## 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
(jakobson@math.mcgill.ca), Iosif Polterovich
(iossif@dms.umontreal.ca) or
Alina Stancu (alina.stancu@concordia.ca)

## WINTER 2014

** Friday, January 10, 13:30-14:30, Burnside 920**

** Igor Khavkine** (University of Trento)

Presymplectic current and the inverse problem of the calculus
of variations

** Abstract:**
The inverse problem of the calculus of variations asks
whether a given system of partial differential equations (PDEs) admits
a variational formulation. I present a reformulation of the problem: a
variational formulation can be recovered (at least for a subsystem) if
and only if the PDE system admits a compatible presymplectic current.
The presymplectic current represents a certain cohomology class in the
on-shell variational bicomplex associated to the PDE. This
construction (following a recent remark due to Bridges, Hydon and
Lawson) is a generalization to PDEs of an old observation of Henneaux
in the context of ordinary differential equations (ODEs). No
constraints on the differential order or number of dependent or
independent variables are assumed. [arXiv:1210.0802]

** Friday, January 24, 13:30-14:30, Burnside 920
**

** Daniel Elton** (University of Lancaster)

Some Analytic Aspects of Dirac Operators in R^n

** Abstract:**
Roughly speaking a Dirac operator is a first order differential
operator whose square is a second order operator such as the Laplacian.
Such operators were first considered in the 1920s in models of
relativistic quantum particles with spin (such as electrons)
but have since turned up in other areas of physics (eg., models
of graphene) and mathematics (especially differential geometry).
In this talk we will review some mathematical aspects of Dirac
operators, principally concentrating on such operators in R^n which
arise in applications in physics.

** Friday, January 31, 13:30-14:30, Burnside 920**

** Line Baribeau** (Laval)

The Spectral Nevanlinna-Pick Problem

** Abstract:**
Let U denote the unit disc, and $\Omega$ the set of complex
$n\times n$ matrices with spectral radius less than 1.
Given $z_1,\dots,z_m\in U$ and $W_1,\dots, W_m\in \Omega$, does
there exist a holomorphic map $F\colon U\to \Omega$ such that
$F(z_i)=W_i$ for $i=1,2,\dots,m$? The problem of finding necessary
and sufficient conditions on the interpolating data for the existence
of $F$ is called the Spectral Nevanlinna-Pick Problem. I will survey
some of the approaches used to attack this still unsolved problem, which
involve several complex variables and set-valued analytic functions.

** Friday, February 7, 13:30-14:30, Burnside 920 **

** Gerasim Kokarev** (Munich, LMU)

Multiplicity bounds for Schrodinger eigenvalues on Riemannian surfaces

** Abstract:**
A classical result by Cheng in 1976, improved later by Besson and
Nadirashvili, says that the eigenvalue multiplicities of the
Schrodinger operator with a smooth potential on a Riemannian surface
are bounded in terms of the eigenvalue index and the genus of a
surface. I will talk about a recent solution to the question asking
whether similar bounds hold for singular potentials. The key ingredient
in the proof is based on the study of the Caratheodory prime ends of
nodal domains.

** Friday, February 21, 13:30-14:30, Burnside 920 **

** Dmitry Jakobson** (McGill)

Conformal invariants on weighted graphs

** Abstract:**
This is joint work with Thomas Ng, Matthew Stevenson and
Mashbat Suzuki. We define the moduli space of conformal structures on
a finite connected simple graph G. We propose a denition of conformally
covariant operators on graphs, motivated by conformally covariant operators
on Riemannian manifolds. We provide examples of such operators, which
include the edge Laplacian and the adjacency
matrix on graphs. In the case where such an operator has a nontrivial
kernel, we construct conformal invariants, motivated by recent results
of Canzani, Gover, Jakobson and Ponge.

** Friday, March 7, 13:30-14:30, Concordia,
Library building, Room 921-04**

** Vassili Nestoridis** (University of Athens)

Universality and regularity of the integration operator

** Abstract:** pdf

** Friday, March 14, 13:30-14:30, Burnside 920 **

** Peter Herbrich** (Dartmouth)

Magnetic Schrodinger Operators and Mane's Critical Value

** Abstract:**
The talk will deal with lifted magnetic fields on covers of
closed manifolds. In particular, the spectra of corresponding
periodic
magnetic Schrodinger operators can be related to Mane's critical
energy values of the corresponding classical Hamiltonian systems.
Namely, if the covering transformation group is amenable, then the
bottom of the spectrum is bounded from above by Mane's critical
value.
In the special case of abelian covers, the spectral analysis reduces
to the study of shifted magnetic potentials on the compact quotient
which parallels the behaviour of Mane's critical value of the
corresponding classical systems. The talk will finish with
examples of
magnetic fields on homogeneous spaces, which facilitate comparisons
between the classical and the quantum data.

** Friday, April 4, 13:30-14:30, Burnside 920 **

** Leonid Parnovski** (University College, London)

Asymptotic behaviour of the spectral function of
multidimentional almost-periodic Schroedinger operators.

** Abstract:**
We study the spectral function (the kernel of the spectral projection)
of Schroedinger operators with almost-periodic potentials and obtain the
complete asymptotic expansion, both on and off the diagonal. This
expansion is obtained using the method of gauge transform. This is a
joint result with Roman Shterenberg.

** Monday, April 14, 13:30-14:30, Burnside 920 **

** Renato Calleja** (IIMAS-UNAM)

Construction of quasi-periodic response solutions for forced systems
with strong damping

** Abstract:**
I will present a method for constructing quasi-periodic response
solutions (i.e. quasi-periodic solutions with the same frequency as
the forcing) for over-damped systems. Our method applies to non-linear
wave equations subject to very strong damping and quasi-periodic
external forcing and to the varactor equation in electronic engineering.
The strong damping leads to very few small divisors which allows to
prove the existence by using a contraction mapping argument requiring
very weak non-resonance conditions on the frequency. This is joint work
with A. Celletti, L. Corsi, and R. de la Llave.

** Friday, May 9, 13:30-14:30, Burnside 920 **

** Hans Christianson** (UNC, Chapel Hill)

From resolvent estimates to damped waves

**Abstract:** The damped wave equation is a prototype nonself-adjoint
PDE, having both advective (wave) component and a diffusive (damping)
component. The associated stationary problem then has both real and imaginary
parts, leading to complex spectrum. The size of the imaginary parts of the
eigenvalues gives information about the rate of energy decay. The interplay
between the advection and the location of the diffusion is expressed in terms of
geometric control theory. In this talk, I will survey a number of recent and
historical results, both with geometric control and lack of geometric control,
and explain how resolvent estimates on scattering manifolds can be "glued" into
damped wave problems. At the end, if there is time, I will briefly explain some
potential new directions in "advection assisted diffusion".

** Monday, May 12, 13:30-14:30, Burnside 920 **

** Jan Cannizzo** (Ottawa)

Invariant measures on graphs and subgroups

** Abstract:**
We will give an introduction to the theory of so-called
invariant random graphs and invariant random subgroups. We will
discuss several recent results, among them the result (joint with
Vadim Kaimanovich) that the boundary action of an invariant random
subgroup is conservative, and also discuss a number of open problems.

** Friday, May 23, 13:30-14:30, Burnside 920 **

** Jared Wunsch** (Northwestern)

The wave trace for conic manifolds

**Abstract:** I will discuss joint work with Austin Ford on the trace of the
half-wave operator for manifolds with conic singularities. We compute the
leading order singularities along closed geodesics that are allowed to
interact with the cone points.

** Friday, June 13, 10:00-11:00, Burnside 1234 **

** Robert Kusner** (Univ of Massachusetts, Amherst)

Linear area growth for CMC surfaces in S^3

**Abstract:**
In 1970, J. Hersch showed that the first eigenvalue lambda_1 of the
Laplacian is maximized for the round metric on S^2 among all metrics with
fixed area.
Different generalizations were obtained by Li, Yang and Yau a decade later.
Using an identity of Reilly, Choi and Wang obtained
a lower bound on lambda_1 for any embedded minimal surface in a 3-manifold
in terms of a (positive) lower bound on the Ricci curvature.
By the Riemann-Roch theorem, this gives an upper area bound for minimal
surfaces in S^3 which is linear in the genus. We sketch similar area
bounds for
CMC surfaces in S^3 and speculate on whether they correspond to similar
lambda_1 bounds.

## ANALYSIS-RELATED TALKS ELSEWHERE,
WINTER 2014

** McGill Mathematical Physics seminar **

** Wednesday, January 8, Burnside 1120, 16:00-19:00**

** Tristan Benoist ** (ENS)

Repeated non demolition measurements and wave function
collapse

** Abstract:** One difficulty in quantum optic experiments
is to measure a system without destroying it. For example to
count a number of photons usually one would need to convert
each photons into an electric signal. To avoid such
destruction one can use non demolition measurements. Instead
of measuring directly the system, quantum probes interact
with it and are then measured. The interaction is tuned
such that a set of system states are stable under the
measurement process.
This situation is typically the one of Serge Haroche's group
experiment which inspired the work I will present. In their
experiment they used atoms as probes to measure the number
of photons inside a cavity without destroying them. With Denis
Bernard and Michel Bauer we studied the convergence of such
repeated non demolition measurements. We proved that the
wave function collapse is obtained as a martingale almost
sure convergence. This convergence is exponentially fast.
We found an explicit rate depending on the limit system
state. We studied the dependency of this rate with respect
to the choice of probes. We also proved that the limit does
not depend on the initial state but only on the measurement
record. These results are useful in the evaluation of the
performance of a given measurement method.
In this talk I will start with a reminder on the description
of measurements in quantum physics. I will then explain
Serge Haroche's group experiment principles. Based on this
example I will present the repeated non demolition measurement
model and show the relation between wave function collapse
and martingale convergence. Finally I will explain how we
proved the exponential speed of convergence, its rate and
the stability of a system state estimation.

** McGill Mathematical Physics seminar **

** Friday, January 10, Burnside 1120, 16:00-19:00**

** Tristan Benoist ** (ENS)

Markovian continuous indirect measurement models for wave
function collapse

** Abstract:**
Quantum trajectories are non linear stochastic differential
equations widely used in the description of continuous
measurements of quantum systems. With Clement Pellegrini we
were interested in finding when these trajectories would
reproduce the wave function collapse corresponding to von
Neumann postulate of quantum physics. Following the work I
had done with Denis Bernard and Michel Bauer, we have found
a non demolition condition for these quantum trajectory equations.
We have proved an equivalence between a stability property for
a set of system states and a martingale property of the system
density matrix diagonal elements. Adding some non degeneracy
conditions, using martingale convergence theorem, we have
found that in the long time limit the state collapse and the
distribution of the limit state is the one of von Neumann
projection postulate.
Using martingale change of measure we have also proved that
the convergence is exponential and have found an explicit
rate depending on the limit state. Moreover we have been able
to show that if one start a computation with an estimate state,
then the limit state is the same as the true one.
In this talk I will start with a presentation of quantum
trajectories both as a model given by constrains imposed by
quantum physics and as an approximation of true physical
situations of continuous measurement. I will then present
the results we obtain with Clement Pellegrini on wave function
collapse, exponential convergence rate and estimation stability.

** Universite de Montreal Analysis seminar **

** Wednesday, January 15, 11:30, Universite de Montreal, Pav.
Andre-Aisenstadt, 2920, chemin de la Tour, SALLE 5340**

** Kirill Datchev ** (MIT)

Taux de decroissance quantique pour des
varietes a bouts hyperboliques

** Abstract:**
Mathematiquement, les taux de decroissance quantique apparaissent
comme des parties imaginaires des poles du prolongement meromorphe
des fonctions de Green. Lorsque l'energie croit, les taux de decroissance
sont lies aux proprietes du flot geodesique et a la structure a l'infini.
Une pointe possede un infini "petit", ce qui ralentit typiquement
la decroissance. Neanmoins, je presenterai une famille d'exemples
pour lesquels les taux de decroissance tendent vers l'infini meme en
presence d'une pointe. Ceci fait partie d'une investigation plus
generale des resonances sur des varietes a bouts hyperboliques.

** McGill Mathematical Physics seminar **

** Wednesday, January 15, Burnside 1120, 16:00-19:00**

** Claude-Allain Pillet** (Toulon)

Non-equilibrium quantum statistical mechanics

** CRM-ISM Mathematics colloquium **

** Friday, January 17, 16:00, UQAM, Pav. Sherbrooke,
200, rue Sherbrooke O., salle SH-3420
**

** Boris Khesin ** (Toronto)

Nondegenerate curves and pentagram maps

** Abstract:**
How many classes of closed nondegenerate curves exist on a sphere?
We are going to see how this geometric problem, solved in 1970,
reappeared along with its generalizations in the context of the
Korteweg-de Vries and Boussinesq equations. Its discrete version
is related to the 2D pentagram map defined by R.Schwartz in 1992.
We will also describe its generalizations, pentagram maps on polygons
in any dimension and discuss their integrability properties.
This is a joint work with Fedor Soloviev.

** Universite de Montreal Analysis Seminar **

** Wednesday, January 22, 11:30-12:30, CRM, Room 6214**

** Joel Fish ** (IAS)

From Gromov to the Moon

**Abstract:**
I will present some recent applications of symplectic geometry
to the restricted three body problem. More specifically, I will discuss
how Gromov's original study of pseudoholomorphic curves in the complex
projective plane has led to the construction of global surfaces of
section, and more generally finite energy foliations, below and slightly
above the first Lagrange point in the regularized planar circular
restricted three body problem. The talk will be accessible to a general
mathematical audience.

** Universite de Montreal Analysis seminar **

** Wednesday, January 29, Time and Room TBA**

** Gerasim Kokarev ** (Munich, LMU)

Direct methods in extremal eigenvalue problems

** Abstract:**
Extremal eigenvalue problems is an actively developing area of
spectral geometry. I will talk about the problems on Riemannian
surfaces related to determining metrics of fixed area which maximise
a given Laplace eigenvalue. I will give a short survey on the subject,
and outline an approach to extremal problems via the direct method
of calculus of variations.

** CRM-ISM Mathematics colloquium **

** Friday, February 7, 16:00, UQAM, Pav. Sherbrooke,
200, rue Sherbrooke O., salle SH-3420 **

** Charles Epstein ** (University of Pennsylvania)

Degenerate Diffusions arising in Population Genetics

** Abstract:**
I will speak on recent work, joint with Rafe Mazzeo and Camelia Pop, on
the analysis of solutions to a class of degenerate diffusion equations that
arise as limits of Markov chain models used in population genetics and
mathematical finance. These equations are naturally defined on spaces with
rather singular boundaries, like simplices and orthants. In addition to basic
existence, uniqueness and regularity results, I will discuss Harnack
inequalities and heat kernel estimates.

** Nirenberg Lectures in Geometric Analysis**

** CRM, May 13-16**

** Alessio Figalli ** (UT Austin)

Stability results for geometric and functional inequalities

Website

## FALL 2013

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Fall 2013 seminar web page was maintained by Alina Stancu.

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