Logic Seminar
Krzysztof Krupinski
University of Illinois at Urbana-Champaign
Small profinite structures and their generalizations
Abstract: A profinite structure in the sense of Newelski is a pair $(X,Aut^*(X))$ consisting of a profinite
topological space $X$ and a closed subgroup $Aut^*(X)$ (called the structural group) of the group of
all homeomorphisms of $X$ respecting the inverse system defining $X$. We say that a profinite
structure $(X,Aut^*(X))$ is small if for every natural number $n>0$, there are only countably many
orbits on $X^n$ under the action of the structural group. In small profinite structures Newelski
introduced a topological notion of independence, which has similar properties to those of forking
independence in stable theories, and developed a counterpart of geometric stability theory in this
context.
I will present this notion of independence and explain why smallness plays an important role here. I will
also give some examples and results concerning small profinite groups regarded as profinite
structures.
Then I will talk about my recent ideas concerning generalizations of small profinite structures to the
case of: 1) non-small profinite structures; 2) 'compact structures' (i.e. $X$ is a compact metric space
and $Aut^*(X)$ is a compact group acting on $X$ continuously); 3) 'Polish structures' (i.e. $X$ is a
Polish space and $Aut^*(X)$ is a Polish group acting on $X$ continuously).
Tuesday April 18, 2006 at 4:00 PM in SEO 427