Continuous logic and functional analysis
Program
Please note that the program below still might change.
| Time |
Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
| 9:30-10:20 |
Welcome coffee |
Berenstein |
Ben Yaacov 3 |
Törnquist |
Sherman |
| 10:30-11:20 |
Solecki |
discussions |
Pestov |
discussions |
Lupini |
| 11:45-12:35 |
Dodos 1 |
Dodos 2 |
Dodos 3 |
Kubis |
Henson |
| 12:35-14:30 |
lunch |
lunch |
lunch |
lunch |
lunch |
| 14:30-15:20 |
Ben Yaacov 1 |
Ben Yaacov 2 |
|
Thomas |
|
| 15:20-16:10 |
discussions |
discussions |
|
discussions |
|
16:10-17:00 |
Todorcevic |
Lopez Abad |
|
Rubin |
|
Abstracts
A. Berenstein, Reflexive representability and stable metrics
C. Ward Henson, Uncountably categorical Banach space structures
Wieslaw Kubis, Metric categories and Fraïssé limits
We propose natural axioms for a category enriched over metric spaces, in order to get a unified approach to universal almost homogeneous structures
like the Gurarii space, where almost homogeneity is with respect to finite-dimensional Banach spaces.
The main problem with the Gurarii space was finding an elementary proof of its uniqueness with respect to an isometry.
Roughly speaking, given a category \(\mathcal K\), we assume that each \(\mathcal K\)-arrow \(f\) has some positive value \(\mu(f)\),
having in mind to measure a ``distortion" of the domain transformed by \(f\).
An isomorphism \(h\) satisfying \(\mu(h) = 0\) can be called an isometry, meaning that it does not distort its domain.
In the category of metric spaces with non-expansive maps, this is the usual notion of a bijective isometry.
The measure \(\mu\) should satisfy some natural axioms, like in the metric space setting.
Next, we assume that each hom-set of \(\mathcal K\) is endowed with a metric \(\rho\),
so that the composition operator becomes a non-expansive map with respect to each coordinate separately.
Finally, some natural conditions on compatibility of \(\mu\) and \(\rho\) should be assumed.
The crucial axiom says that arrows can be ``corrected" in the sense that given \(\varepsilon > 0\) and a \(\mathcal K\)-arrow \(f\)
with \(\mu(f)\) small enough, there exist \(\mathcal K\)-arrows \(g,h\) such that \(\mu(g)\) and \(\mu(h)\) are arbitrarily small and \(\rho(g \circ f, h) < \varepsilon\).
We are able to prove the following: If \(\mathcal K\) satisfies an ``approximate" variant of the amalgamation and joint embedding property, and has a ``dense set"
of countably many arrows,
then \(\mathcal K\) has a unique ``approximate" Fraïssé limit \(U\). What is more important, we are able to prove its uniqueness with respect to isometry.
Some examples, different from metric spaces, will be given.
Martino Lupini, Continuous logic and sofic groups
Jordi Lopez Abad, On the Banach-Saks property
Vladimir Pestov,
On amenability and property (T) of infinite dimensional groups (tentative)
Matti Rubin, The reconstruction problem for locally convex metrizable
topological vector spaces (joint with V.P. Fonf)
A space-group pair is a pair \(\langle X,G \rangle\), where \(X\) is a topological space, and
\(G\) is a subgroup of \(H(X)\) – The group of all auto-homeomorphisms of X.
A class \(K\) of space-group pairs is faithful if for every \(\langle X,G \rangle, \langle Y, H \rangle \in K\)
and an isomorphism \(\varphi \colon G \cong H \) between the groups \(G\) and \(H\), there is a
homeomorphism \( \tau \colon X \cong Y\) between \(X\) and \(Y\) such that \( \varphi(g) = \tau \circ g \circ \tau^{-1} \) for
every \(g \in G\).
Theorem A. Let \(K\) be the space of all space-group pairs \(X,G\), where
(1) \(X\) is an open subset of a locally convex metric topological vector space,
(2) \(G\) contains all locally uniformly continuous auto-homeomorphisms of \(X\).
Then \(K\) is faithful.
A bigger and more natural question in this topic is whether a result similar to Theorem A, is true for the class of all locally convex (not necessarily metrizable)
topological vector spaces. This is a long-standing open question. The simplest open special case of this bigger question is whether the class
\( K_{prd} \colon= \{\langle \mathbb R^{\lambda}, H(\mathbb R^{\lambda} \rangle |\lambda \text{ is an infinite cardinal}\}\)
is faithul.
The proof of theorem A is based on a more general theorem that has other applications. In particular, an application to normed spaces.
David Sherman, Some model theoretic results about operator algebras
Slawomir Solecki, An abstract approach to Ramsey theory with applications
Simon Thomas, A Descriptive View of Unitary Group Representations
Stevo Todorcevic, A density theorem for products of finite sets
Asger Törnquist, A Fraïssé theoretic view of the Poulsen simplex