Jordi Lopez Abad: Ramsey properties of embeddings between finite dimensional normed spaces

Speaker: Jordi Lopez Abad (ICMAT Madrid and University of Sao Paolo)

Date:  24-October-2014

Time: 13:30- 15:00

Place:  Fields Institute, Room 210

Title: Ramsey properties of embeddings between finite dimensional normed  spaces.

Abstract:  Given $d\le m$, let $E_{m,n}$ be the set of all $m\times d$ matrices $(a_{i,j})$ such that

(a) $\sum_{j=1}^d |a_{i,j}|\le 1$ for every $1\le i\le m$.

(b) $\max_{i=1}^m |a_{i,j}|=1$ for every $1\le j\le d$.

These matrices  correspond to the linear isometric embeddings from the normed space $\ell_\infty^d:=(\mathbb{R}^d,\| \cdot \|_\infty)$ into $\ell_\infty^d$, in their unit bases.

We will discuss and give (hints of) a proof of the following new approximate Ramsey result:

For every integers $d$, $m$ and $r$ and every $\varepsilon>0$ there exists $n$ such that for every coloring of $E_{d,n}$ into $r$-many colors there is $A\in E_{m,n}$ and a color $i<r$ such that
\begin{equation*}
A \cdot  E_{d,m}\subseteq (c^{-1}(i))_\varepsilon.
\end{equation*}
Its proof uses the Graham-Rothschild Theorem on partitions of finite sets.
We extend this result, first for embeddings between \emph{polyhedral} normed spaces, and finally for arbitrary finite dimensional normed spaces to get the following:

For every finite dimensional normed spaces $E$ and $F$, every $\theta>1$ and $\varepsilon>0$, and every integer $r$, there is some $n$ such that for every coloring of $\mathrm{Emb}_{\theta^2}(F,\ell_\infty^n)$ into $r$-many colors there is $T\in \mathrm{Emb}_{\theta}(G,\ell_\infty^n)$ and some color $i<r$ such that $T\circ \mathrm{Emb}_\theta(F,G)\subseteq (c^{-1}(i))_{\theta^2-1+\varepsilon}$.

As a consequence, we obtain that the group of linear isometries of the Gurarij space is extremely amenable. A similar result for positive isometric embeddings gives that the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex is the Poulsen simplex itself.

This a joint work (in progress) with Dana Bartosova (University of Sao Paulo) and Brice Mbombo (University of Sao Paulo)

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