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

Seminars are usually held on Mondays or Fridays. 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 online on zoom, organized
jointly with Laval University in Quebec City. Please contact one of the
organizers for the seminar zoom links.**

** The talks are recorded
and posted on the CRM
Youtube channel, on Mathematical
Analysis Lab playlist **

## 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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