CRM/Montreal/Quebec Analysis Seminar

CRM/MONTREAL/QUEBEC ANALYSIS SEMINAR

The seminar webpage has moved here.

Seminars are usually held on Fridays from 2:30 to 3:30. In person seminars in Montreal are held at Concordia, McGill or Universite de Montreal; in person seminars in Quebec City are held at Laval.
To attend a zoom session, and for suggestions, questions etc. please contact Galia Dafni (galia.dafni@concordia.ca), Alexandre Girouard (alexandre.girouard@mat.ulaval.ca), Dmitry Jakobson (dmitry.jakobson@mcgill.ca), Damir Kinzebulatov (damir.kinzebulatov@mat.ulaval.ca) or Maxime Fortier Bourque (maxime.fortier.bourque@umontreal.ca)

Montreal Analysis seminar is currently held in hybrid mode, organized jointly with Laval University in Quebec City. Please contact one of the organizers for the seminar zoom links.

Some of the talks are recorded and posted on the CRM Youtube channel, on Mathematical Analysis Lab playlist

WINTER 2023

Friday, January 13, 14:30-15:30 Eastern time, hybrid seminar at CRM room 4336-4384
Cyril Letrouit (MIT)
Nodal sets of eigenfunctions of sub-Laplacians
Abstract: Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studied extensively over the past decades. In a recent work, we initiated the study of nodal sets of eigenfunctions of hypoelliptic operators on compact manifolds, focusing on sub-Laplacians (e.g. on compact quotients of the Heisenberg group). Fixing an arbitrary sub-Laplacian, some of our results hold for any eigenfunction, and others hold when averaging over random linear combinations of eigenfunctions. Our results show that nodal sets behave in an anisotropic way which can be analyzed with standard tools from sub-Riemannian geometry such as sub-Riemannian dilations, nilpotent approximation and desingularization at singular points. This is a joint work with S. Eswarathasan.

Friday, January 20, 14:30-15:30 Eastern time, hybrid seminar at CRM room 4336-4384
Carlo Pagano (Concordia University)
Universal Optimality of Hexagonal lattices
Abstract: It is well-known that the densest packing of the plane comes from the hexagonal lattice. A continuous version of this question is the universal optimality, asking whether the hexagonal lattice is the unique minimum configuration for each Gaussian potential. This deeper question is currently wide open. In the special case one restricts to the case of lattice configurations this can be rephrased as saying that the hexagonal lattice is the unique minimum for theta functions. This special case was established by Montgomery using a technique tailored to the space of lattice, which does not seem to extend to higher spaces of configuration. I will present a new proof of Montgomery's theorem obtained jointly with Naser Sardari and explain how our new strategy promises to adapt to higher spaces of configurations, which is ongoing work in progress.

Friday, February 17, 14:30-15:30 Eastern time, hybrid seminar at CRM room 4336-4384
Olivier Lafitte (Institut Galilée, Université Paris 13)
Reflection coefficient of a fractional reflector
Abstract: In seismology and in oil exploration, the diffraction or refraction of sound by an interface where there is a jump of sound velocity, formulae are well known. However, little is known when the velocity is continuous and has a jump in one of its derivatives (even a fractional one) This talk defines and give an estimate for the reflection \(R\) coefficient for solutions of the Helmholtz equation \[(\Delta u -c^{-2}(1+l(((x_{1})_+^{\alpha}))\partial^2_{t^2}u=0 \] that is \(e^{-i\omega t+ i k_2 x_2+i\sqrt{\frac{\omega^2}{c^2}-k_2^2}x_1}+R.e^{-i\omega t+ ik_2x_2-i\sqrt{\frac{\omega^2}{c^2}-k_2^2}x_1} \) for \(x_1<0\) and \(Tu^{>}\) for \(x_1>0\). This passes through the precise definition of an 'outgoing at infinity wave' \(u^{>}\) and its precise expression, using the limiting absorption principle. The leading order term of \(R\) is a Fourier multiplier, whose principal symbol will be given.

Friday, March 17, 13:30 to 14:30 Eastern time, Concordia University, Conference room LB 921-4
Andrew Comech (Texas A&M University)
Limiting absorption principle and virtual levels of operators in Banach spaces
Abstract : Virtual levels, also known as threshold resonances, admit several equivalent characterizations:
1. there are corresponding "virtual states" from a space "slightly weaker" than L^2;
2. there is no limiting absorption principle in their vicinity (e.g. no weights such that the "sandwiched" resolvent is uniformly bounded);
3. an arbitrarily small perturbation can produce an eigenvalue.
We develop a general approach to virtual levels in Banach spaces and provide applications to Schroedinger operators with nonselfadjoint potentials and in any dimension, deriving optimal estimates on the resolvent (including the lower dimensional cases). This is a joint work with Nabile Boussaid (Université Franche-Comte Besançon).

Friday, March 24, 14:00 to 15:00 Eastern time, CRM room 5340
Wilhelm Schlag (Yale University)
Lyapunov exponents, Schrödinger cocycles, and Avila's global theory
Abstract: In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years. By now, an extensive mathematical theory has developed around Anderson's predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or Diophantine rotation on the circle as base dynamics. In this setting, Artur Avila discovered about a decade ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the circle. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some recent results with Rui HAN (Louisiana) connecting Avila's notion of acceleration (the slope of the complexified Lyapunov exponent in the imaginary direction) to the number of zeros of the determinants of finite volume Hamiltonians relative to the complex toral variable. This connection allows one to answer questions arising in the supercritical case of Avila's global theory concerning the measure of the second stratum, Anderson localization on this stratum, as well as settle a conjecture on the Hölder regularity of the integrated density of states.

Friday, March 31, 14:00 to 15:00 Eastern time, McGill University, Burnside Hall, room 1104
Jessica Lin (McGill University)
Quantitative Homogenization of the Invariant Measure for Nondivergence Form Elliptic Equations
Abstract: In this talk, I will first give an overview of stochastic homogenization for nondivergence form elliptic equations, and its probabilistic counterpart, the study of quenched invariance principles for nonreversible diffusion processes. I will then present new quantitative homogenization results for the parabolic Green Function (fundamental solution) and for the unique ergodic invariant measure. This invariant measure is a solution of the adjoint equation in doubly divergence form, satisfying certain integrability conditions. I will discuss the implications of these homogenization results, such as heat kernel bounds on the parabolic Green function, regularity results for the adjoint equation, and quantitative ergodicity for the environmental process. This talk is based on joint work with Scott Armstrong and Benjamin Fehrman.

Friday, April 14, 13:00 to 14:00 Eastern time, McGill University, Burnside Hall, room 1104
Renato Lucà (Basque Center for Applied Mathematics)
Probabilistic pointwise convergence for the Schrödinger equation
Abstract: The problem of (Lebesgue) almost everywhere pointwise convergence of solutions of the Schrödinger equation to the initial data, when the initial data are in Sobolev Spaces, is very classic. It has been proposed by Carleson in the 80's and it has been only recently solved in any dimension combining the fundamental contributions of Bourgain, Du-Guth-Lee and Du-Zhang. The goal of this talk is to present some improved results for randomized initial data in both the linear and non linear settings.

Friday, April 28, 14:00 to 15:00 Eastern time, McGill University, Burnside Hall, room 1104
Annalisa Panati (Centre de Physique Théorique, Luminy, and University of Toulon)
Entropic Fluctuations in Quantum Two-time Measurement Framework
Abstract: Non-equilibrium statistical mechanics has seen some impressive developments in the last three decades, thank to the pioneering works of Evans, Cohen, Morris and Searles on the violation of the second law, soon followed by the ground-breaking formulation of the Fluctuation Theorem by Gallavotti and Cohen for entropy fluctuation in the early nineties. Their work was by vast literature, both theoretical and experimental.
The extension of these results to the quantum setting has turned out to be surprisingly challenging and it is still an undergoing effort. Kurchan's seminal work (2000) showed the measurement role has to be taken in account, leading to the introduction of the so called two-time measurement statistics(also known as full counting statistics). However introducing this framework leads to surprising phenomena with no classical counterpart. In this talk, I will present some work in progress, where we attempt to introduce a quantum equivalent of Gallavotti-Cohen (steady) entropic functional and compare it with the Evans-Searls (transient) entropic functional. We show that, due to the invasive measurement role, the situation differs considerably to its classical counterpart. We are able to obtain general results using functional and spectral analysis and operator algebras tools. Under more restrictive hypothesis, we can extend our analysis to theoexperimentally accessible indirect measurement framework (through an ancilla), using resonance analysis.
Joint work with T. Benoist, L. Bruneau, V. Jaksic, C.A.Pillet.

Friday, May 12, 14:00 to 15:00 Eastern time, CRM room 4336-4384
Dominique Maldague (MIT)
A sharp square function estimate for the moment curve in R^3
Abstract: I will present recent work which proves a sharp L^7 square function estimate for the moment curve (t , t^2, t^3) in R^3 using ideas from decoupling theory. In the context of restriction theory, in which we consider functions with specialized Fourier support, this is the only known sharp square function estimate with a non-even L^p exponent (p=7). The basic set-up is to consider a function f with Fourier support in a small neighborhood of the moment curve. Then partition the neighborhood into box-like subsets and form a square function in the Fourier projections of f onto these box-like regions. We will use a combination of recent tools including the "high-low" method and wave envelope estimates to bound f in L^7 by the square function of f in L^7.

Wednesday, May 17, 14:30 to 15:30 Eastern time, Concordia, LB 921-4
Cody Stockdale (Clemson)
A different approach to endpoint weak-type estimates for Calderón-Zygmund operators
Abstract: The weak-type (1,1) estimate for Calderón-Zygmund operators is fundamental in harmonic analysis. We investigate weak-type inequalities for Calderón-Zygmund singular integral operators using the Calderón-Zygmund decomposition and ideas inspired by Nazarov, Treil, and Volberg. We discuss applications of these techniques in the Euclidean setting, in weighted settings, for multilinear operators, for operators with weakened smoothness assumptions, and in studying the dimensional dependence of the Riesz transforms.

Friday, May 26, 14:30 to 15:30 Eastern time, Online
Denis Grebenkov (École Polytechnique)
Probabilistic insights onto the Dirichlet-to-Neumann operator
Abstract: In this talk, I show how the spectral properties of the Dirichlet-to-Neumann operator can be used to describe various characteristics of diffusion-controlled reactions, i.e., of reflected Brownian motion in an Euclidean domain with appropriate stopping conditions. For instance, one can derive a spectral expansion for the probability flux density that determines the joint probability law for the stopping time and location of the stochastic process. I discuss the advantages of this approach as compared to conventional spectral expansions based on the eigenfunctions of the Laplace operator. In particular, the Dirichlet-to-Neumann operator allows one to disentangle the diffusive dynamics in an Euclidean domain from surface reactions on its boundary. Several conjectures on spectral properties of this operator and related open problems will be presented.

FALL 2022

Tuesday, September 13, 15:00-16:00 Eastern time, hybrid seminar at CRM room 5340/5380/5388
Dmitry Jakobson (McGill University)
Nodalsets and negative eigenvalues in conformal geometry
Abstract: We discuss conformal invariants that arise from eigenfunctions with eigenvalue of the conformal Laplacian (and other conformally covariant operators). We also discuss some results about negative eigenvalues of the conformal Laplacian, as well as some generalizations for manifolds with boundary and discrete Laplacians (as time permits)

Friday, September 16, 14:30-15:30 Eastern time, hybrid seminar at Université Laval VCH-2870
Karol Szczypkowskis(Polytechnika Wroclawska)
Relativistic stable operators with critical potentials
Abstract: We prove sharp local in time heat kernel estimates for the relativistic stable operators perturbed by critical (Hardy) potentials. We discuss Hardy's inequality and blow-up of solutions. Other non-local operators with critical perturbations will be discussed.

Friday, October 7, 14:30-15:30 Eastern time, hybrid seminar at Université Laval VCH-2870
Panagiotis Polymerakisd(Université Laval)
Spectral estimates for Riemannian submersions
d Abstract: In this talk we will survey some results on the behavior of the spectrum under Riemannian submersions. In the first part of the talk, we will show that if the submersion has closed fibers of bounded mean curvature, then the base manifold has discrete spectrum if and only if so does the total space. In the second part, we will consider submersions arising from actions of Lie groups and investigate how the behavior of the spectrum is intertwined with algebraic properties of the group.

Friday, October 21, 14:30-15:30 Eastern time, hybrid seminar at the.CRM,.room 4336-4384
Dmitry Faifman (Tel Aviv University)
A quasianalytic property of families in the image of integral transforms on higher grassmannians
Abstract: We will consider certain integral operators on higher grassmannians that appear naturally in convex geometry, as well as in representation theory: the Radon and cosine transforms. The image of such operators is often a rather small subspace of all functions, and can be explicitly described in terms of its \(SO(n)\)-components. We will describe a quasianalytic-type property exhibited by those images, allowing to uniquely determine a function from its values on a small set. This allows us to sharpen classical uniqueness theorems of Funk and Alexandrov in geometric tomography, and of Klain and Schneider in valuation theory. Similar results hold for more general families of functions and global sections of bundles appearing as representations of \(GL_n(R)\).

Thursday, October 27, 14:30-15:30 Eastern time, hybrid seminar.at.Université Laval VCH-3840
Catherine Sulem (University.of.Toronto)
A Hamiltonian approach to nonlinear modulation of surface.water.waves in the presence of linear shear currents.
Abstract:.This is.a study of the water wave problem in a two-dimensional domain in the.presence of.constant vorticity. The goal is to describe the effects of uniform.shear flow.on the modulation of weakly nonlinear quasi-monochromatic surface.waves..Starting from the Hamiltonian formulation of this problem and using.techniques.of Hamiltonian transformation theory, we derive a Hamiltonian,.high-order.Nonlinear Schrodinger equation (often referred to as Dysthe.equation) for the.time evolution of the wave envelope. Consistent with previous.studies, we.observe that the uniform shear flow tends to enhance or weaken the.modulational.instability of Stokes waves depending on its direction and.strength. This model.is tested against direct numerical simulations of the full.Euler equations and.against a related Dysthe equation recently derived by.Curtis, Carter and Kalisch.(2018). This is a joint work with P. Guyenne and A..Kairzhan.

Friday, October 28, 14:30-15:30 Eastern time, hybrid seminarnatnUniversité Laval VCH-2870
Edward Bierstone (UniversitynofnToronto)
Extension and division of \(\mathcal{C}^m\) semialgebraicnfunctionsn
Abstract: I will discuss \(\mathcal{C}^m\) Whitneynproblems wherenthe given data is semialgebraic and the solution is to bensemialgebraic; innparticular, questions concerning extension to \(\mathbb{R}^n\)nofn\(\mathcal{C}^m\) semialgebraic functions defined on a closed subset,nandn\(\mathcal{C}^m\) semialgebraic solutions of systems of linear equationsnwhosencoefficients are semialgebraic functions. Positive answers are known fornforn\(n=2\) (Fefferman-Luli, 2021) and for general \(m,\, n\) modulo a certainnlossnof differentiability (Bierstone-Campesato-Milman, 2021). I will try tondescribenthe methods of both results. It is not yet evident whether positivenanswersnpreserving the differentiability class should be expected, in general.n

Friday, November 4, 14:00-15:00 Eastern time (note the earlier time!),lZoomlseminar
Eli Liflyand (Bar Ilan University)
Wienerlalgebrasland trigonometric series in a coordinated fashion
lAbstract: Letl\(W_0(\mathbb R)\) be the Wiener Banach algebra of functionslrepresentable bylthe Fourier integrals of Lebesgue integrable functions. It islproven in thelpaper that, in particular, a trigonometriclseriesl\(\sum\limits_{k=-\infty}^\infty c_k e^{ikt}\) is the Fourier series oflanlintegrable function if and only if there exists a \(\phi\in W_0(\mathbblR)\)lsuch that \(\phi(k)=c_k\), \(k\in\mathbb Z\). If \(f\in W_0(\mathbb R)\),lthenlthe piecewise linear continuous function \(\ell_f\) definedlbyl\(\ell_f(k)=f(k)\), \(k\in\mathbb Z\), belongs to \(W_0(\mathbb R)\) aslwell.lMoreover, \(\|\ell_f\|_{W_0}\le \|f\|_{W_0}\). Similar relationslarelestablished for more advanced Wiener algebras. These results arelsupplemented bylnumerous applications. In particular, new necessary andlsufficient conditionslare proved for a trigonometric series to be a Fourierlseries and new propertieslof \(W_0\) are established. This is a joint work withlR. Trigub.

Friday, November 11, 14:30-15:30 Eastern time, hybrid seminar at thebCRM,broom 4336-4384
Jérôme Vétoisb(McGillbUniversity)
Sign-changing blowing-up solutions to the Yamabebequation on abclosed Riemannian manifold
Abstract: In this talk, Ibwill discussbthe question of existence of families of sign-changing solutions tobthe Yamabebequation, which blow up in the sense that their maximum values tendbto infinity.bIt is known that in the case of positive solutions, there does notbexist anybblowing-up families of solutions to this problem in dimensions lessbthan 25,bexcept in the case of manifolds conformally equivalent to the roundbsphereb(Khuri, Marques and Schoen, 2009). I will present a construction showingbthebexistence of a non-round metric on spherical space forms of dimensionsbgreaterbthan 10 for which there exist families of sign-changing blowing-upbsolutions tobthis problem. Moreover, the solutions we construct have the lowestbpossibleblimit energy level. As a counterpart, we will see that such solutionsbdo notbexist at this energy level in dimensions less than 10. This is a jointbwork withbBruno Premoselli (Université Libre de Bruxelles).

Friday, November 18, 14:30-15:30 Eastern time, hybrid seminar at the)CRM,)room 4336-4384
Sergey Tikhonov (ICREA - CRM)Barcelona)
)Hardy-Littlewood inequalities for Fourier transforms
)Abstract: We)discuss classical Hardy-Littlewood-Paley inequalities for)Fourier)coefficients/transforms as well as their possible extensions for any \(1)< p <)\infty\).

Friday, December 16, 14:30-15:30 Eastern time, hybrid seminar)at)Université Laval VCH-2870
Hugues Auvray)(Université)Paris-Sud)
Noyaux de Bergman sur les surfaces de)Riemann)épointées
Abstract: Dans des travaux en)commun avec)X. Ma (Paris 7) et G. Marinescu (Cologne), nous obtenons des)asymptotiques)raffinées pour des noyaux de Bergman calculées)à partir de)données singulières. On travaille sur le)complémentaire)d'un nombre fini de points, vus comme singularités,)dans une surface de)Riemann compacte, que l'on munit d'une métrique)étendant la)métrique cusp de Poincaré autour des)singularités ; on se)donne également un fibré en droites)holomorphe polarisant pour)cette métrique. J'expliquerai dans cet)exposé, en me concentrant)sur l'aspect analytique des résultats,)comment une description)avancée du modèle (sur le disque)unité)épointé) et des techniques de localisation dans un)contexte)à poids permettent de décrire les noyaux de)Bergman)associés à de telles surfaces de Riemann, et ce jusque)aux)singularités.