A,B,C such that
A is congruent to B (i.e., there is an isometry of the sphere which
sends A to B), B is congruent to C, and A is congruent to (B union C);
this result was the precursor of the Banach-Tarski paradox.
Later, R. Robinson characterized the systems of congruences like this
which could be realized by partitions of the sphere with rotations
witnessing the congruences.  The pieces involved were nonmeasurable.

In this talk, we consider the problem of which systems of
congruences can be satisfied using open subsets of the sphere (or
related spaces); of course, these open sets cannot form a partition of
the sphere, but they can be required to cover `most of' the sphere in
the sense that their union is dense.  While some cases of this problem
are simple geometrical dissections, others involve complex iterative
constructions and results from the theory of free groups.
(A sample question: Can one find five disjoint nonempty open subsets
A,B,C,D,E of the sphere such that the unions of pairs (A union B),
(A union C), ... are all congruent?  Can one find six?)
Many interesting questions remain open.