2020 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 Iosif Polterovich (iossif@dms.umontreal.ca)

After a break due to COVID-19, Montreal Analysis seminar has resumed 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

FALL 2020

Friday, September 11, 9:00 Eastern Time, zoom seminar
Jose Maria Martell (ICMAT)
Uniform rectifiability and elliptic operators satisfying a Carleson measure condition
Abstract: In this talk I will study the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. Our setting is that of domains having an Ahlfors regular boundary and satisfying the so-called interior Corkscrew and Harnack chain conditions (these are respectively scale-invariant/quantitative versions of openness and path-connectivity) and we show that for the class of Kenig-Pipher uniformly elliptic operators (operators whose coefficients have controlled oscillation in terms of a Carleson measure condition) the solvability of the $L^p$-Dirichlet problem with some finite $p$ is equivalent to the quantitative openness of the exterior domains or to the uniform rectifiablity of the boundary. Joint work with S. Hofmann, S. Mayboroda, T. Toro, and Z. Zhao.

Friday, October 16, 13:00 Eastern Time (to be confirmed), zoom seminar
Guangyu Xi (University of Maryland)
Title TBA

SPRING/SUMMER 2020

Joint with Geometric Analysis Seminar
Wednesday, April 29, 13:30-14:30, Zoom seminar
Julian Scheuer (Freiburg)
Concavity of solutions to elliptic equations on the sphere
Abstract: An important question in PDE is when a solution to an elliptic equation is concave. This has been of interest with respect to the spectrum of linear equations as well as in nonlinear problems. An old technique going back to works of Korevaar, Kennington and Kawohl is to study a certain two-point function on a Euclidean domain to prove a so-called concavity maximum principle with the help of a first and second derivative test.
To our knowledge, so far this technique has never been transferred to other ambient spaces, as the nonlinearity of a general ambient space introduces geometric terms into the classical calculation, which in general do not carry a sign.
In this talk we have a look at this situation on the unit sphere. We prove a concavity maximum principle for a broad class of degenerate elliptic equations via a careful analysis of the spherical Jacobi fields and their derivatives. In turn we obtain concavity of solutions to this class of equations. This is joint work with Mat Langford, University of Tennessee Knoxville.

Friday, May 1, starts at 13:00 (Eastern time), on zoom
Video link:
13:00. Alexandre Girouard (Universite Laval)
Homogenization of Steklov problems with applications to sharp isoperimetric bounds, part I
Abstract: The question to find the best upper bound for the first nonzero Steklov eigenvalue of a planar domain goes back to Weinstock, who proved in 1954 that the first nonzero perimeter-normalized Steklov eigenvalue of a simply-connected planar domain is 2*pi, with equality iff the domain is a disk. In a recent joint work with Mikhail Karpukhin and and Jean Lagacé, we were able to let go of the simple connectedness assumption. We constructed a family of domains for which the perimeter-normalized first eigenvalue tends to 8π. In combination with Kokarev's bound from 2014, this solves the isoperimetric problem completely for the first nonzero eigenvalue. The domains are obtained by removing small geodesic balls that are asymptotically densely periodically distributed as their radius tends to zero. The goal of this talk will be to survey recent work on homogenisation of the Steklov problem which lead to the above result. On the way we will see that many spectral problems can be approximated by Steklov eigenvalues of perforated domains. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than 2*pi. This talk is based on joint work with Antoine Henrot (U. de Lorraine), Mikhail Karpukhin (UCI) and Jean Lagacé(UCL).
13:50. Jean Lagacé (UCL)
Homogenization of Steklov problems with applications to sharp isoperimetric bounds, part II.
Abstract: Traditionally, deterministic homogenisation theory uses the periodic structure of Euclidean space to describe uniformly distributed perturbations of a PDE. It has been known for years that it has many applications to shape optimization. In this talk, I will describe how the lack of periodic structure can be overcome to saturate isoperimetric bounds for the Steklov problem on surfaces. The construction is intrinsic and does not depend on any auxiliary periodic objects or quantities. Using these methods, we obtain the existence of free boundary minimal surfaces in the unit ball with large area. I will also describe how the intuition we gain from the homogenization construction allows us to actually construct some of them, partially verifying a conjecture of Fraser and Li. This talk is based on joint work with Alexandre Girouard (U. Laval), Antoine Henrot (U. de Lorraine) and Mikhail Karpukhin (UCI).

Thursday, May 7, starts at 13:00 (Eastern time), Zoom seminar
Video link
Steve Zelditch (Northwestern)
Spectral asymptotics for stationary spacetimes
Abstract: We explain how to formulate and prove analogues of the standard theorems on spectral asymptotics on compact Riemannian manifolds -- Weyl's law and the Gutzwiller trace formula-- for stationary spacetimes. As a by-product we prove a semi-classical Weyl law for the Klein-Gordon equation where the mass is the inverse Planck constant.

Friday, May 15, 14:30-15:30 Eastern time, Zoom seminar
Video link
Malik Younsi (Hawaii)
Holomorphic motions, conformal welding and capacity
Abstract: The notion of a holomorphic motion was introduced by Mane, Sad and Sullivan in the 1980's, motivated by the observation that Julia sets of rational maps often move holomorphically with holomorphic variations of the parameters. Even though the original motivation for their study came from complex dynamics, holomorphic motions have found over the years to be of fundamental importance in other related areas of Complex Analysis, such as the theory of Kleinian groups and Teichmuller theory for instance. Holomorphic motions also played a central role in the seminal work of Astala on distortion of dimension and area under quasiconformal mappings. In this talk, I will first review the basic notions and results related to holomorphic motions, including quasiconformal mappings and the (extended) lambda lemma. I will then present some recent results on the behavior of logarithmic capacity and analytic capacity under holomorphic motions. As we will see, conformal welding (of quasicircle Julia sets) plays a fundamental role. This is joint work with Tom Ransford and Wen-Hui Ai.

Friday, May 22, 11:00-12:00 Eastern time, Zoom seminar
Video link
Jeff Galkowski (UCL)
Viscosity limits for 0th order operators
Abstract: In recent work, Colin de Verdiere--Saint-Raymond and Dyatlov--Zworski showed that a class of zeroth order pseudodifferential operators coming from experiments on forced waves in fluids satisfies a limiting absorption principle. Thus, these operators have absolutely continuous spectrum with possibly finitely many embedded eigenvalues. In this talk, we discuss the effect of small viscosity on the spectra of these operators, showing that the spectrum of the operator with small viscosity converges to the poles of a certain meromorphic continuation of the resolvent through the continuous spectrum. In order to do this, we introduce spaces based on an FBI transform which allows for the testing of microlocal analyticity properties. This talk is based on joint work with M. Zworski.

Thursday, May 28, 12:00-13:00 Eastern time, Zoom seminar
Video link
Blair Davey (CUNY)
A quantification of the Besicovitch projection theorem and its generalizations
Abstract: The Besicovitch projection theorem asserts that if a subset E of the plane has finite length in the sense of Hausdorff and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every linear projection of E to a line will have zero measure. As a consequence, the probability that a line dropped randomly onto the plane intersects such a set E is equal to zero. Thus, the Besicovitch projection theorem is connected to the classical Buffon needle problem. Motivated by the so-called Buffon circle problem, we explore what happens when lines are replaced by more general curves. We discuss generalized Besicovitch theorems and, as Tao did for the classical theorem (Proc. London Math. Soc., 2009), we use multi-scale analysis to quantify these results. This work is joint with Laura Cladek and Krystal Taylor.

Wednesday, June 3, 13:30-14:30, Zoom seminar
Joint seminar with geometric analysis
Video link
Sagun Chanillo (Rutgers)
Bourgain-Brezis inequalities, applications and Borderline Sobolev inequalities on Riemannian Symmetric spaces of non-compact type.
Abstract: Bourgain and Brezis discovered a remarkable inequality which is borderline for the Sobolev inequality in Eulcidean spaces. In this talk we obtain these inequalities on nilpotent Lie groups and on Riemannian symmetric spaces of non-compact type. We obtain applications to Navier Stokes eqn in 2D and to Strichartz inequalities for wave and Schrodinger equations and to the Maxwell equations for Electromagnetism. These results were obtained jointly with Jean Van Schaftingen and Po-lam Yung.

Thursday, June 11, 12:30-13:30 Eastern time, zoom seminar
Video link
Spyros Alexakis (Toronto)
Singularity formation in Black Hole interiors
Abstract: The prediction that solutions of the Einstein equations in the interior of black holes must always terminate at a singularity was originally conceived by Penrose in 1969, under the name of "strong cosmic censorship hypothesis". The nature of this break-down (i.e. the asymptotic properties of the space-time metric as one approaches the terminal singularity) is not predicted, and remains a hotly debated question to this day. One key question is the causal nature of the singularity (space-like, vs null for example). Another is the rate of blow-up of natural physical/geometric quantities at the singularity. Mutually contradicting predictions abound in this topic. Much work has been done under the assumption of spherical symmetry (for various matter models). We present a stability result for the Schwarzschild singularity under polarized axi-symmetric perturbations of the initial data, joint with G. Fournodavlos). One key innovation of our approach is a certain new way to treat the Einstein equations in axial symmetry, which should have broader applicability.

Friday, June 19, 12:00-13:00 Eastern time, zoom seminar
Video link
Alexander Strohmaier (Leeds)
Scattering theory for differential forms and its relation to cohomology
Abstract: I will consider spectral theory of the Laplace operator on a manifold that is Euclidean outside a compact set. An example of such a setting is obstacle scattering where several compact pieces are removed from $R^d$. The spectrum of the operator on functions is absolutely continuous. In the case of general $p$-forms eigenvalues at zero may exist, the eigenspace consisting of L^2-harmonic forms. The dimension of this space is computable by cohomological methods. I will present some new results concerning the detailed expansions of generalised eigenfunctions, the scattering matrix, and the resolvent near zero. These expansions contain the L^2-harmonic forms so there is no clear separation between the continuous and the discrete spectrum. This can be used to obtain more detailed information about the L^2-cohomology as well as the spectrum. If I have time I will explain an application of this to physics. (joint work with Alden Waters)

Special seminar on the occasion of the 65th birthday of N. Nadirashvili
Friday, June 26, starts at 10:00am Eastern time, zoom seminar
10:00-10:50am Eastern time. Alexandr Logunov (Princeton)
Nodal sets, Quasiconformal mappings and how to apply them to Landis' conjecture.
Abstract: A while ago Nadirashvili proposed a beautiful idea how to attack problems on zero sets of Laplace eigenfunctions using quasiconformal mappings, aiming to estimate the length of nodal sets (zero sets of eigenfunctions) on closed two-dimensional surfaces. The idea have not yet worked out as it was planned. However it appears to be useful for Landis' Conjecture. We will explain how to apply the combination of quasiconformal mappings and zero sets to quantitative properties of solutions to $\Delta u + V u =0 on the plane, where $V$ is a real, bounded function. The method reduces some questions about solutions to Shrodinger equation $\Delta u + V u =0$ on the plane to questions about harmonic functions. Based on a joint work with E.Malinnikova, N.Nadirashvili and F. Nazarov.
Video link
11:00-11:50am, Eastern time. Vladimir Sverak (Minneapolis)
Liouville theorems for the Navier-Stokes equations
Abstract: Assume u is a smooth, bounded, and divergence-free field on R^3 satisfying the steady Navier-Stokes equation -\Delta u +u\nabla u + \nabla p=0 (for a suitable function p). Does u have to be constant? We still don't know. Interesting things are known and Nikolai made important contributions to our knowledge concerning this question. Similar problems can also be considered for various model equations. The lecture will concern various aspects of this problem.
12:15-13:30, Eastern time: Zoom banquet

Wednesday, July 15, 11:00 Eastern time, zoom seminar
Michael Magee (Durham University)
The spectral gap of a random hyperbolic surface
Abstract: On a compact hyperbolic surface, the Laplacian has a spectral gap between 0 and the next smallest eigenvalue if and only if the surface is connected. The size of the spectral gap measures both how highly connected the surface is, and the rate of exponential mixing of the geodesic flow on the surface. There is an analogous concept of spectral gap for graphs, with analogous connections to connectivity and dynamics. Motivated by theorems about the spectral gap of random regular graphs, we proved that for any $\epsilon > 0$, a random cover of a fixed compact connected hyperbolic surface has no new eigenvalues below 3/16 - $\epsilon$, with probability tending to 1 as the covering degree tends to infinity. The number 3/16 is, mysteriously, the same spectral gap that Selberg obtained for congruence modular curves. The talk is intended to be accessible to graduate students and is based on joint works with Frédéric Naud and Doron Puder.

Friday, August 21 (changed from August 7!), 11:00 Eastern time, zoom seminar
Mike Wilson (University of Vermont)
Perturbed Haar function expansions
Abstract

Friday, August 28, 12:00 (noon) Eastern time, zoom seminar
Malabika Pramanik (UBC)
Restriction of eigenfunctions to sparse sets on manifolds
Abstract: Given a compact Riemannian manifold $(M, g)$ without boundary, we consider the restriction of Laplace-Beltrami eigenfunctions to certain subsets $\Gamma$ of the manifold. How do the Lebesgue $L^p$ norms of these restricted eigenfunctions grow? Burq, Gerard, Szvetkov and independently Hu studied this question when $\Gamma$ is a submanifold. In ongoing joint work with Suresh Eswarathasan, we extend earlier results to the setting where $\Gamma$ is an arbitrary Borel subset of $M$. Here differential geometric methods no longer apply. Using methods from geometric measure theory, we obtain sharp growth estimates for the restricted eigenfunctions that rely only on the size of $\Gamma$. Our results are sharp for large $p$, and are realized for large families of sets $\Gamma$ that are random and Cantor-like.