November 2021 - Banach - mathdept.okstate.edu

Banach spaces webinar on Nov 19 by Krzysztof Święcicki
by Sari, Bunyamin 18 Nov '21

18 Nov '21

The next Banach spaces webinar is on Friday November 19 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Title: No dimension reduction for doubling spaces of $\ell_q$ for $q>2$. Abstract: We'll provide a new elementary proof for the impossibility of dimension reduction for doubling subsets of $\ell_q$ for $q>2$. This is done by constructing a family of diamond graph-like objects based on the construction by Bartal, Gottlieb, and Neiman. We'll compare our approach with previous results and discuss their advantages and disadvantages. One noteworthy consequence of our proof is that it can be naturally generalized to obtain embeddability obstructions into non-positively curved spaces or asymptotically uniformly convex Banach spaces. Based on the work with Florent Baudier and Andrew Swift. For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Best regards, Bunyamin Sari

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Banach spaces webinar on Nov 12 by Florent Baudier (Texas A&M)
by Sari, Bunyamin 11 Nov '21

11 Nov '21

Hello, The next Banach spaces webinar is on Friday November 12 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Title: Umbel convexity and the geometry of trees Speaker: Florent Baudier (Texas A&M) Abstract. Markov convexity is a powerful invariant, introduced by Lee, Naor and Peres more than 15 years ago, which is related to the geometry of (locally finite) trees and (quantitative) uniformly convex renormings. In a joint work with Chris Gartland we introduced new metric invariants capturing the geometry of countably branching trees. Our main invariant, called umbel convexity, was inspired by Markov convexity and shares many of its desirable features. Most notably, it provides lower bounds on the distortion/compression required when embedding countably branching trees, and it is stable under certain nonlinear quotients. I will explain the close relationship between umbel convexity and Rolewicz's property $\beta$ renormings. If time permits, I will discuss the notion of umbel cotype, a relaxation of umbel convexity, and its relevance to the geometry of Heisenberg groups. 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 November 5 by Abraham Rueda Zoca (Universidad de Murcia)
by Sari, Bunyamin 04 Nov '21

04 Nov '21

Hello, The next Banach spaces webinar is on Friday November 5 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Title: $L$-orthogonal elements and spaces of operators Speaker: Abraham Rueda Zoca (Universidad de Murcia) Abstract. Given a Banach space $X$, we say that an element $u\in X^{**}$ is $L$-orthogonal if, for every $x\in X$, it follows that $$\Vert x+u\Vert=\Vert x\Vert+\Vert u\Vert.$$ In 1989, G. Godefroy proved that a Banach space $X$ admits an equivalent renorming with non-zero $L$-orthogonal elements if, and only if, $X$ contains an isomorphic copy of $\ell_1$. Moreover, G. Godefroy and N. J. Kalton proved (in 1989 too) that a separable space $X$ has non-zero $L$-orthogonal elements if, and only if, the following condition holds: \begin{center} For every finite-dimensional subspace $F$ of $X$ and every $\varepsilon>0$ there exists $x\in S_X$ so that $\Vert y+\lambda x\Vert\geq (1-\varepsilon)(\Vert y\Vert+\vert\lambda\vert)$ holds for every $y\in F$ and every $\lambda\in\mathbb R$. \end{center} In this talk we will examine the validity of this theorem for non-separable Banach spaces. For this, and for other results of the structure of the set of $L$-orthogonal elements, the Banach spaces of linear bounded operators between two Banach spaces will play a crucial role. The author was supported by Juan de la Cierva-Formaci\'on fellowship FJC2019-039973, by MTM2017-86182-P (Government of Spain, AEI/FEDER, EU), by MICINN (Spain) Grant PGC2018-093794-B-I00 (MCIU, AEI, FEDER, UE), by Fundaci\'on S\'eneca, ACyT Regi\'on de Murcia grant 20797/PI/18, by Junta de Andaluc\'ia Grant A-FQM-484-UGR18 and by Junta de Andaluc\'ia Grant FQM-0185. 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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