Monthly Archives: November 2014

The minimal distortion needed to embed a binary tree into I will cover theorems very similar to those in the previous talk, but the methods will be different, with a more combinatorial flavour. 1. Ramsey theory 1.1. Friends and strangers … Continue reading →

Lower bounding distortion via metric invariants Let and be metric spaces. Recall that is the infimum of bi-Lipschtz distortions of maps . Typically, is a Banach space and is combinatorial (finite metric space). When , one denotes by Theorem 1 … Continue reading →

1. Super-reflexivity 1.1. Uniform convexity A Banach space is uniformly convex if the midpoint of two points on the unit sphere which are sufficiently far apart is deep inside the ball. Example. Hilbert spaces, spaces with are uniformly convex. Uniform … Continue reading →

Non-embeddability of Urysohn space Theorem 1 (Pestov 2008) Urysohn space does not embed uniformly in any super-reflexive Banach space. 1. Urysohn space 1.1. Universality Urysohn space is the universal complete separable metric space. It contains an isometric copy of every … Continue reading →

Assouad’s embedding theorem 1. Motivation 1.1. Doubling constant Recall that the doubling constant of a metric space is the least such that every -ball can be covered by -balls. Example. and its subspaces are doubling. 1.2. An open problem Open … Continue reading →

Wall structures and coarse embeddings Lots of spaces have Property A, great! This provides many coarse embeddings in Hilbert space. Nevertheless, there exist spaces which coarsely embed but do not satisfy Property A. I will describe the best known obstruction … Continue reading →

1. Metric aspects F\o lner’s criterium shows that amenability is a coarse equivalence invariant of groups. A-T-menability is not. 1.1. Yu’s Property A From now on, we deal with metric spaces . is assumed to be uniformly discrete. Yu pursued … Continue reading →

F\o lner type sets, Property A and coarse embeddings 1. Banach-Tarski paradox Theorem 1 (Hausdorff 1914, Banach-Tarski 1924) There exists 4 disjoint subsets of the 2-sphere such that they constitute a partition of ; there are rotations and such that … Continue reading →

Stochastic decompositions of metric spaces This has applications to many problems : Ramsey decompositions, distributed computing… This decompositions are a basic tool for various parts of mathematics and design of algorithms, so it is good to know how to produce … Continue reading →

1. Markov type and compression, continued 1.1. Markov type, an example has Markov type 2, and this is not obvious. Let be a finite set with kernel , a stochastic matrix. It acts on functions on . Reversibility means that … Continue reading →