Réunion d'été SMC 2025
Ville de Québec, 6 - 9 juin 2025
Complex analysis, Harmonic analysis and Operator theory
Org: Marcu-Antone Orsoni (Université Laval) et Pierre-Olivier Parisé (Université du Québec à Trois-Rivières)
[PDF]
Org: Marcu-Antone Orsoni (Université Laval) et Pierre-Olivier Parisé (Université du Québec à Trois-Rivières)
[PDF]
- ABDEL RAHMAN AL-ABDALLAH, Brandon University
- SHAHRIAR ASLANI, University of Toronto
- ILIA BINDER, University of Toronto
Schr\"odinger operators with small quasiperiodic potentials: comb domains [PDF]
- SHAHRIAR ASLANI, University of Toronto
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In the talk, we discuss a characterization of the spectra of Schr\"odinger operators with small quasiperiodic analytic potentials in terms of their comb domains. We will also discuss action variables motivated by the integrable Korteweg-de Vries (KdV) system. The talk is based on joint work with D.~Damanik, M.~Goldstein, and M.~Lukic.
- ALEX BRUDNYI, University of Calgary
- ALMAZ BUTAEV, University of The Fraser Valley
- ANA ČOLOVIĆ, Washington University in St. Louis
- SETAREH ESKANDARI, Umea University
- PAUL GAUTHIER, Université de Montréal
- DAMIR KINZEBULATOV, Université Laval
- POORNENDU KUMAR, University of Manitoba
- YU-RU LIU, University of Waterloo
- JAVAD MASHREGHI, Université Laval
- MAËVA OSTERMANN, Université de Lille
- THOMAS RANSFORD, Université Laval
Double-layer potentials, configuration constants and applications to numerical ranges [PDF]
- ALMAZ BUTAEV, University of The Fraser Valley
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We consider estimates $\|p(T)\| \leq K \sup_{z \in \Omega} |p(z)|$, where $T$ is a Hilbert-space operator, $p$ is a polynomial and $\Omega$ is a compact convex set containing the numerical range of $T$. We show that the well-known Crouzeix--Palencia bound $K \leq 1 + \sqrt{2}$ can be improved to $K \leq 1 + \sqrt{1 + a(\Omega)}$, where $a(\Omega)$ is what we call the analytic configuration constant of $\Omega$. The latter is an analytic analogue of the classical configuration constant $c(\Omega)$ arising in the theory of double-layer potentials. A celebrated result of Neumann, dating back to 1877, states that $c(\Omega)<1$ unless $\Omega$ is a triangle or a quadrilateral. Among other results, we prove that, in our case, we always have $a(\Omega)<1$. Consequently, equality never holds in the Crouzeix--Palencia bound. (Joint work with Bartosz Malman, Javad Mashreghi and Ryan O'Loughlin.)
- ERIC SAWYER, Mc Master University
- RASUL SHAFIKOV, University of Western Ontario
Meromorphic convexity on Stein manifolds [PDF]
- RASUL SHAFIKOV, University of Western Ontario
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Rational convexity of compact subsets in complex Euclidean spaces is
important in the approximation theory. I will discuss generalizations of
rational convexity to Stein manifolds. I will then give a
characterization of this type of convexity for a certain class of compacts
in the spirit of Duval-Sibony, Guedj, and Nemirovski. This is joint work
with B. Boudreaux and P. Gupta.
- WILLIAM VERREAULT, University of Toronto
Fourier Decay of GMC Measures [PDF]
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The Fourier coefficients of multiplicative chaos measures appear naturally in the study of random matrices, QFTs, and even number theory. The harmonic analysis of the canonical GMC measure on the unit circle allowed Garban and Vargas to show that the associated Fourier coefficients tend to 0. The next step is to ask how fast this decay occurs, which corresponds to the Fourier dimension studied in fractal analysis. We compute the exact Fourier dimension of the circle-GMC measure, thereby proving a conjecture of Garban-Vargas based on a fourth moment computation. Our arguments are elementary, relying on a construction of an auxiliary, scale-invariant Gaussian field.
- QUN WANG, University of Toronto
- MAHISHANKA WITHANACHCHI, University of Calgary
- NINA ZORBOSKA, University of Manitoba
- MAHISHANKA WITHANACHCHI, University of Calgary