Clinton Conley: Realizing abstract systems of congruence II

Mathematical logic seminar – Oct 9 2018
Time: 3:30pm – 4:30 pm

Room: Wean Hall 8220

Speaker: Clinton Conley
Department of Mathematical Sciences
CMU

Title: Realizing abstract systems of congruence II

Abstract:

An abstract system of congruence (ASC) is simply an equivalence relation
on the power set of a finite set F satisfying some nondegeneracy
conditions. Given such an ASC and an action of a group G on a set X, a
realization of the ASC is a partition of X into pieces indexed by F such
that whenever two subsets A, B are asc-equivalent, the corresponding
subsets XA and XB of X can be translated to one another in the action.
Familiar notions like paradoxical decompositions can be easily formalized
and refined by the ASC language. Wagon, upon isolating this notion,
characterized those ASCs which can be realized by rotations of the sphere.
He asks whether there is an analogous characterization for realizing ASCs
using partitions with the property of Baire. We provide such a
characterization. This is joint work with Andrew Marks and Spencer Unger.

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