## 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)

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