## 2019-20 Montreal Analysis Seminar

Seminars are usually held on Mondays or Fridays at Concordia,
McGill or Universite de Montreal

For suggestions, questions etc. please contact Dmitry Jakobson
(dmitry.jakobson@mcgill.ca), Iosif Polterovich
(iossif@dms.umontreal.ca) or
Galia Dafni (galia.dafni@concordia.ca)

## WINTER 2020

**Joint seminar with geometric analysis**

**Friday, January 17, 13:30-14:30, McGill, Burnside Hall, Room 1104**

**Henrik Matthiesen** (University of Chicago)

Handle attachment and the normalised first eigenvalue

** Abstract:**
I will discuss asymptotic lower bounds of the first eigenvalue for two
constructions of attaching degenerating handles to a given closed
Riemannian surface. One of these constructions is relatively simple but
often fails to strictly increase the first eigenvalue normalized by area.
Motivated by this negative result, we then give a much more involved
construction that always strictly increases the first eigenvalue normalized
by area.
As a consequence we obtain the existence of a metric that maximizes the
first eigenvalue among all unit area metrics on a given closed surface.
This is based on joint work with Anna Siffert.

**Friday, January 31, 13:30-14:30, Concordia, Library building,
Room LB 921-4**

**Alexey Kokotov** (Concordia)

Flat conical Laplacian in the square of the canonical bundle and its
regularized determinants

** Abstract:**
We discuss two natural definitions of the determinant of the Dolbeault
Laplacian acting in the square of the canonical bundle over a compact
Riemann surface equipped with flat conical metric given by the modulus of
a holomorphic quadratic differential with simple zeroes. The first one
uses the zeta-function of some special self-adjoint extension of the
Laplacian (initially defined on smooth sections vanishing near the zeroes
of the quadratic differential), the second one is an analog of
Eskin-Kontsevich-Zorich (EKZ) regularization of the determinant of the
conical Laplacian acting in the trivial bundle. In contrast to the
situation of operators acting in the trivial bundle, these two
regularizations turn out to be essentially different. Considering the
regularized determinant of the Laplacian as a functional on the moduli
space of quadratic differentials with simple zeroes on compact Riemann
surfaces of a given genus, we derive explicit expressions for this
functional for the both regularizations. The expression for the EKZ
regularization is closely related to the well-known explicit expressions
for the Mumford measure on the moduli space of compact Riemann surfaces.

**Friday, February 14, 13:30-14:30, McGill, Burnside Hall, Room 1104**

**Bradley Siwick** (Chemistry and Physics, McGill)

STRUCTURE AND DYNAMICS WITH ULTRAFAST ELECTRON MICROSCOPES
… or how to make atomic-level movies of fundamental processes in
molecules and materials

** Abstract:**
In this talk I will describe how combining ultrafast lasers and electron
microscopes in novel ways makes it possible to directly ‘watch’ the
time-evolving structure of condensed matter and the couplings between
carrier and lattice degrees of freedom on the fastest timescales open to
atomic motion [1-4]. By combining such measurements with complementary
(and more conventional) spectroscopic probes we can now develop
structure-property relationships for materials under even very far from
equilibrium conditions [2].
I will assume no familiarity with ultrafast lasers or electron microscopes
and highlight opportunities for theoretical/computational developments
that support these experiments. Rigorous modeling of the processes
described is currently intractable.

[1] Morrison et al Science 346 (2014) 445

[2] Otto et al, PNAS, 116 (2019) 450

[3] Stern et al, Phys. Rev. B 97 (2018) 165416

[4] Rene de Cotret et al, Phys. Rev. B 100 (2019) 214115

** Joint with Symplectic Geometry Seminar**

**Friday, February 28, 13:40-14:40, UQAM, Pavillon President
Kennedy, Salle PK-5115**

**Egor Shelukhin** (Universite de Montreal)

Smith theory in Floer homology, persistence, and dynamics

** Abstract:**
Recent years have seen a renewed interest in a classical topological
inequality that originated in the work of P. A. Smith, extended to the
framework of Floer homology. We describe such inequalities in the setting
of persistence modules obtained from Floer homology, and their recent
applications to questions in Hamiltonian dynamics. In particular, we show
that for a class of symplectic manifolds including complex projective
spaces, a Hamiltonian diffeomorphism with more fixed points, counted
suitably, than the dimension of the ambient homology, must have an infinite
number of simple periodic points. This is a higher-dimensional homological
generalization of a celebrated result of Franks from 1992, as conjectured
by Hofer and Zehnder in 1994. Time permitting, we may discuss further,
very recent, related results.

** CANCELLED**

** Joint with Geometris Analysis Seminar**

**Friday, March 20, 13:30-14:30, McGill, Burnside Hall, Room 1104**

**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.

** CANCELLED**

**Friday, March 27, 14:30-15:30, McGill, Burnside Hall, Room 1104**

**Alexandre Girouard** (Universite Laval)

Title TBA

** CANCELLED**

**Joint seminar with geometric analysis**

**Friday, April 17, Time and room TBA**

**Sagun Chanillo** (Rutgers)

Title TBA

** CANCELLED**

**Friday, April 24 (to be confirmed), Time and room TBA**

**Alexei Penskoi** (HSE, Moscow)

Title TBA

## FALL 2019

** Friday, September 6, 13:30-14:30, McGill, Burnside Hall, Room 1104**

**Reem Yassawi** (Open University)

Measure non-rigidity for linear cellular automata

**Abstract:**
pdf

** Friday, September 20, 13:30-14:30, McGill, Burnside Hall, Room 1104
**

**Damir Kinzebulatov** (Laval)

Heat kernel bounds and desingularizing weights for non-local
operators

**Abstract:**
In 1998, Milman and Semenov introduced the method of desingularizing
weights in order to obtain sharp two-sided bounds on the heat kernel of
the Schroedinger operator with a potential having critical-order
singularity at the origin. In this talk, I will discuss the method of
desingularizing weights in a non-symmetric, non-local situation. In
particular, I will talk about sharp two-sided bounds on the heat kernel
of the fractional Laplacian perturbed by a Hardy drift.
The crucial ingredient of the desingularization method is a weighted
L^1->L^1 estimate on the semigroup, leading to the weighted Nash initial
estimate. Milman and Semenov established this estimate appealing to the
Stampacchia criterion in L^2. These arguments becomes quite problematic
in the non-local non-symmetric situation (e.g. for a strong enough
singularity of the drift, there is only L^p theory of the operator
for p>2). The core of the talk will be the discussion of a new approach
to the proof of this estimate.
Joint with Yu.A.Semenov and K.Szczypkowsi (arxiv:1904.07363)

** Monday, November 4, 13:30-14:30, McGill, Burnside Hall, Room 1104**

**Stephane Sabourau** (U. Paris-Est)

Systolically extremal metrics on nonpositively curved surfaces

** Abstract:**
The regularity of systolically extremal surfaces (i.e., surfaces of
minimal area with fixed systole) is a delicate problem already discussed
by M. Gromov in the 80's. We propose to study the problem of
systolically extremal metrics in the context of generalized metrics of
nonpositive curvature. A natural approach would be to work in the class
of Alexandrov surfaces of finite total curvature, where one can exploit
the tools of the completion provided in the context of Radon measures as
studied by Reshetnyak and others. However the generalized metrics in
this sense still don't have enough regularity. Instead, we develop a
more hands-on approach and show that, for each genus, every systolically
extremal nonpositively curved surface is piecewise flat with finitely
many conical singularities. Joint work with M. Katz.

** Friday, November 15, 14:30-15:30, Universite de Montreal,
Pavillon Andre-Aisenstadt, Room 5183. **

**Almaz Butaev** (U. Calgary)

Extension problem on subspaces of BMO on domains

** Abstract:**
In joint work with Galia Dafni, we discuss the extension problem for some
subspaces of functions of bounded mean oscillation (BMO). Based on the
extension operator of Jones we construct a universal extension in the sense
that it simultaneously extends certain natural subspaces of BMO. The
presented results will show an interplay between approximation, extension
and geometric properties of the domain.

** Spectral Geometry Seminar**

** Tuesday, November 26, 14:00-15:00, Universite de Montreal,
Pavillon Andre-Aisenstadt, Room 5448. **

**Olivier Lafitte** (CRM)

Precise descriptions of bands of the Airy-Schrodinger operator on the
real line

** Abstract:**
Joint work with Hakim Boumaza, LAGA, Université Paris 13
In this talk, we present recent results on band spectrum generated by a
Schrodinger operator with a non C^1 potential for which one has
eigenfunctions described by special functions. This generalizes a result
Harrell (1979) and in particular we are able to have a precise estimate on
the validity regime of the semi-classical behavior as well as the exact
width of each band.
The ongoing work on a multiple-wells potential will be as well presented.

** Friday, November 29, 14:00-15:00, Concordia, Library Building,
Room LB921-4. **

**Ritva Hurri-Syrjanen** (U. of Helsinki)

On the John-Nirenberg inequalities

** Abstract:**
The goal of my talk is to address some inequalities which Fritz John
and Louis
Nirenberg proved to be valid for certain functions defined in a cube.
I will discuss the
validity of similar inequalities for functions dened in an arbitrary
bounded domain.
My talk is based on joint work with Niko Marola and Antti
Vahakangas.

** Friday, December 13, 13:30-14:30, McGill, Burnside Hall, Room 1104**

**Jean-Philippe Burelle** (U. Sherbrooke)

Higher Teichmuller and higher rank Schottky groups

**Abstract:**
Schottky groups are the simplest and most classical examples of
Kleinian groups, that is,
of discrete subgroups of Mobius transformations. I will explain
several generalisations of this notion
to subgroups of higher rank Lie groups. One of these generalisations
leads to an explicit description
of positive representations of surfaces with non-empty boundary, a type
of higher Teichmuller representation
introduced by Fock and Goncharov in 2003. I will show how this
description allows the construction
of fundamental domains for an open domain of discontinuity in the
projective space or the sphere, depending
on the dimension. This talk will feature joint work with N. Treib,
F. Kassel and V. Charette.

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