 
Summary: UIC Model Theory Seminar, April 18, 2006
Small profinite structures and their generalizations
Krzysztof KrupiŽnski
University of Illinois at UrbanaChampaign
A profinite structure in the sense of Newelski is a pair (X, Aut(X)) con
sisting of a profinite topological space X and a closed subgroup Aut(X)
(called the structural group) of the group of all homeomorphisms of X re
specting 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 Xn under the action of the structural group. In
small profinite structures Newelski introduced a topological notion of inde
pendence, 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) nonsmall profinite structures; 2) 'com
pact structures' (i.e. X is a compact metric space and Aut(X) is a compact
