Abstract:
This survey discusses the problem of describing properties of the class of metric spaces in which the Uryson construction of a universal homogeneous metric space (for this class) can be carried out axiomatically. One of the main properties of this kind is the possibility of gluing together two metrics given on closed subsets and coinciding on their intersection. The uniqueness problem for a (countable or complete) homogeneous space universal in a given class of metric spaces is
discussed. The problem of extending a Clifford translation of a compact subset of an (ultrametric) Uryson space to a Clifford translation of the entire Uryson space is studied.
N. D. Lebedeva, A. M. Petrunin, “All-set-homogeneous spaces”, St. Petersburg Math. J., 35:3 (2024), 473–476
Jessica Popowicz, Aleksander Ivanov, “An amalgamation property for metric groups”, ADM, 33:1 (2022), 140
Ishiki Y., “An Embedding, An Extension, and An Interpolation of Ultrametrics”, P-Adic Numbers Ultrametric Anal. Appl., 13:2 (2021), 117–147
Sabok M., “Completeness of the isomorphism problem for separable C*-algebras”, Invent. Math., 204:3 (2016), 833–868
Aleksander Ivanov, Barbara Majcher-Iwanow, “An amalgamation property for metric spaces”, Algebra Discrete Math., 22:2 (2016), 233–239
Dana Ashkenazi, Zvi Lotker, “The Quasicrystals Discovery as a Resonance of the Non-Euclidean Geometry Revolution: Historical and Philosophical Perspective”, Philosophia, 2013
Cherlin G., “Two problems on homogeneous structures, revisited”, Model Theoretic Methods in Finite Combinatorics, Contemporary Mathematics, 558, 2011, 319–415
The L. Nguyen Van, “Structural Ramsey Theory of Metric Spaces and Topological Dynamics of Isometry Groups Introduction”, Memoirs of the American Mathematical Society, 206:968 (2010), 1
Nguyen Van Thé Lionel, Sauer N.W., “The Urysohn sphere is oscillation stable”, Geom. Funct. Anal., 19:2 (2009), 536–557
Nguyen Van The Lionel, “Ramsey degrees of finite ultrametric spaces, ultrametric Urysohn spaces and dynamics of their isometry groups”, European J. Combin., 30:4 (2009), 934–945
Lopez-Abad J., Nguyen Van Thé L., “The oscillation stability problem for the Urysohn sphere: a combinatorial approach”, Topology Appl., 155:14 (2008), 1516–1530
Melleray J., “Some geometric and dynamical properties of the Urysohn space”, Topology Appl., 155:14 (2008), 1531–1560
Brodskiy N., Dydak J., Higes J., Mitra A., “Dimension zero at all scales”, Topology Appl., 154:14 (2007), 2729–2740
Vershik A.M., “Universality and randomness for the graphs and metric spaces”, Frontiers in Number Theory, Physics and Geometry I - ON RANDOM MATRICES, ZETA FUNCTIONS, AND DYNAMICAL SYSTEMS, 2006, 245–266
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