October 2015 - Banach - mathdept.okstate.edu

Abstract of a paper by Xiao Chun Fang and Marat Pliev
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Narrow Orthogonally Additive Operators on Lattice-Normed Spaces" by Xiao Chun Fang and Marat Pliev. Abstract: The aim of this article is to extend results of M.~Popov and second named author about orthogonally additive narrow operators on vector lattices. The main object of our investigations are an orthogonally additive narrow operators between lattice-normed spaces. We prove that every $C$-compact laterally-to-norm continuous orthogonally additive operator from a Banach-Kantorovich space $V$ to a Banach lattice $Y$ is narrow. We also show that every dominated Uryson operator from Banach-Kantorovich space over an atomless Dedekind complete vector lattice $E$ to a sequence Banach lattice $\ell_p(\Gamma)$ or $c_0(\Gamma)$ is narrow. Finally, we prove that if an orthogonally additive dominated operator $T$ from lattice-normed space $(V,E)$ to Banach-Kantorovich space $(W,F)$ is order narrow then the order narrow is its exact dominant $\ls T\rs$. Archive classification: math.FA Mathematics Subject Classification: 46B99. 47B99 Remarks: 16 pages Submitted from: martin.weber(a)tu-dresden.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.09189 or http://arXiv.org/abs/1509.09189

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Abstract of a paper by Alexandr Andoni, Assaf Naor, and Ofer Neiman
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Snowflake universality of Wasserstein spaces" by Alexandr Andoni, Assaf Naor, and Ofer Neiman. Abstract: For $p\in (1,\infty)$ let $\mathscr{P}_p(\mathbb{R}^3)$ denote the metric space of all $p$-integrable Borel probability measures on $\mathbb{R}^3$, equipped with the Wasserstein $p$ metric $\mathsf{W}_p$. We prove that for every $\varepsilon>0$, every $\theta\in (0,1/p]$ and every finite metric space $(X,d_X)$, the metric space $(X,d_{X}^{\theta})$ embeds into $\mathscr{P}_p(\mathbb{R}^3)$ with distortion at most $1+\varepsilon$. We show that this is sharp when $p\in (1,2]$ in the sense that the exponent $1/p$ cannot be replaced by any larger number. In fact, for arbitrarily large $n\in \mathbb{N}$ there exists an $n$-point metric space $(X_n,d_n)$ such that for every $\alpha\in (1/p,1]$ any embedding of the metric space $(X_n,d_n^\alpha)$ into $\mathscr{P}_p(\mathbb{R}^3)$ incurs distortion that is at least a constant multiple of $(\log n)^{\alpha-1/p}$. These statements establish that there exists an Alexandrov space of nonnegative curvature, namely $\mathscr{P}_{\! 2}(\mathbb{R}^3)$, with respect to which there does not exist a sequence of bounded degree expander graphs. It also follows that $\mathscr{P}_{\! 2}(\mathbb{R}^3)$ does not admit a uniform, coarse, or quasisymmetric embedding into any Banach space of nontrivial type. Links to several longstanding open questions in metric geometry are discussed, including the characterization of subsets of Alexandrov spaces, existence of expanders, the universality problem for $\mathscr{P}_{\! 2}(\mathbb{R}^k)$, and the metric cotype dichotomy problem. Archive classification: math.MG math.FA Submitted from: naor(a)math.princeton.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.08677 or http://arXiv.org/abs/1509.08677

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Analytic and Probabilistic Techniques in Modern Convex Geometry, Nov 7-9, 2015
by Pivovarov, Peter 30 Sep '15

30 Sep '15

Dear Colleagues: The Mathematics Department at the University of Missouri-Columbia is pleased to host a conference on Analytic and Probabilistic Techniques in Modern Convex Geometry, dedicated to Alexander Koldobsky on the occassion of his 60th birthday, November 7-9, 2015. We aim to bring together experienced and early-stage researchers to discuss the latest developments on slicing inequalities for convex sets, geometry of high-dimensional measures, affine isoperimetric inequalities and non-asymptotic random matrix theory. Information is available at http://www.bengal.missouri.edu/~pivovarovp/APTMCG/index.html Funding is still available to cover the local and travel expenses of a limited number of participants. Graduate students, postdoctoral researchers, and members of underrepresented groups are particularly encouraged to apply for support. Please register online or contact Peter Pivovarov at pivovarovp(a)missouri.edu. A poster session will be held for researchers to display their work. Graduate students are particularly encouraged to submit a poster. Yours sincerely, Peter Pivovarov on behalf of the organizers: Grigoris Paouris Peter Pivovarov Mark Rudelson Artem Zvavitch

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