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Conference on Geometric Functional Analysis and its Applications

Autumn 2014 events Conference

Dear colleague,

we would like to announce the upcoming Conference on Geometric Functional Analysis and its Applications, to be held in Besançon from the 27th to the 31st of October 2014.

The following main speakers have agreed to deliver a plenary lecture:

Fernando Albiac
On the fundamental theorem of calculus in the lack of local
Motivated by the attempt to develop tools that can be applied to the study of the nonlinear geometry of quasi-Banach spaces, we embark on a journey to explore which integration methods work for functions mapping in quasi-Banach spaces and investigate the interplay of integration and differentiation in this setting.

Florent Baudier
Metric geometry of stable metric spaces and applications
The Lipschitz, coarse, uniform, and strong geometries of stable metric spaces shall be browsed. Recent developments on embeddability of proper metric spaces will be discussed, in particular a strengthened notion of strong embeddability will be introduced. We will also present applications, or connections, of the above with geometric group theory and topology.
Part of this work was carried out in collaboration with G. Lancien.
Alejandro Chávez-Domínguez
Lipschitz $p$-convex and $q$-concave maps
The notions of $p$-convexity and $q$-concavity are mostly known because of their importance as a tool in the study of isomorphic properties of Banach lattices, but they also play a role in several results involving linear maps between Banach spaces and Banach lattices. In this talk we introduce Lipschitz versions of these concepts, dealing with maps between metric spaces and Banach lattices, and start by proving nonlinear versions of two well-known factorization theorems through $L_p$ spaces due to Maurey/Nikishin and Krivine. We also show that a Lipschitz map from a metric space into a Banach lattice is Lipschitz $p$-convex if and only if its linearization is $p$-convex. Furthermore, we elucidate why there is such a close relationship between the linear and nonlinear concepts by proving characterizations of Lipschitz $p$-convex and Lipschitz $q$-concave maps in terms of factorizations through $p$-convex and $q$-concave Banach lattices, respectively, in the spirit of the work of Raynaud and Tradacete.
Chavez Dominguez

Robert Deville
Lipschitz embedding of metric spaces into $c_0$
Let $(M,d)$ be a separable metric space and $1<\lambda\le 2$. If $E$ is a non empty subset of $M\times M$, we denote $\widetilde{E}$ the smallest rectangle containing $E$, $\delta(E):=\inf\{d(x,y);(x,y)\in E\}$ and $diam(E)=\sup\{d(x,y);\,(x,y)\in E\}$.
Following G. Lancien and N. Kalton, we say that $(M,d)$ has the property $\Pi(\lambda)$ if, for all balls $B_1$ and $B_2$ of $M$ with radii $r_1$ and $r_2$ and for every non empty subset $E$ of $B_1\times B_2$ such that $\delta(E)>\lambda(r_1+r_2)$,
we can partition $E$ into finitely many subsets $E_1,\cdots,E_N$ satisfying

$$ \text{ for each }n ,\quad diam(E_n)< \lambda\delta(\widetilde{E}_n) $$

We prove that property $\Pi(\lambda)$ is equivalent to the existence of a mapping
$f=(f_n):M\to c_0$ such that for all $x\ne y\in M$, we have $d(x,y)<\Vert f(x)-f(y)\Vert$,
and there exists a sequence $(\lambda_n)$ of scalars satisfying, for all $n$, $\lambda_n<\lambda$ and $f_n$ is $\lambda_n$-Lipschitzian.

Stephen Dilworth
Greedy bases and the greedy constant
A greedy basis is one for which the greedy algorithm yields the best $n$-term approximation up to a multiplicative constant called the greedy constant. Greedy bases are characterized as being unconditional and `democratic’, where the latter is a symmetry condition on constant coefficient vectors. We review the theory of greedy bases, including duality and `bidemocratic bases’ and recent results on improving the greedy and democratic constants by renorming.

Valentin Ferenczi
On singular twistings of Banach spaces
We obtain sufficient conditions for singularity of a twisted sum $X \oplus_{\Omega_{\theta}} X$ induced by an interpolation scheme $X=(X_0,X_1)_{\theta}$. As consequences we recover and generalize the singularity of Kalton-Peck sums of sequence spaces; discover the "disjoint" singularity of Kalton-Peck sums of function spaces; and also construct singular sums without the use of a lattice structure. Joint work with J. Castillo and M. González.

Bill Johnson
Some unrelated results in non separable Banach space theory
I’ll discuss three recent papers, one with Gideon Schechtman, the second with Amir Bahman Nasseri, Schechtman, and Tomasz Tkocz, the third with Tomasz Kania and Schechtman. The papers are unrelated but connected in the sense that they are all concerned with the structure of classical non separable Banach spaces; namely, $L_p$, $\ell_\infty$, and $\ell_\infty^c$ (the space of bounded functions that have countable support), respectively.

Denka Kutzarova
Lenses and Asymptotic Midpoint Uniform Convexity
As a continuation of a paper on property (beta) of Rolewicz we consider a geometric property of the unit ball of a Banach space, namely when the Kuratowski index of noncompactness of some special type of lenses tends to zero uniformly in the elements of the unit sphere. We give an analytic characterization of this new property (AMUC) and show that isometrically it is weaker than asymptotic uniform convexity (AUC). Let us point out that if the diameters of these lenses tend uniformly to zero, this characterizes uniform convexity (Laakso, Tyson-Wu). Finally, we show that an AMUC space with an unconditional basis admits an equivalent AUC norm.
This is a joint work with S.J. Dilworth, N. Lovasoa Randrianarivony, J.P. Revalski and N.V. Zhivkov.
Pavlos Motakis
The stabilized set of $p$’s in Krivine’s theorem can be disconnected
J. L. Krivine’s theorem states that for every Banach space $X$ with a basis, there exists a $p\in[1,\infty]$ such that $\ell_p$ is finitely block represented in $X$. The set of all such $p$’s is called the Krivine set of $X$. As it was proved by H.P. Rosenthal, this set is stabilized on some block subspace $Y$ of $X$, i.e. the Krivine set of $Y$ and the corresponding one of any of its further block subspaces coincide. The form of such a stabilized Krivine set has been a subject of study, since Rosenthal asked whether it always had to be a singleton. This question was answered negatively by E. Odell and Th. Schlumprecht by constructing a space having $[1,\infty]$ as its stabilized Krivine set. The question that followed was if such a stabilized Krivine set had to be an interval, which was asked by P. Habala and N. Tomczak-Jaegermann as well as by E. Odell. We answer this question in the negative direction by constructing, for every $F\subset [1,\infty]$ which is either finite or consists of an increasing sequence and its limit, a reflexive Banach space $X$ with an unconditional basis such that for every infinite dimensional block subspace $Y$ of $X$, the Krivine set of $Y$ is precisely $F$. This construction also addresses some open problems concerning spreading models.
This is a joint work with K. Beanland and D. Freeman.


Beata Randrianantoanina
On an isomorphic Banach-Mazur rotation problem and maximal norms in Banach spaces
We prove that the spaces $\ell_p$, $1 < p < \infty, p\ne 2$, and all infinite-dimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. This is a step towards the solution of the Banach-Mazur rotation problem, which asks whether a separable Banach space with a transitive norm has to be isometric or isomorphic to a Hilbert space.
We obtain this as a consequence of a new property of almost transitive spaces with a Schauder basis, namely we prove that in such spaces the unit vector basis of $\ell_2^2$ belongs to the two-dimensional asymptotic structure and we obtain some information about the asymptotic structure in higher dimensions.

Further, we prove that the spaces $\ell_p$, $1 < p < \infty$, $p\ne 2$, have continuum different renormings with 1-unconditional bases each with a different maximal isometry group, and that every symmetric space other than $\ell_2$ has at least a countable number of such renormings. On the other hand we show that the spaces $\ell_p$, $1 < p < \infty$, $p\ne 2$, have continuum different renormings each with an isometry group which is not contained in any maximal bounded subgroup of the group of isomorphisms of $\ell_p$.

Joint work with S.J. Dilworth.


Yves Raynaud
Banach ultraroots of certain Banach lattices
Let $\mathcal C$ be an axiomatizable class of Banach lattices, that is, this class is closed under Banach lattice isomorphisms and ultraproducts, and the complementary class is closed under ultrapowers. We show that if linear isometric embeddings of members of $\mathcal C$ in their ultrapowers preserve disjointness, the class $\mathcal C^\mathcal B$ of Banach spaces obtained by forgetting the Banach lattice structure is still axiomatizable. Moreover if $\mathcal C$ coincides with its ``script class’’ $\mathcal{SC}$, so does $\mathcal C^\mathcal B$ with $\mathcal{SC}^\mathcal B$. This allows to give new examples of axiomatizable classes of Banach spaces.

Gideon Schechtman
Metric $X_p$ inequalities
A new nonlinear inequality of the flavour of the nonlinear version of the inequalities for type and cotype will be presented. This is a nonlinear extension of a linear inequality that was proved by Johnson, Maurey, Schechtman and Tzafriri in 1979 (and resembles the $X_p$ inequality of Rosenthal). The formulation (and proof) of the new inequality completes the search for bi-Lipschitz invariants that serve as an obstruction to the embeddability of $L_p$ spaces into each other, the previously understood cases of which were metric notions of type and cotype, which however fail to certify the nonembeddability of $L_q$ into $L_p$ when $2 < q < p$. Among the consequences of the new inequality are new quantitative restrictions on the bi-Lipschitz embeddability into $L_p$ of snowflakes of $L_q$ and integer grids in $\ell_q^n$, for $2 < q < p$.

Joint work with Assaf Naor.

Thomas Schlumprecht
The algebra of bounded linear operators on $\ell_p\oplus \ell_q$ has infinitely many closed ideals
In this joint work with A. Zsak, we solve a problem stated by A. Pietsch and show that the algebra of bounded linear operators on $\ell_p \oplus \ell_q$ contains infinitely many closed ideals. More precisely, we show that there is a chain of sub ideals whose cardinality is that of the continuum.

Jarno Talponen
Smoothness of quasihyperbolic balls in convex domains of Banach spaces
Quasihyperbolic (QH) metric is a weighted metric on a path-connected metric space motivated by looking at invariants of Mobius tranformations of the unit disk. It is used for analyzing quasiconformal mappings.

In this talk the metric is given on an open convex non-trivial subset of a Banach space. Many properties of the underlying Banach space are visible in the QH geometry.

The study of the following problem was initiated by F.W. Gehring and M. Vuorinen in the 1970s. Is a quasihyperbolic ball $B_{qh}(x,r)$ in a convex domain $\Omega \subset \mathbb{R}^n$ necessarily $C^1$-smooth?
It turns out that the answer is affirmative for uniformly smooth Banach spaces in place of $\mathbb{R}^n$. To our knowledge the question was open even for $\mathbbR^2$ and in fact some people conjectured that the answer should be negative for a strip of a plane. There is a preprint in the ArXiv.

Alain Valette
The Kadison–Singer Problem
In 1959, R.V. Kadison and I.M. Singer asked whether each pure state of the algebra of bounded diagonal operators on $\ell^2$ admits a unique state extension to $B(\ell^2)$. The positive answer was given in June 2013 by A. Marcus, D. Spielman and N. Srivastava, who took advantage of a series of translations of the original question, due to C. Akemann, J. Anderson, P. Casazza, N. Weaver,. . . Ultimately, the problem boils down to an estimate of the largest zero of the expected characteristic polynomial of the sum of independent random variables taking values in rank-one positive matrices in the algebra of $n$-by-$n$ matrices. In turn, this is proved by studying a special class of polynomials in $d$ variables, the so-called real stable polynomials. The talk will highlight the main steps in the proof.

The purpose of this meeting is to bring together researchers with common interest in the field, and to give an opportunity to deliver a short talk. The meeting will offer many possibilities for informal discussions.
Graduate students and others beginning their mathematical career are encouraged to participate.
This conference is preceded by Autumn School on Nonlinear Geometry of Banach Spaces and Applications in the Jura mountains near Besançon and is part of the special trimester "Geometric and non-commutative methods in functional analysis" which will be organized at Université Franche-Comté.

The registration for both events is now closed. The deadline for the registration is September 30, 2014.
If you want to deliver a short talk, please submit the abstract by September 20, 2014.

Kindly inform your colleagues and students interested in this field.

We are looking forward to meeting you in Franche-Comté

Gilles Lancien, Tony Procházka
gilles.lancien(at) antonin.prochazka(at)