Learn More
(A) We study in this paper some connections between the Fra¨ıssé theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. A prime concern of topological dynamics is the study of continuous actions of (Hausdorff) topological groups G on (Hausdorff) compact spaces(More)
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)
This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, uni-tary groups U (n) and symmetric groups S n , n ∈ N. Hyperlinear groups come from theory of operator algebras (Connes'(More)
We offer a theoretical validation of the curse of dimensionality in the pivot-based indexing of datasets for similarity search, by proving, in the framework of statistical learning, that in high dimensions no pivot-based indexing scheme can essentially outperform the linear scan. A study of the asymptotic performance of pivot-based indexing schemes is(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) We study in this paper some connections between the Fra¨ıssé theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. A prime concern of topological dynamics is the study of continuous actions of (Haus-dorff) topological groups G on (Hausdorff) compact spaces(More)