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Tuesday, March 14, 2006

**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 of 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).