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https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks
Recent Events in en-usThu, 15 Apr 2021 09:31:15 PDT3600Matrix weights: On the way to the linear bound
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/20
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/20Mon, 18 Jul 2016 09:40:00 PDT
In recent years, the attempts to prove sharp bounds for Calderon-Zygmund operators on weighted $L^p$ spaces in terms of the $A_p$-characteristic of the weight has been an im- portant driving force in Harmonic Analysis. After the work of many authors, this culminated with the proof of the conjectured linear bound for p = 2 for all Calderon-Zygmund operators by Tuomas Hyt\”onen in 2010. Recently, the question of the validity of the linear bound for all Calderon-Zygmund operators in the matrix-weighted setting has attracted some interest. In the talk, I want to present the reduction of this question to the case of Haar multipliers and dyadic paraproducts. I also want to talk about the remaining obstacles, some of which have recently been resolved, and focus on the matrix techniques being used. This is joint work with Andrei Stoica.
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Sandra PottApplied Harmonic Analysis meets Sparse Regularization of Operator Equations
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/19
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/19Tue, 19 Jul 2016 09:40:00 PDT
Sparse regularization of operator equations has already shown its effectiveness both theoretically and practically. The area of applied harmonic analysis offers a variety of systems such as wavelet systems which provide sparse approximations within certain model situations which then allows to apply this general approach provided that the solution belongs to this model class. However, many important problem classes in the multivariate situation are governed by anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shear layers in solutions of transport dominated equations. Since it was shown that the (isotropic) wavelet systems are not capable of sparsely approximating such anisotropic features, the need arose to introduce appropriate anisotropic representation systems. Among various suggestions, shearlets are the most widely used today. Main reasons for this are their optimal sparse approximation properties within a suitable model situation in combination with their unified treatment of the continuum and digital realm, leading to faithful implementations. In this talk, we will first provide an introduction to sparse regularization of operator equations, followed by an introduction to the area of applied harmonic analysis, in particular, discussing the anisotropic representation system of shearlets and presenting the main theoretical results. We will then analyze the effectiveness of using shearlets for sparse regularization of exemplary operator equations both theoretically and numerically.
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Gitta KutyniokFactorizations of Kernels and Reproducing Kernel Hilbert Spaces
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/18
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/18Tue, 19 Jul 2016 11:00:00 PDT
In this talk we will explain a series of results concerning reproducing kernel Hilbert spaces, related to the factorization of their kernels. In particular, we will talk about (trivial) isometric multipliers for a large class of reproducing kernel Hilbert spaces. We will then discuss a particular type of dilation, as well as a classification of Brehmer/Beurling type invariant subspaces of natural reproducing kernel Hilbert spaces. This is joint work with R. Kumari, S. Sarkar and D. Timotin.
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Jaydeb SarkarRandom and pseudo-random Taylor series
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/17
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/17Wed, 20 Jul 2016 09:40:00 PDT
We study entire functions represented by random and pseudo-random Taylor series. We prove that in many situations, their angular zero distribution is determined by certain autocorrelations of the coefficient multiplier sequence. This is a joint work with Alon Nishry and Mikhail Sodin.
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Alexander BorichevMultiple singular values of Hankel operators and weak turbulence in the cubic Szeg\H{o} equation
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/16
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/16Wed, 20 Jul 2016 11:40:00 PDT
We establish an inverse spectral result on compact Hankel operators on the unit sphere. Namely, we describe the set of symbols of compact Hankel operators having a prescribed sequence of singular values. It is done by constructing a one-to-one correspondence between a symbol of a compact Hankel operator and its sequence of singular values as well as some additional spectral parameters. \\ This one-to-one correspondence plays the role of a non linear Fourier transform for some hamiltonian equation: the cubic Szeg\H{o} equation. It allows to obtain explicit formulae of the solutions and to prove a wave turbulence phenomenon: for a dense $G_\delta$ of initial data, solutions develop large oscillations on small space scales. It is from joint works with Patrick G\'erard.
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Sandrine GrellierConfigurations in sets big and small
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/15
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/15Mon, 18 Jul 2016 11:40:00 PDT
When does a given set contain a copy of your favorite pattern (for example, specially arranged points on a line or spiral, or the vertices of a polyhedron)? Does the answer depend on how thin the set is in some quantifiable sense? Problems involving identification of prescribed configurations under varying interpretations of size have been vigorously pursued both in the discrete and continuous setting, often with spectacular results that run contrary to intuition. Yet many deceptively simple questions remain open. I will survey the literature in this area, emphasizing some of the landmark results that focus on different aspects of the problem.
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Malabika PramanikEntropic and Displacement interpolation of probability distributions: geometric and computational aspects
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/14
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/14Thu, 21 Jul 2016 09:40:00 PDT
We will discuss two problems with a long history and a timely presence. Optimal mass transport (OMT) was posed as a problem in 1781 by Gaspar Monge. It provides a natural geometry for interpolating distributions (displacement interpolation) and for modeling flows. As such it has been the cornerstone of many recent developments in physics, probability theory, and image processing. The Schrödinger bridge problem (SBP) was posed by Erwin Schrödinger in 1931, in an attempt to provide a classical interpretation of quantum mechanics. It is rooted in statistical mechanics and large deviations theory, and provides an alternative model for flows of the distribution of particles (entropic interpolation -Schrödinger bridge). We will explain the relation between the two problems, their practical relevance in the control of particles, ensembles, thermal noise, time-series analysis, images interpolation, etc., and we will present a computational approach based on the Hilbert metric. The talk is based on joint work withYongxin Chen (Mechanical Engineering, University of Minnesota) and Michele Pavon (Department of Mathematics, University of Padova).
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Tryphon GeorgiouThe functional calculus for commuting row contractions
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/13
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/13Fri, 22 Jul 2016 09:40:00 PDT
A commuting row contraction is a $d$-tuple of commuting operators $T_1,\dots,T_d$ such that $\sum_{i=1}^d T_iT_i^* \le I$. Such operators have a polynomial functional calculus which extends to a norm closed algebra of multipliers $\A_d$ on Drury-Arveson space. We characterize those row contractions which admit an extension of this map to a weak-$*$ continuous functional calculus on the full multiplier algebra. In particular, we show that completely non-unitary row contractions are always absolutely continuous, in direct parallel with the case of a single contraction. This is based on the detailed structure of the dual space of $\A_d$. Finally, we consider refinements of this question for row contractions that are annihilated by a given ideal. This is joint work with Rapha\"el Clou\^atre.
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Kenneth DavidsonRational Schur-Agler functions on polynomially-defined domains
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/12
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/12Tue, 19 Jul 2016 11:40:00 PDT
\documentclass[12pt]{amsart} \begin{document} When $p(z_1,\ldots , z_d)$ is a polynomial in $d$ variables that can be represented as $$ p(z_1,\ldots , z_d) = p_0 \det \left( I - K (\oplus_{j=1}^d z_j I_{n_j})\right),$$ where $p_0\neq 0$ and $K$ is a $(\sum_{j=1}^d n_j) \times (\sum_{j=1}^d n_j)$ contraction, then the rational inner function $$ f(z_1,\ldots , z_d)= \frac{\left( \prod_{j=1}^d z_j^{n_j} \right) \overline{p (1/\bar z_1, \ldots , 1/\bar z_d)}}{p(z_1, \ldots , z_d)} $$ is in the Schur-Agler class of the polydisk; that is, if $(T_1, \ldots , T_d)$ are commuting strict contractions then $ \| f(T_1, \ldots , T_d)\| \le 1$. The converse question,`` is every rational inner function in the Schur-Agler class of the polydisk necessarily of the above form?" led to questions regarding finite dimensional realizations of rational Schur-Agler functions, determinantal representations of stable polynomials, rational inner functions that are not Schur-Agler, and so forth. In this work we study these questions in Schur-Agler classes defined via a matrix-valued polynomial $\mathbf{P}$, leading to domains of the type $$\mathcal{D}_\mathbf{P}:= \{ z=(z_1,\ldots , z_d ) \in \mathbb{C}^d \ : \ \mathbf{P}(z)^*\mathbf{P}(z)< I \} . $$ Aside from the polydisk this general setting also includes the unit ball $\mathbb{B}^d$, and more generally, Cartan's classical domains. Using methods of Free Noncommutative Analysis, Systems Theory, and Algebraic Geometry, several new results were obtained. This talk is based on joint work with A. Grinshpan, D. S. Kaliuzhnyi-Verbovetskyi, and V. Vinnikov. \end{document}
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Hugo WoerdemanMultiple operator integrals and their applications.
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/11
https://openscholarship.wustl.edu/iwota2016/plenarytalks/plenarytalks/11Thu, 21 Jul 2016 11:40:00 PDT
Multiple operator integrals are multilinear transformations on operators that naturally arise as remainders of Taylor approximations of operator functions and also as extensions of Schur multipliers to a multilinear case. Theory of multiple operator integrals has been developing for over 60 years and has accumulated a number of important results and interesting applications. We will discuss recently established analytic properties of these transformations and their applications to perturbation theory.
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Anna Skripka