March 2021 - Banach - mathdept.okstate.edu

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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Banach spaces webinar on Friday 3/5 by Antonio Avilés López (Universidad de Murcia)
by Sari, Bunyamin 04 Mar '21

04 Mar '21

Hello, The next Banach spaces webinar is on Friday March 5 at 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Speaker: Antonio Avilés López, Universidad de Murcia Title: Sequential octahedrality and L-orthogonal elements Abstract: Given a Banach space $X$, we consider the following two isometric properties, variations on the notion of octahedrality that can be traced back to the work of B. Maurey: 1. There is an element $e^{**}$ in the sphere of the bidual such that $\|e^{**}+x\| = 1 + \|x\|$ for every $x\in X$. 2. There is a sequence $(e_n)$ in the sphere of $X$ such that $\lim_n \|e_n+x\| = 1 + \|x\|$ Uncountable sums provide examples that 1 does not imply 2. But the converse is unclear. It is natural to conjecture that a weak$^*$-cluster point of the sequence $(e_n)$ would give the desired $e^{**}$. This turns out to be independent of the usual axioms of set theory. The proof involves understanding different kinds of ultrafilters that may or may not exist, as well as a filter version of the Lebesgue dominated convergence theorem, similar to those considered by V. Kadets and A. Leonov. This is a joint work (in progress) with G. Mart\'{\i}nez Cervantes and A. Rueda Zoca. 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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