## 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
**

**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**

**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**

**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.

** 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**

**Friday, July 15, Time TBA, zoom seminar**

**Michael Magee** (Durham University)

Title TBA

**Friday, August 7, Time TBA, zoom seminar**

**Mike Wilson** (University of Vermont)

Perturbed Haar function expansions

**Friday, August 28, Time TBA, zoom seminar**

**Malabika Pramanik** (UBC)

Title TBA

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