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This is an account of one man's view of the current perspective of theory of topological groups. We survey some recent developments which are, from our viewpoint, indicative of the future directions, concentrating on actions of topologi-cal groups on compacta, embeddings of topological groups, free topological groups, and 'massive' groups (such as groups of(More)
In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso (U) of isome-tries of Urysohn's universal complete separable metric space U, equipped with the compact-open topology, acts upon an arbitrary(More)
We suggest that the curse of dimensionality affecting the similarity-based search in large datasets is a manifestation of the phenomenon of concentration of measure on high-dimensional structures. We prove that, under certain geometric assumptions on the query domain Ω and the dataset X, if Ω satisfies the so-called concentration property, then for most(More)
— A topological group G is extremely amenable if every continuous action of G on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one.(More)
—Exchangeable random variables form an important and well-studied generalization of i.i.d. variables, however simple examples show that no nontrivial concept or function classes are PAC learnable under general exchangeable data inputs X1, X2,. . .. Inspired by the work of Berti and Rigo on a Glivenko–Cantelli theorem for exchangeable inputs, we propose a(More)