## 2020-21 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)

**
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 2021

** Friday, January 15, 14:00-15:00 Eastern time, zoom seminar **

** Anush Tserunyan** (McGill)

Ergodic theorems along trees

** Abstract:**
In the classical pointwise ergodic theorem for a probability measure
preserving (pmp) transformation $T$, one takes averages of a given
integrable function over the intervals $\{x, T(x), T^2(x), \hdots, T^n(x)\}$
in front of the point $x$. We prove a “backward” ergodic theorem for a
countable-to-one pmp $T$, where the averages are taken over subtrees of the
graph of T that are rooted at $x$ and lie behind $x$ (in the direction of
$T^{-1}$). Surprisingly, this theorem yields forward ergodic theorems for
countable groups, in particular, one for pmp actions of free groups of
finite rank, where the averages are taken along subtrees of the standard
Cayley graph rooted at the identity. This strengthens Bufetov’s theorem
from 2000, which was the most general result in this vein. This is joint work
with Jenna Zomback.

** Friday, February 12, 11:00 Eastern time, zoom seminar **

** Amir Vig** (CRM, McGill)

Spectral invariants and Birkhoff Billiards

** Abstract:**
Consider a smooth, bounded, strictly convex domain in the plane. There
are two games one can play. The classical one is billiards, in which a
billiard ball orbits around the domain and reflects elastically at the
boundary. The “quantum” analogue involves the study of wave propagation in
the domain and understanding the frequencies at which such waves oscillate.
In this talk, we discuss recent progress on the inverse spectral problem of
determining a billiard table from its Laplace spectrum. In particular, we
introduce a new class of spectral invariants for a generic class of billiard
tables obtained from an explicit Hadamard-Riesz type parametrix for the
wave propagator, microlocally near geodesic loops of small rotation number.
These same techniques also allow us to prove (infinitesimal) Robin spectral
rigidity of the ellipse, when both the boundary and boundary conditions
are allowed to deform simultaneously. Finally, we mention ongoing work
together with Vadim Kaloshin to cancel singularities in the wave trace for
special types of domains.

** Friday/Saturday, February 26-27**

**Zoom conference on the occasion of Alexander Shnirelman's
birthday**

Conference web page

** Friday, March 5, 11:00 Eastern time, zoom seminar **

** Benjamin Jaye** (Georgia Tech)

Multi-scale analysis of Jordan curves

** Abstract:**
In this talk we will describe how one can detect regularity in Jordan
curves through analysis of associated geometric square functions. We will
particularly focus on the resolution of a conjecture of L. Carleson.
Joint work with Xavier Tolsa and Michele Villa
(arXiv:1909.08581).

** Friday, March 19, 14:00 Eastern time, zoom seminar **

** Thomas Ransford** (Laval)

Failure of approximation of odd functions by odd polynomials

** Abstract:**
We construct a Hilbert holomorphic function space $H$ on the unit disk
such that the polynomials are dense in $H$, but the odd polynomials are
not dense in the odd functions in $H$. As a consequence, there exists a
function $f$ in $H$ that lies outside the closed linear span of its Taylor
partial sums $s_n(f)$, so it cannot be approximated by any triangular
summability method applied to the $s_n(f)$. We also show that there exists
a function $f$ in $H$ that lies outside the closed linear span of its
radial dilates $f_r, r < 1$. (Joint work with Javad Mashreghi and
Pierre-Olivier Parise).

** Friday, March 26, 14:00 Eastern time, zoom seminar **

** Dmitry Faifman** (Tel Aviv)

A Funk perspective on convex geometry

** Abstract:**
The Funk metric in the interior of a convex set is a lesser-known
cousin of the Hilbert metric. The latter generalizes the Beltrami-Klein
model of hyperbolic geometry, and both have straight segments as geodesics,
thus constituting solutions of Hilbert's 4th problem alongside normed
spaces. Unlike the Hilbert metric, the Funk metric is not projectively
invariant. I will explain how, nevertheless, the Funk metric gives rise
to many projective invariants, which moreover enjoy a duality extending
results of Holmes-Thompson and Alvarez Paiva on spheres of normed spaces
and Gutkin-Tabachnikov on Minkowski billiards. I will also discuss how
the maximal volume problem in Funk geometry yields an extension of the
Blaschke-Santalo inequality.

** Friday, April 2, 11:30 Eastern time, zoom seminar **

** Hakim Boumaza** (Paris)

Integrated density of states of the periodic Airy-Schrödinger operator

** Abstract:**
In this talk I present, in the semiclassical regime, an explicit formula
for the integrated density of states of the periodic Airy-Schrodinger
operator on the real line. The potential of this Schrödinger operator is
periodic, continuous and piecewise linear. For this purpose, the spectrum
of the Schrödinger operator whose potential is the restriction of the
periodic Airy-Schrödinger potential to a finite number of periods is
studied. We prove that all the eigenvalues of the operator corresponding
to the restricted potential are in the spectral bands of the periodic
Airy-Schrodinger operator and none of them are in its spectral gaps.
In the semiclassical regime, we count the number of these eigenvalues in
each of the spectral bands. Note that in these results there are explicit
constants which characterize the semiclassical regime.
This is joint work with Olivier Lafitte (USPN - CRM Montreal).

** Friday, April 9, 14:00 Eastern time, zoom
seminar **

** Konstantin Khanin** (Toronto)

On Stationary Solutions to the Stochastic Heat Equation

** Abstract:**
I plan to discuss the problem of uniqueness of global solutions
to the random Hamilton-Jacobi equation. I will formulate several conjectures
and present results supporting them. Then I will discuss a new
uniqueness result
for the Stochastic Heat equation in the regime of weak disorder.

** Friday, April 16, 10:00 Eastern time,
zoom seminar **

** Nicolai Krylov** (Minnesota)

A review of some new results in the theory of linear
elliptic equations with drift in $L_{d}$

** Abstract:**
We present an overview of
recent results related to
the Aleksandrov type estimates with power of summability
of the free term $d_0 < d$, the Harnack inequality
for $u\in W^{2}_{d_{0},loc}$, Holder continuity
of $L$-harmonic and $L$-caloric functions.
Under the assumption that the main coefficients
are almost in VMO (and $b\in L_{d}$) we also present the results about
solvability of the elliptic
equations in $W^{2}_{d_{0}}$ in domains and in the whole space.
A few relates issues are discussed as well.

** Friday, April 23, 14:00 Eastern time,
zoom seminar **

** Vitali Vougalter** (Toronto)

Solvability of some integro-differential equations
with anomalous diffusion and transport

** Abstract:** The work deals with the existence of solutions of an
integro-differential equation in the case of the anomalous diffusion with
the negative Laplace operator in a fractional power in the presence of the
transport term. The proof of existence of solutions is based on a fixed point
technique. Solvability conditions for elliptic operators without Fredholm
property in unbounded domains are used. We discuss how the introduction of
the transport term impacts the regularity of solutions.

** Friday, April 30, 14:00 Eastern time, zoom seminar **

** Krzysztof Bogdan** (Wroclaw)

Optimal Hardy identities and inequalites for the fractional Laplacian on
$L^p$

** Abstract:**
We will present a route from symmetric Markovian semigroups to
Hardy inequalities, to nonexplosion and contractivity results for
Feynman-Kac semigroups on $L^p$. We will focus on the fractional Laplacian
on $\mathbb{R}^d$, in which case the constants, estimates of the
Feynman-Kac semigroups and tresholds for contractivity and explosion are
sharp.
Namely we will discuss selected results from arXiv:2103.06550, joint
with Bartl omiej Dyda, Tomasz
Grzywny, Tomasz Jakubowski, Panki Kim, Julia Lenczewska, Katarzyna
Pietruska-Pa\l uba or Dominika Pilarczyk.

** Wednesday, May 5, 13:30 Eastern time, zoom seminar
(Joint with the Geometric Analysis Seminar)**

**Frederic Naud** (Paris)

The spectral gap of random hyperbolic surfaces

** Abstract:**
We will start with a survey on (some very recent) results on the low
spectrum of random compact hyperbolic surfaces, for various models
including discrete and continuous Teichmuller spaces. We will then give
some ideas of the proofs by emphasizing the analogy with radom graphs.
We will also dicuss non compact situations where similar results on
resonances can be obtained. Joint works with Michael Magee and Doron Puder.

** Friday, May 7, 14:00 Eastern time, zoom seminar **

** Nages Shanmugalingam** (U. Cincinnati)

Prime ends for domains in metric measure spaces and their use in
potential theory and QC theory.

** Abstract:**
Prime ends were first developed by Caratheodory in order to understand
the boundary behavior of conformal mappings from the disk. As such, the
construction
of Caratheodory and Ahlfors worked for simply connected planar domains,
but had
to be modified for more general domains. In this talk we will focus
on a construction
in the setting of domains in metric spaces, and describe their use in
potential theory
(Dirichlet problem for the p-energy minimizers) and in studying boundary
behavior
of QC maps.

** Friday, May 14, 11:00 Eastern time, zoom seminar **

** Michael Lipnowski** (McGill)

Towards optimal spectral gaps in large genus.

** Abstract:**
I'll discuss recent joint work with Alex Wright (arXiv: 2103.07496)
showing that typical large genus hyperbolic surfaces have first
Laplacian eigenvalue at least 3/16−epsilon.

** Wednesday, May 19, 13:30 Eastern time, zoom seminar
(Joint with the Geometric Analysis Seminar)**

**Laurent Moonens** (Paris-Saclay)

Solving the divergence equation with measure data in non-regular domains

** Abstract:**
In this talk, we shall present a recent joint work with E. Russ
(Grenoble) concerning the equation $\mathrm{div}\,v=\mu$ in a (rather
general) open domain $\Omega$, where $\mu$ is a (signed) Radon measure
in $\Omega$ satisfying $\mu(\Omega)=0$. We show in particular that, under
mild assumptions on the geometry of $\Omega$ (and some assumptions
on $\mu$), one can provide a constructive way to build solutions $v$
in a weighted $L^\infty$ space enjoying weak Neumann-type boundary
conditions.

** Friday, May 28, 14:00 Eastern time, zoom seminar **

** Li Chen** (Louisiana State)

BV functions and some functional inequalities on nested fractals

** Abstract:** In this talk, we discuss functions of bounded
variations (BV) on fractals which satisfy sub-Gaussian heat kernel bounds.
We also prove isoperimetric inequalities and Poincare inequalities on
some nested fractals. Our proofs use heat kernel methods and the key
ingredient is a weak Bakry-Emery curvature type condition.
The talk are based on joint works with Patricia Alonso-Ruiz, Fabrice
Baudoin, Luke Rogers, Nageswari Shanmugalingam and Alexander Teplyaev.

## 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 9, 13:00 Eastern time, zoom seminar **

** Arick Shao** (Queen Mary university)

On Controllability of Waves and Geometric Carleman Estimates

** Abstract:**
We consider the question of exact (boundary) controllability of wave
equations: whether one can steer their solutions from any initial state
to any final state using appropriate boundary data. In particular, we
discuss new and fully general results for linear wave equations on
time-dependent domains with moving boundaries. We also discuss the novel
geometric Carleman estimates that are the main tools for proving these
controllability results.

**Friday, October 16, 14:30 Eastern Time,
zoom seminar**

** Guangyu Xi** (University of Maryland)

On the Smoluchowski-Kramers approximation of infinite dimensional systems
with state-dependent damping

** Abstract:**
We study a class of stochastic nonlinear damped wave equations endowed with
Dirichlet boundary conditions. We show that the Smoluchowski-Kramers
approximation describes the convergence of solutions of the stochastic
wave equations to the solution of a quasilinear stochastic parabolic
equation endowed with Dirichlet boundary conditions. This is a joint
work with Sandra Cerrai from UMD.

**Friday, October 23, 14:30 Eastern time,
zoom seminar**

** Mohammad Shirazi** (McGill)

Schiffer operator corresponding to Riemann surfaces with finitely many
borders

** Abstract:**
In this talk, a brief history of the Schiffer operator on the complex
plane, on the Riemann sphere, and (some type of) Riemann surfaces will
be reviewed. Some of my thesis' results concerning the development of
this operator on Riemann surfaces (open, with finitely many borders, each
homeomorphic to S^1) will be presented. We will see when the Schiffer
operator is a bounded isomorphism.

**Friday, November 13, 14:30 Eastern time, zoom seminar **

**Hien Nguyen** (Iowa State)

Strangling of the fundamental gap in hyperbolic space

** Abstract:**
For the Laplace operator with Dirichlet boundary conditions on convex
domains in $H^n, n ≥ 2$, we prove that the product of the fundamental
gap with the square of the diameter can be arbitrarily small for domains
of any diameter. This property distinguishes hyperbolic spaces from
Euclidean and spherical ones, where the quantity is bounded below by
$3 \pi^2$.

**Friday, November 20, 12noon Eastern time, zoom seminar **

**Daniel Peralta-Salas** (ICMAT)

Optimal domains for the first curl eigenvalue

** Abstract:**
The classical Faber-Krahn inequality for the first eigenvalue of the
Dirichlet Laplacian shows that the ball is the unique optimal domain.
In this talk I will explore the analogous problem for the curl operator:
for a fixed volume, what is the optimal domain for the first positive
(or negative) eigenvalue of curl? In spite of being one of the most
important vector-valued operators, this question is
rather unexplored and remains wide open. In this talk I will show that, even
taking into account that the first eigenvalue is uniformly lower bounded
in terms
of the volume, there are no axisymmetric smooth optimal domains for the
curl that satisfy a mild technical assumption. In particular, this rules
out the existence of optimal axisymmetric domains with a convex section.
This is based on joint work with Alberto Enciso.

**Friday, November 27, 14:30 Eastern time, zoom seminar **

**Alex Brudnyi** (Calgary)

On the stable rank of algebras of bounded holomorphic functions

** Abstract:**
The concept of the stable rank introduced by Bass plays an important
role in some stabilization problems of algebraic K-theory analogous to
that of dimension in topology. Despite a simple definition, the stable
rank is often quite difficult to calculate even for relatively
uncomplicated rings. We present examples of algebras for which the stable
rank is computed. Next, we consider a similar problem for algebras of
bounded holomorphic functions on Riemann surfaces. The central result in
this area is a theorem of Treil asserting that the Bass stable rank of
the algebra of bounded holomorphic functions on the open unit disk is 1.
We discuss some generalizations of Treil's result.

**Friday, December 11, 9:30-10:30 Eastern time, zoom seminar **

**Ryan Gibara** (Laval)

Boundedness and continuity for rearrangements on spaces defined by
mean oscillation

** Abstract:**
In joint work with Almut Burchard and Galia Dafni, we study the boundedness
and continuity of rearrangement operators on the space BMO of functions of
bounded mean oscillation. Improved bounds are obtained for the BMO-seminorm
of the decreasing rearrangement, and the symmetric decreasing rearrangement
is shown to be bounded on BMO. Both of these rearrangements are shown to
be discontinuous as maps on BMO, but sufficient normalisation conditions
are established to guarantee continuity on the subspace VMO of functions of
vanishing mean oscillation.

**Friday, December 11, 11:00-12:00 Eastern time, zoom seminar **

**Renaud Raquepas** (McGill)

Entropy production in nondegenerate diffusions: large times and small
noises

** Abstract:**
Entropy production (EP) is a key quantity from thermodynamics which
quantifies the irreversibility of the time evolution of physical systems.
I will start the presentation with a general introduction to the different
approaches to defining EP. Then, I will focus on the context of
nondegenerate diffusions and I will describe the large-deviation
properties of EP as time goes to infinity. I will also explain how the
behaviour of the corresponding rate function boils down to the study of
the smallest eigenvalue for a family of differential operators with a
small parameter.

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