October 2013 - Banach - mathdept.okstate.edu

Abstract of a paper by Olivier Guedon
by alspach＠math.okstate.edu 15 Oct '13

15 Oct '13

This is an announcement for the paper "Concentration phenomena in high dimensional geometry" by Olivier Guedon. Abstract: The purpose of this note is to present several aspects of concentration phenomena in high dimensional geometry. At the heart of the study is a geometric analysis point of view coming from the theory of high dimensional convex bodies. The topic has a broad audience going from algorithmic convex geometry to random matrices. We have tried to emphasize different problems relating these areas of research. Another connected area is the study of probability in Banach spaces where some concentration phenomena are related with good comparisons between the weak and the strong moments of a random vector. Archive classification: math.FA Remarks: This paper is written after a plenary talk given in August 2012 at the "Journ\'ees MAS" organized in Clermont Ferrand. To appear in ESAIM Proceedings Submitted from: olivier.guedon(a)univ-mlv.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.1204 or http://arXiv.org/abs/1310.1204

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Abstract of a paper by Mathieu Meyer, Carsten Schuett, and Elisabeth M. Werner
by alspach＠math.okstate.edu 15 Oct '13

15 Oct '13

This is an announcement for the paper "Dual affine invariant points" by Mathieu Meyer, Carsten Schuett, and Elisabeth M. Werner. Abstract: An affine invariant point on the class of convex bodies in R^n, endowed with the Hausdorff metric, is a continuous map p which is invariant under one-to-one affine transformations A on R^n, that is, p(A(K))=A(p(K)). We define here the new notion of dual affine point q of an affine invariant point p by the formula q(K^{p(K)})=p(K) for every convex body K, where K^{p(K)} denotes the polar of K with respect to p(K). We investigate which affine invariant points do have a dual point, whether this dual point is unique and has itself a dual point. We define a product on the set of affine invariant points, in relation with duality. Finally, examples are given which exhibit the rich structure of the set of affine invariant points. Archive classification: math.FA Submitted from: elisabeth.werner(a)case.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.0128 or http://arXiv.org/abs/1310.0128

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Abstract of a paper by Felix Cabello Sanchez, Joanna Garbulinska, and Wieslaw Kubis
by alspach＠math.okstate.edu 15 Oct '13

15 Oct '13

This is an announcement for the paper "Quasi-Banach spaces of almost universal disposition" by Felix Cabello Sanchez, Joanna Garbulinska, and Wieslaw Kubis. Abstract: We show that for each $p\in(0,1]$ there exists a separable $p$-Banach space $\mathbb G_p$ of almost universal disposition, that is, having the following extension property: for each $\epsilon>0$ and each isometric embedding $g:X\to Y$, where $Y$ is a finite dimensional $p$-Banach space and $X$ is a subspace of $\mathbb G_p$, there is an $\epsilon$-isometry $f:Y\to \mathbb G_p$ such that $x=f(g(x))$ for all $x\in X$. Such a space is unique, up to isometries, does contain an isometric copy of each separable $p$-Banach space and has the remarkable property of being ``locally injective'' amongst $p$-Banach spaces. We also present a nonseparable generalization which is of universal disposition for separable spaces and ``separably injective''. No separably injective $p$-Banach space was previously known for $p<1$. Archive classification: math.FA Mathematics Subject Classification: 46A16, 46B04 Remarks: 22 pages Submitted from: kubis(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.7649 or http://arXiv.org/abs/1309.7649

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Abstract of a paper by Matthew Tarbard
by alspach＠math.okstate.edu 15 Oct '13

15 Oct '13

This is an announcement for the paper "Operators on Banach spaces of Bourgain-Delbaen type" by Matthew Tarbard. Abstract: We begin by giving a detailed exposition of the original Bourgain-Delbaen construction and the generalised construction due to Argyros and Haydon. We show how these two constructions are related, and as a corollary, are able to prove that there exists some $\delta > 0$ and an uncountable set of isometries on the original Bourgain-Delbaen spaces which are pairwise distance $\delta$ apart. We subsequently extend these ideas to obtain our main results. We construct new Banach spaces of Bourgain-Delbaen type, all of which have $\ell_1$ dual. The first class of spaces are HI and possess few, but not very few operators. We thus have a negative solution to the Argyros-Haydon question. We remark that all these spaces have finite dimensional Calkin algebra, and we investigate the corollaries of this result. We also construct a space with $\ell_1$ Calkin algebra and show that whilst this space is still of Bourgain-Delbaen type with $\ell_1$ dual, it behaves somewhat differently to the first class of spaces. Finally, we briefly consider shift-invariant $\ell_1$ preduals, and hint at how one might use the Bourgain-Delbaen construction to produce new, exotic examples. Archive classification: math.FA Remarks: Oxford University DPhil Thesis Submitted from: matthew.tarbard(a)sjc.ox.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.7469 or http://arXiv.org/abs/1309.7469

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Conference Announcement - First Brazilian Workshop in Geometry of Banach Spaces: August 2014
by valentin ferenczi 07 Oct '13

07 Oct '13

1st ANNOUNCEMENT OF BWB 2014 First Brazilian Workshop in Geometry of Banach Spaces August 25-29, 2014 Maresias, São Paulo State, Brazil. This is the 1st announcement for the First Brazilian Workshop in Geometry of Banach Spaces, organized by the University of São Paulo (USP), in the week August 25-29, 2014. This international conference will take place at the Beach Hotel Maresias, on the coast of São Paulo State, in Maresias. The scientific program will focus on the theory of geometry of Banach spaces, with emphasis on the following directions: linear theory of infinite dimensional spaces and its relations to Ramsey theory, homological theory and set theory; nonlinear theory; and operator theory. The webpage of the Workshop may be found at http://www.ime.usp.br/~banach/bwb2014/ Registration will start in early 2014. Additional scientific, practical and financial information will be given at that time. Plenary speakers: S. A. Argyros (Nat. Tech. U. Athens) J. M. F. Castillo (U. Extremadura) P. Dodos (U. Athens) G. Godefroy (Paris 6) R. Haydon (U. Oxford) W. B. Johnson (Texas A&M) P. Koszmider (Polish Acad. Warsaw) G. Pisier (Paris 6 & Texas A&M) C. Rosendal (U. Illinois Chicago) G. Schechtman (Weizmann Inst.) Th. Schlumprecht (Texas A&M) S. Todorcevic (Paris 7 & U. Toronto) Scientific committee J. M. F. Castillo (U. Extremadura) V. Ferenczi (U. São Paulo) R. Haydon (U. Oxford) W. B. Johnson (Texas A&M) G. Pisier (Paris 6 & Texas A&M) Th. Schlumprecht (Texas A&M) S. Todorcevic (Paris 7 & U. Toronto) We are looking forward to meeting you next year in Brazil, F. Baudier, C. Brech, V. Ferenczi, E. M. Galego, and J. Lopez-Abad.

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