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- Vladimir Pestov
- 1997

TOPOLOGICAL DYNAMICS

(A) We study in this paper some connections between the Fräıssé 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 X.… (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 topological groups on compacta, embeddings of topological groups, free topological groups, and ‘massive’ groups (such as groups of… (More)

(A) We study in this paper some connections between the Fräıssé 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 X.… (More)

We show that every abelian Polish group is the topological factor-group of a closed subgroup of the full unitary group of a separable Hilbert space with the strong operator topology. It follows that all orbit equivalence relations induced by abelian Polish group actions are Borel reducible to some orbit equivalence relations induced by actions of the… (More)

- Vladimir Pestov
- Inf. Process. Lett.
- 2000

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 highdimensional 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)

- Vladimir Pestov
- 2005

We prove that the isometry group Iso (U) of the universal Urysohn metric space U equipped with the natural Polish topology is a Lévy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups, exhibiting the phenomenon of concentration of measure. This strengthens an earlier result by Vershik… (More)

- Ilya Volnyansky, Vladimir Pestov
- 2009 Second International Workshop on Similarity…
- 2009

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)

- Vladimir Pestov
- 1997

We exhibit abelian topological groups admitting no nontrivial strongly continuous irreducible representations in Banach spaces. Among them are some abelian Banach–Lie groups and some monothetic subgroups of the unitary group of a separable Hilbert space.

- Vladimir Pestov
- 2005

This is a selection of open problems dealing with “large” (non locally compact) topological groups and concerning extreme amenability (fixed point on compacta property), oscillation stability, universal minimal flows and other aspects of universality, and unitary representations. A topological group G is extremely amenable, or has the fixed point on… (More)