## 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)

2020 Zoom Seminars