## 2021-22 CRM/MONTREAL/QUEBEC ANALYSIS ZOOM SEMINARS

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

## FALL 2022

Tuesday, September 13, 15:00-16:00 Eastern time, hybrid seminar at CRM room 5340/5380/5388
Dmitry Jakobson (McGill University)
Nodal sets and negative eigenvalues in conformal geometry
Abstract: We discuss conformal invariants that arise from eigenfunctions with eigenvalue 0 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 Szczypkowski (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 Polymerakis (Université Laval)
Spectral estimates for Riemannian submersions
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 seminar at Université Laval VCH-2870
Edward Bierstone (University of Toronto)
Extension and division of $$\mathcal{C}^m$$ semialgebraic functions
Abstract: I will discuss $$\mathcal{C}^m$$ Whitney problems where the given data is semialgebraic and the solution is to be semialgebraic; in particular, questions concerning extension to $$\mathbb{R}^n$$ of $$\mathcal{C}^m$$ semialgebraic functions defined on a closed subset, and $$\mathcal{C}^m$$ semialgebraic solutions of systems of linear equations whose coefficients are semialgebraic functions. Positive answers are known for for $$n=2$$ (Fefferman-Luli, 2021) and for general $$m,\, n$$ modulo a certain loss of differentiability (Bierstone-Campesato-Milman, 2021). I will try to describe the methods of both results. It is not yet evident whether positive answers preserving the differentiability class should be expected, in general.
Friday, November 4, 14:00-15:00 Eastern time (note the earlier time!), Zoom seminar
Eli Liflyand (Bar Ilan University)
Wiener algebras and trigonometric series in a coordinated fashion
Abstract: Let $$W_0(\mathbb R)$$ be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proven in the paper that, in particular, a trigonometric series $$\sum\limits_{k=-\infty}^\infty c_k e^{ikt}$$ is the Fourier series of an integrable function if and only if there exists a $$\phi\in W_0(\mathbb R)$$ such that $$\phi(k)=c_k$$, $$k\in\mathbb Z$$. If $$f\in W_0(\mathbb R)$$, then the piecewise linear continuous function $$\ell_f$$ defined by $$\ell_f(k)=f(k)$$, $$k\in\mathbb Z$$, belongs to $$W_0(\mathbb R)$$ as well. Moreover, $$\|\ell_f\|_{W_0}\le \|f\|_{W_0}$$. Similar relations are established for more advanced Wiener algebras. These results are supplemented by numerous applications. In particular, new necessary and sufficient conditions are proved for a trigonometric series to be a Fourier series and new properties of $$W_0$$ are established. This is a joint work with R. Trigub.
Friday, November 11, 14:30-15:30 Eastern time, hybrid seminar at the CRM, room 4336-4384
Jérôme Vétois (McGill University)
Sign-changing blowing-up solutions to the Yamabe equation on a closed Riemannian manifold
Abstract: In this talk, I will discuss the question of existence of families of sign-changing solutions to the Yamabe equation, which blow up in the sense that their maximum values tend to infinity. It is known that in the case of positive solutions, there does not exist any blowing-up families of solutions to this problem in dimensions less than 25, except in the case of manifolds conformally equivalent to the round sphere (Khuri, Marques and Schoen, 2009). I will present a construction showing the existence of a non-round metric on spherical space forms of dimensions greater than 10 for which there exist families of sign-changing blowing-up solutions to this problem. Moreover, the solutions we construct have the lowest possible limit energy level. As a counterpart, we will see that such solutions do not exist at this energy level in dimensions less than 10. This is a joint work with Bruno 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 $$1 < p < \infty$$.

## WINTER 2022

Friday, February 11, 14:30-15:30 Eastern time, zoom seminar
Chris Bishop (Stony Brook)
Dessins and Dynamics
Abstract: After defining harmonic measure on a planar domain, I will discuss "true trees", i.e., trees drawn in the plane so that every edge has equal harmonic measure and so that these measures are symmetric on each edge. True trees on the 2-sphere are a special case in Grothendieck's theory of dessins d'enfant, where a graph on a topological surface induces a conformal structure on that surface. I will recall the connection between dessins, equilateral triangulations and branched coverings (Belyi's theorem). I will also describe some recent applications of these ideas to holomorphic dynamics: approximating sets by polynomial Julia sets, finding meromorphic functions with prescribed postcritical orbits, constructing finite type dynamical systems on hyperbolic Riemann surfaces, building wandering domains for entire functions, and estimating the fractal dimensions of transcendental Julia sets. There will be many pictures and few proofs.
Friday, February 25, 14:30-15:30 Eastern time, zoom seminar
Jane Wang (Indiana University)
Slope Gap Distributions of Veech Translation Surfaces
Abstract: Translation surfaces are surfaces that are locally Euclidean except at finitely many points called cone points, an example being the regular octagon with opposite sides identified (the vertices are identified and become a single cone point). A saddle connection is then a straight trajectory that begins and ends at a cone point. It is known that on almost every translation surface, the set of angles of saddle connections on the surface is equidistributed in the circle. A finer notion of how random the saddle connection directions are is given by something called the gap distribution of the surface. In this talk, we will explain what the slope gap distribution of a translation surface is and survey some known results about slope gap distributions, including how one can use properties of the horocycle flow to compute the slope gap distributions of special translation surfaces called Veech surfaces. We'll then discuss recent results showing that the slope gap distributions of Veech surfaces have to satisfy some nice analytic properties. This project is joint work with Luis Kumanduri and Anthony Sanchez.
Friday, March 25, 14:30-15:30 Eastern time, zoom seminar
Kevin Pilgrim (Indiana University)
Conformal surface embeddings and extremal length
Abstract: Given two Riemann surfaces with boundary and a homotopy class of topological embeddings between them, we show there is a conformal embedding in the homotopy class if and only if the extremal length of every simple multi-curve is decreased under the embedding. For applications to dynamical systems, we need an additional fact: if the ratio is bounded above away from one, then it remains so under passing to any finite cover. I will also briefly mention how under natural conditions the technique of quasiconformal surgery promotes so-called rational-like maps f:f^{-1}(S)→S, where f^{-1}(S)⊂S are planar Riemann surfaces, to rational maps. This is joint work of Jeremy Kahn, Kevin M. Pilgrim, and Dylan P. Thurston; https://arxiv.org/abs/1507.05294
Friday, April 8, 14:30-15:30 Eastern time, zoom seminar
Blake Keeler (McGill University)
The Two-Point Weyl Law on Manifolds without Conjugate Points
Abstract: In this talk, we discuss the asymptotic behavior of the spectral function of the Laplace-Beltrami operator on a compact Riemannian manifold $$M$$ with no conjugate points. The spectral function, denoted $$\Pi_\lambda(x,y),$$ is defined as the Schwartz kernel of the orthogonal projection from $$L^2(M)$$ onto the eigenspaces with eigenvalue at most $$\lambda^2$$. In the regime where $$(x,y)$$ is restricted to a sufficiently small neighborhood of the diagonal in $$M\times M$$, we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for $$\Pi_\lambda$$ and its derivatives of all orders. This generalizes a result of Bérard which established an on-diagonal estimate for $$\Pi_\lambda(x,x)$$ without derivatives. Furthermore, when $$(x,y)$$ avoids a compact neighborhood of the diagonal, we obtain the same logarithmic improvement in the standard upper bound for the derivatives of $$\Pi_\lambda$$ itself. We also discuss an application of these results to the study of monochromatic random waves.
Friday, April 22, 14:30-15:30 Eastern time, zoom seminar
Norm Levenberg (Indiana University)
Zeros of Random Polynomial Mappings in Several Complex Variables
Abstract: We discuss some results on random polynomials with an eye towards obtaining universality results under the most general assumptions on the random coefficients. In particular, we generalize and strengthen some previous results on asymptotic distribution of normalized zero measures and currents associated to random polynomials and random polynomial mappings in several complex variables. The talk is based on joint work with Turgay Bayraktar and Tom Bloom.
Friday, May 6, 14:30-15:30 Eastern time, zoom seminar
Stefano Decio (Norwegian University of Science and Technology)
Zeros of sums of eigenfunctions
Abstract: A guiding principle in the study of eigenfunctions of the Laplace-Beltrami operator is that their properties should resemble those of polynomials. In this light, I will discuss oscillations and zeros for linear combinations of Laplace eigenfunctions on Riemannian manifolds. In particular, I will prove that zeros become dense in the manifold if not too many eigenfunctions are summed. The talk will also feature some open questions on eigenfunctions sums.
Friday, May 20, 14:30-15:30 Eastern time, hybrid talk at CRM and on Zoom
Álvaro Romaniega (ICMAT)
Nodal sets of monochromatic waves from a deterministic and random point of view
Abstract: In this talk we present recent results on the nodal set (i.e., the zero level set) of monochromatic waves (i.e., solutions of the Helmholtz equation) on the Euclidean space. Following the breakthrough work of F. Nazarov and M. Sodin, a growing literature gives us powerful probabilistic results for the number of connected components of the nodal set of random monochromatic waves. The aim of this talk is to explore the properties of these standard random monochromatic waves and, consequently, define a more general class of random monochromatic waves depending on a parameter $$s\in\mathbb{R}$$, which includes the standard definition as a particular case. This parameter controls some regularity (of the Fourier transform) and decay properties of these waves. Given that, we study the structure of the nodal set depending on that parameter from a deterministic and from a random point of view. Finally, we show how to construct deterministic realizations or examples of monochromatic waves satisfying the probabilistic Nazarov-Sodin volumetric growth for the number of connected components of the nodal set and similarly for the volume of the nodal set. This is a joint work with A. Enciso, D. Peralta-Salas and A. Sartori.

## FALL 2021

Friday, October 22, 14:30-15:30 Eastern time, zoom seminar
Yannick Sire (Johns Hopkins)
Some results on harmonic maps with free boundary and beyond
Abstract: The theory of harmonic maps with free boundary is an old topic in geometric analysis. I will report on recent results on their Ginzburg-Landau approximation, regularity theory, and their heat flow. I will also describe several models in the theory of liquid crystals where the heat flow of those maps appears, emphasizing on some well-posedness issues and some hints on the construction of blow-up solutions. Several important results in geometric analysis such as extremal metrics for the Steklov eigenvalues for instance make a crucial use of such maps. I’ll give some open problems and will try to explain how to attack few open questions in the field using tools recently developed.
Friday, October 29, 14:30-15:30 Eastern time, zoom seminar
Michael Roysdon (Tel Aviv)
On measure theoretic projection bodies
Abstract: pdf
Friday, November 12, 14:30-15:30 Eastern time, zoom seminar
Maxime Fortier Bourque (Universite de Montreal)
The extremal length systole of the Bolza surface
Abstract: The extremal length of a curve on a Riemann surface is a conformal invariant that has a nice geometric description but is not so simple to compute in practice. The extremal length systole is defined as the infimum of the extremal lengths of all non-contractible closed curves. I will discuss joint work with Didac Martinez-Granado and Franco Vargas Pallete in which we compute the extremal length systole of the Bolza surface, the most symmetric surface of genus two. The calculation involves certain identities for elliptic integrals called the Landen transformations. We also prove that the Bolza surface is a local maximizer for the extremal length systole and conjecture that it is the unique global maximizer.
Friday, November 19, 14:30-15:30 Eastern time, zoom seminar
Dimitrios Ntalampekos (Stony Brook)
Rigidity theorems for circle domains
Abstract: A circle domain $$\Omega$$ in the Riemann spherehisha domain each of whose boundary components is either a circle or a point. Ahcircle domain $$\Omega$$ is called conformally rigid if every conformal maphfrom $$\Omega$$ onto another circle domain is the restriction of a Mobiushtransformation. In this talk I will present some new rigidity theorems forhcircle domains satisfying a certain quasihyperbolic condition. As a corollary, John and Holder circle domains are rigid. This provides new evidence for a conjecture of He and Schramm, relating rigidity and conformal removability. This talk is based on joint work with Malik Younsi.
Friday, November 26, 14:30-15:30 Eastern time, zoom seminar
Suresh Eswarathasan (Dalhousie)
Fractal uncertainty principle for discrete Cantor sets for random alphabets.
Abstract: The fractalhruncertainty principle (FUP) introduced by Dyatlov-Zahl '16 has seen somehrpowerful applications in the last few years and become a hot topic in harmonichranalysis. In this talk, we study the FUP for discrete Cantor sets from ahrprobabilistic perspective. We show that randomizing our alphabets gives ahrquantifiable improvement over the current "zero" and "pressure" bounds. In turn, this provides the best possible exponent when the Cantor sets enjoy either the strongest Fourier decay or additive energy assumptions. This is joint work with Xiaolong Han (Cal. State Northridge)

2020 Zoom Seminars