Banach

banach@mathdept.okstate.edu

2549 discussions

Announcement of a Talk by Harrison Gaebler
by Bentuo Zheng (bzheng) 26 May '21

26 May '21

Hello, The next Banach spaces webinar is on Friday May 14 at 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Speaker: Harrison Gaebler, University of Kansas Title: Asymptotic Geometry of Banach Spaces that have a Well-Behaved Riemann Integral Abstract: Banach spaces for which Riemann integrability implies Lebesgue almost everywhere continuity are said to have the Property of Lebesgue, or to be ``PL-spaces." It is an open problem to derive a full characterization of PL-spaces. In this talk, I will first give a brief overview of Riemann and Darboux integrability for Banach-valued functions, and I will then introduce the Property of Lebesgue with some relevant examples. I will next show how the Property of Lebesgue is connected to the asymptotic geometry (both global and local) of the underlying Banach space, and I will present three new results in this direction that are to appear later this year in Real Analysis Exchange. Finally, I will discuss two possibilities for future research on characterizing PL-spaces, and a connection between the Property of Lebesgue and the distortion of the unit sphere as well. For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach

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Talk Annoucement
by Bentuo Zheng (bzheng) 20 May '21

20 May '21

Hello, The next Banach spaces webinar is on Friday May 21 at 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Speaker: Hugh Wark (York, England) Title: Equilateral sets in large Banach spaces Abstract: A subset of a Banach space is called equilateral if the distances between any two of its distinct points are the same. In this talk it will be shown that there exist non separable Banach spaces with no uncountable equilateral sets and indeed non separable Banach spaces with no infinite equilateral sets. For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach

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Announcement
by Bentuo Zheng (bzheng) 11 May '21

11 May '21

Dear Subscribers, I added a link to a documentary movie about Stefan Banach (suggested by Per Enflo). https://bentuo.wixsite.com/Banach Best regards, Bentuo Zheng

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Banach spaces webinar on Friday May 7 by Jordi López Abad (UNED)
by Sari, Bunyamin 06 May '21

06 May '21

Hello, The next Banach spaces webinar is on Friday May 7 at 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Speaker: Jordi López Abad, UNED Title: A note on Pelczynski's universal basis space Abstract: We prove that the isometry group of a renorming of the Pelczynski's universal basis space is extremely amenable. To do this, we see that the class of finite dimensional normed spaces is a complemented Fraïssé class with the approximate Ramsey property. This is a joint work with Jamal Kawach. For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Thank you, and best regards, Bunyamin Sari

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Banach spaces webinar on Friday 4/30 by Beata Randrianantoanina (Miami University in Ohio)
by Sari, Bunyamin 29 Apr '21

29 Apr '21

Hello, The next Banach spaces webinar is on Friday April 30 at 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Speaker: Beata Randrianantoanina (Miami University in Ohio) Title: On $L_1$-embeddability of unions of $L_1$-embeddable metric spaces and of twisted unions of hypercubes Abstract: Let $\mathcal{E}$ be a class of metric spaces, $(X,d)$ be a metric space, and $A,B$ be metric subspaces of $X$ such that $X=A\cup B$ and $(A,d), (B,d)$ embed bilipschitzly into spaces $E_A,E_B\in \mathcal{E}$ with distortions $D_A, D_B$, respectively. Does this imply that there exists a constant $D$ depending only on $D_A, D_B$, and the class $\mathcal{E}$, so that $(X,d)$ embeds bilipschitzly into some space $E_X\in \mathcal{E}$ with distortion $D$? This question was answered affirmatively for the class $\mathcal{E}$ of all ultrametric spaces by Mendel and Naor in 2013, and for the class $\mathcal{E}$ of all Hilbert spaces by K. Makarychev and Y. Makarychev in 2016. K. Makarychev and Y. Makarychev in 2016 conjectured that the answer is negative when $\mathcal{E}$ is a class of $\ell_p$-spaces for any fixed $p\notin\{2,\infty\},$ in particular for $p=1$. In this connection, Naor in 2015 and Naor and Rabani in 2017 asked whether the metric space known as ``twisted union of hypercubes'', first introduced by Lindenstrauss in 1964, and also considered by Johnson and Lindenstrauss in 1986, embeds into $\ell_1$. In this talk I will show how to embed general classes of twisted unions of $L_1$-embeddable metric spaces into $\ell_1$, including twisted unions of hypercubes whose metrics are determined by concave functions of the $\ell_1$-norm, and discuss some related results (joint work with Mikhail I. Ostrovskii). For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Thank you, and best regards, Bunyamin Sari

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Banach spaces webinar on Friday 4/23 by March Boedihardjo (UCLA)
by Sari, Bunyamin 21 Apr '21

21 Apr '21

Hello, The next Banach spaces webinar is on Friday April 23 at 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Speaker: March Boedihardjo (UCLA) Title: Spectral norms of Gaussian matrices with correlated entries Abstract: We give a non-asymptotic bound on the spectral norm of a d×d matrix X with centered jointly Gaussian entries in terms of the covariance matrix of the entries. In some cases, this estimate is sharp and removes the sqrt(log d) factor in the noncommutative Khintchine inequality. Joint work with Afonso Bandeira. For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Thank you, and best regards, Bunyamin Sari

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Banach spaces webinar on Friday 4/9 by Valentin Ferenczi (Universidade de São Paulo)
by Sari, Bunyamin 06 Apr '21

06 Apr '21

Hello, The next Banach spaces webinar is on Friday April 9 at 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Speaker: Valentin Ferenczi, Universidade de São Paulo Title: There is no largest proper operator ideal Abstract: An operator ideal $U$ (in the sense of Pietsch) is proper if $Space(U)$, the class of spaces $X$ for which $Id_X \in U$, is reduced to the class of finiite-dimensional spaces. Equivalently, $U$ is proper if $U(X)$ is a proper ideal of $L(X)$ whenever $X$ is infinite dimensional (where $U(X)$ denotes the set of operators on $X$ which belong to $U$). We answer a question posed by Pietsch in 1979 by proving that there is no largest proper operator ideal. Our proof is based on an extension of the construction by Aiena-Gonz\'alez (2000), of an improjective but essential operator on Gowers-Maurey's shift space (1997), through a new analysis of the algebra of operators on powers of the shift space. Supported by FAPESP, project 2016/25574-8, and CNPq, grant 303731/2019-2 For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Thank you, and best regards, Bunyamin Sari

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Banach spaces webinar on Friday 3/26 by Yuval Wigderson (Stanford)
by Sari, Bunyamin 24 Mar '21

24 Mar '21

Hello, The next Banach spaces webinar is on Friday March 26 at 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Speaker: Yuval Wigderson (Stanford) Title: New perspectives on the uncertainty principle Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson. For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Thank you, and best regards, Bunyamin Sari

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Banach spaces webinar on Friday 3/19 by Paul Müller (JKU Linz)
by Sari, Bunyamin 17 Mar '21

17 Mar '21

Hello, The next Banach spaces webinar is on Friday March 19 at 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Speaker: Paul Müller (JKU Linz) Title: Complex Convexity Estimates, Extensions to $R ^n$, and log-Sobolev Inequalities. Abstract. The talk is based on joint work with P.Ivanishvili (North Carolina State University), A. Lindenberger (JKU) and M. Schmuckenschlaeger (JKU). We extend complex uniform convexity estimates to $R^n$ and determine the corresponding best constants. Furthermore we provide the link to log-Sobolev inequalities on the unit-sphere of $R^n$ and discuss several open conjectures related to our work. For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Thank you, and best regards, Bunyamin Sari

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Banach spaces webinar on Friday 3/12 by Johann Langemets (University of Tartu)
by Sari, Bunyamin 11 Mar '21

11 Mar '21

Hello, The next Banach spaces webinar is on Friday March 12 at 9AM Central time. Please join us at https://yorku.zoom.us/j/99330056697?pwd=NlBnTERWTGRPbDQyYitnc0k1bTNqZz09<https://nam04.safelinks.protection.outlook.com/?url=https%3A%2F%2Fyorku.zoo…> (Please note the new zoom link. It shouldn’t ask for a passcode but if it does use Passcode: 036383) Speaker: Johann Langemets (University of Tartu) Title: A characterization of Banach spaces containing $\ell_1(\kappa)$ via ball-covering properties Abstract: In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of $\ell_1$ if and only if it can be equivalently renormed to be octahedral. It is known that octahedral norms can be characterized by means of covering the unit sphere by a finite number of balls. This observation allows us to connect the theory of octahedral norms with ball-covering properties of Banach spaces introduced by L. Cheng in 2006. Following this idea, we extend G. Godefroy's result to higher cardinalities. We prove that, for an infinite cardinal $\kappa$, a Banach space $X$ contains an isomorphic copy of $\ell_1(\kappa^+)$ if and only if it can be equivalently renormed in such a way that its unit sphere cannot be covered by $\kappa$ many open balls not containing $\alpha B_X$, where $\alpha\in (0,1)$. We also investigate the relation between ball-coverings of the unit sphere and octahedral norms in the setting of higher cardinalities. This is a joint work with S. Ciaci and A. Lissitsin. For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Thank you, and best regards, Bunyamin Sari

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