ANALYSIS  SEMINAR (UTK)

Abstract: I recently introduced the concept of pseudo-Riesz bases, extending the influential idea of near-Riesz bases first proposed by J. Holub in the 1990s. I established that pseudo-Riesz bases can be characterized as sequences with a Fredholm synthesis operator. Unlike Riesz bases, they are neither spanning nor independent, yet they retain valuable expansion properties. I have also developed some perturbation results for our Pseudo-Riesz bases.A well-known result by Hrusˇcˇev, Nikol’ski ̆i, and Pavlov characterizes Riesz bases of non-harmonic complex exponentials in terms of the invertibility of an appropriate Toeplitz operator. I prove a similar result for pseudo-Riesz bases of non-harmonic complex exponentials, aiming to establish Kadec-type perturbation results.

Abstract: The main theme of this talk is the study of mappings—primarily continuously differentiable and Lipschitz—that are critical everywhere, in the sense that the rank of their derivative is small at every point. Such mappings arise naturally in a variety of contexts across analysis, geometry, and topology. I will discuss problems related to approximation, homotopy, Heisenberg groups and analysis on metric spaces.

Abstract: A natural question in the bi-Lipschitz geometry of trees is whether a large class of geodesic trees admits a single universal element, that is, a fixed tree into which every member of the class embeds in a bi-Lipschitz manner. Furthermore, Kinnenberg asked if every quasiconformal tree of Assouad dimension < 2 admits a  bi-Lipchitz embedding into R^2. Chrontsios-Garitsis, Ioannidis, and Vellis prove that the class of quasiconformal trees with uniform separation can quasisymmetrically embed into a universal element T, and moreover, this T admits a bi-Lipschitz embedding into R^2. In this talk, we will show that such a universal element cannot be found if one replaces quasisymmetric maps with bi-Lipschitz maps, not even in the case of the geodesic trees. This answers a question of the aforementioned authors.

This is joint work with Sylvester Eriksson-Bique.

Abstract: We compare classical techniques involving orthogonality with some more recent approaches involving "almost orthogonality."  Orthogonality methods are very precise and powerful when they apply, but they are limited to L^2 or other Hilbert spaces.  Almost orthogonality is a much more flexible and robust tool, for example applying to L^p for 1<p<infinity and to Hardy, Holder, and Sobolev spaces.  There is a close interplay between these results and Calderon-Zygmund theory, going in both directions: using vector-valued Calderon-Zygmund theory to obtain characterizations of function spaces, and using almost orthogonal decompositions to prove L^2 boundedness for non-convolution Calderon-Zygmund operators.

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