Réunion d'hiver SMC 2019

Toronto, 6 - 9 décembre 2019

Analyse complexe et théorie des opérateurs
Org: Ilia Binder (University of Toronto), Damir Kinzebulatov (Université Laval) et Javad Mashreghi (Université Laval)
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IEVGEN BILOKOPYTOV, University of Manitoba Maximum modulus principle for multipliers [PDF]

In this talk I will show that (not necessarily holomorphic) multipliers of a wide class of normed spaces of continuous functions over a connected Hausdorff topological space cannot attain their multiplier norms, unless they are constants.

THOMAS BLOOM, University of Toronto Asymptotic zero distribution of random orthogonal polynomials [PDF]

We consider random polynomials of the form $H_n(z)=\sum_{j=0}^n \xi_jq_j(z)$ where the $\{\xi_j\}$ are i.i.d. non-degenerate complex random variables, and the $\{q_j(z)\}$ are orthogonal polynomials ( deg $q_j=j$) with respect to an appropriate compactly supported measure in the plane.The problem is to understand (probabilistically) the behavior of the zeros of $H_n$ as $n\rightarrow\infty.$

Study of the Kac Ensemble, (when $q_j(z)=z^j$) goes back to the 1950's.

I will present recent results on the almost sure convergence and the convergence in probability of the zeros. This is work of Ibraguimov and Zaporozhets, D.Dauvergne and myself.

ALEX BRUDNYI, University of Calgary TOWARD SOLUTION OF THE MULTIVARIATE CORONA PROBLEM [PDF]

We present a new approach to the multivariate corona problem for the algebra of bounded holomorphic functions on an open polydisk.

CHENG CHU, Université Laval Which de Branges-Rovnyak spaces have complete Nevanlinna-Pick property? [PDF]

Complete Nevanlinna-Pick kernels are related to Nevanlinna-Pick interpolation problem. Many properties of the Hardy space carry over to other spaces with complete Nevanlinna-Pick property. A natural question is to decide which reproducing kernel Hilbert spaces have complete Nevanlinna-Pick property. We characterize the de Branges-Rovnyak spaces with complete Nevanlinna-Pick property. Our method relies on the general theory of reproducing kernel Hilbert spaces.

GALIA DAFNI, Concordia University Extension and approximation for VMO on a domain [PDF]

In joint work with Almaz Butaev, we study the question of extension of functions of vanishing mean oscillation on a domain $\Omega \subset {\mathbb R}^n$ to VMO(${\mathbb R}^n$). For the case of bounded mean oscillation (BMO), it was shown by P. Jones (1980) that a bounded extension if possible if and only if $\Omega$ is a uniform domain. By suitably modifying Jones' extension we are able to use a single operator to extend BMO and VMO as well as other function spaces. This question turns out to be related to approximation of VMO functions by Lipschitz functions. We also study the question in the local setting.

RICHARD FOURNIER, Dawson College and CRM (Montréal) On a polynomial inequality of Schur [PDF]

In this talk, I shall compare a polynomial inequality due to Dryanov, Fournier and Ruscheweyh (see J. Approx.Theory 136(2005), 84–90) to another one attributed to Isaï Schur ( see J. Approx.Theory 182(2014), 103–109) and observe that they indeed are independent one of each other.

PAUL GAUTHIER, Université de Montréal Universality, polynomial approximation and the Riemann Hypothesis [PDF]

Two observations. Firstly, the universality of the Riemann Hypothesis easily yields a conjecture equivalent to the Riemann Hypothesis. Secondly, we recall an interesting and plausible conjecture of Andersson on polynomial approximation, which has been partially confirmed with much effort and which (if correct) suggests the Riemann Hypothesis fails.

ADI GLUCKSAM, University of Toronto Growth of measurably entire function and related questions [PDF]

Let $T$ be the action of the complax plain on the space of entire functions defined by translations, i.e $T_w$ takes the entire function $f(z)$ to the entire function $f(z+w)$. B.Weiss showed in `97 that there exists a probability measure defined on the space of entire functions, which is invariant under this action. In this talk I will present (almost) optimal bounds on the minimal possible growth of functions in the support of such measures, and discuss other growth related problems inspired by this work. The talk is partly based on a joint work with L. Buhovsky, A.Logunov, and M. Sodin.

ISAO ISHIKAWA, RIKEN On the boundedness of composition operators on reproducing kernel Hilbert spaces with analytic positive definte functions [PDF]

In this talk,I will explain our result which says boundedness of composition operators of maps implies the maps are affine maps in certain situations. Our problem originally comes from the applied mathematics. Composition operators (Koopman operators) are classically investegated in the theory of function space and complex analysis, but, they have been getting popular in the context of machine leraning and data analysis these days. Besides, reproducing kernel Hilbert spaces with analytic positive definite functions on euclidean spaces are utilized in many fields in engeneering and statistics. On the other hand, although it is important to prove the relation between the properties of maps and those of composition operators of the maps to guruantee theoretical validity, such relation is currently not known very well. In some important situation, we prove that a map become an affine map if its composition operator is bounded on an RKHS associated with analytic positive definite functions on euclidean spaces. This is the joint work with Masahiro Ikeda (RIKEN/Keio University) and Yoshihiro Sawano (Tokyo metropolitan University/RKEN

JAVAD MASHREGHI, Laval

ERIC SAWYER, McMaster University Two weight testing theory [PDF]

We report on recent results in the theory of two weight testing, aka T1 theorems, including joint works with Tuomas Hytonen, Kangwei Li, Chun-Yen Shen, Ignacio Uriarte-Tuero, Robert Rahm and Brett Wick

RASUL SHAFIKOV, University of Western Ontario Polynomially and rationally convex embeddings [PDF]

I will discuss recent developments in polynomially and rationally convex embeddings of real submanifolds into complex Euclidean spaces.

MOHAMMAD SHIRAZI, University of Manitoba Grunsky Operator and Inequality for Open Riemann Surfaces with Finite Borders [PDF]

Consider an open Riemann surface $\Sigma$ of genus $g>0$ with $n>1$ borders, each one homeomorphic to the unit circle. The surface $\Sigma$ can be described as a compact Riemann surface $\mathcal{R}$ of the same genus $g$, from which $n$ simply-connected domains $\Omega_1, \dots, \Omega_n$, removed; that is, $\Sigma=\mathcal{R}\backslash \cup cl(\Omega_k)$. Fix conformal maps $f_k$ from the unit disc $\mathbb{D}$ onto $\Omega_k, k=1, \dots, n$. We may assume each $f_k$ has a quasiconformal extension to an open neighbourhood of $\mathbb{D}$. Let $\mathbf{f}=\left(f_1, \dots, f_n\right)$.

I will define the Grunsky operator $Gr_{\mathbf{f}}$ corresponding to $\mathbf{f}$ (equivalently to $\Sigma$) on some Dirichlet spaces when all the boundary curves are quasicircles in $\mathcal{R}$. I will show that the norm of the Grunsky operator is less than or equal to one. This is a generalization of the classical Grunsky inequalities from the planar case to bordered Riemann surfaces described above.

Joint work with E. Schippers and W. Staubach.

IGANCIO URIARTE-TUERO, Michigan State University Two weight norm inequalities for singular and fractional integral operators in $R^n$. [PDF]

I will report on recent advances on the topic, related to proofs of T1 type theorems in the two weight setting for Calder\'{o}n-Zygmund singular and fractional integral operators, with side conditions, and related counterexamples. Mostly joint work with Eric Sawyer and Chun-Yen Shen.

The talk will be self-contained and provide a general overview of the area plus some recent advances. Eric Sawyer's talk in this same session will provide more specifics on the latest advances.

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