Analysis Seminar
Seminars are usually held on Mondays or Fridays at Concordia or at McGill
For suggestions, questions etc. please contact Galia Dafni
(gdafni@mathstat.concordia.ca), Dmitry Jakobson
(jakobson@math.mcgill.ca), Ivo Klemes (klemes@math.mcgill.ca)
or Alexander Shnirelman (shnirel@mathstat.concordia.ca)
SUMMER 2007
Tuesday, May 29, 2pm-3pm
Burnside 1234
Tadashi Tokieda (Cambridge)
How to choose a Lagrangian
Monday, June 4, 2pm-3pm
Burnside 920
V. Roytburd (Rensselaer Polytechnic Institute)
From BS to KS and beyond: Dissipativity and pattern formation
Abstract. Attempts to capture salient features of cellular flame
instabilities have
led to a variety of beautiful mathematical models of a very geometrical
nature. The most famous of them is the Kuramoto---Sivashinsky (KS) equation.
Its lesser relative is the Burgers---Sivashinsky (BS) equation (which is
just a linearly forced Burgers equation). The linear dispersion relations
for both equations admit exponential mode growth for a range of long waves.
Nonetheless the equations are dissipative due to the nonlinear mixing. For
the purposes of this talk, dissipativity means that the eventual time
evolution of solutions is confined to a bounded (actually compact) absorbing
set. The principal subject of this talk is yet another, recently introduced
model of quasi-steady development of cellular flames, the Quasi-Steady
equation (joint work with M. Frankel, IUPUI). In a sense, QS is intermediate
between BS and KS, as its dispersion relation coincides with that for BS for
short waves, and is virtually identical to that of KS for long waves.
Similarly to KS, QS demonstrate a very rich dynamical behavior (note that BS
has more or less trivial dynamics). The proof of dissipativity and
generalizations to elliptic pseudo-differential operators will be discussed.
Monday, June 11, 14:00-15:00
Burnside 920
Adrian Butscher (Toronto)
New constructions of submanifolds of the sphere which are critical
points of the volume functional
Abstract: If one searches for k-dimensional submanifolds
with critical k-dimensional volume in a Riemannian manifold, then one is
led towards elliptic partial differential equations involving the mean
curvature vector of the submanifold. I will present new
constructions of volume-critical submanifolds of the sphere in two
contexts: hypersurfaces with constant mean curvature in spheres of
any dimension; and Legendrian submanifolds in spheres of odd
dimension that are stationary under variations preserving the
contact structure. These are constructed by solving the associated
elliptic PDE using singular perturbation theory. I will then
highlight some of the analytic and geometric similarities between
these two contexts.
Friday, June 15, 14:00-15:00
Burnside 920
Xinan Ma (Univ. of Science and Tech. of China)
A generalization of Brascamp-Lieb Theorem on log concavity to
$\Sigma_{2}$ operator
Abstract: In this talk I shall consider the generalized of
Brascamp-Lieb's theorem on
(JFA76) the log convcavity of first eigenfunction of the Dirichlet
eigenvalue problem on eulidean convex domain, more precisely we consider
the operator of second elementry symmetric functon of the hessian in three
dimension case.As an application we get the Brunn-Minkowski inequality and
describe the equality case. This is a joint works with Xu Lu.
July 16-18, Burnside 920
Vojkan Jaksic and Eugene Kritchevski (McGill)
Preparatory lectures for F. Germinet's course on July 18-23.
Monday July 16, 14:00-16:00 Jaksic: The spectral theorem
Tuesday July 17, 14:00-16:00 Jaksic: The ergodic theorem
Wednesday July 18, 14:00-16:00 Kritchevski: Weyl sequences and
Borel-Cantelli lemmas
July 18-23, Burnside 920
F. Germinet (Cergy-Pontoise)
Mini-course: A basic introduction to Random Schroedinger operators
Wendesday July 18, 10:00-12:30. Lecture 1
Thursday July 19, 10:00-12:30. Lecture 2
Friday July 20, 10:00-12:30. Lecture 3
Saturday July 21, 10:00-12:30. Lecture 4
Monday July 23, 10:00-12:30. Lecture 5
July 23, 14:00-15:00
Concordia, LB 921-4
I. Markina (U. Bergen, Norway)
Tangential Cauchy-Riemann operator on quaternionic Sigel upper half space
Abstract
July 23, 15:30-16:30
Concordia, LB 921-4
A. Vasiliev (U. Bergen, Norway)
Tangential slit solutions to the Loewner equation
Abstract: We consider the Loewner differential equation
generating the unit disk
or the upper half-plane onto itself minus a single slit. Marshall and Rohde
recently proved that if a slit is a non-tangential quasiarc, then the
corresponding Loewner
equation contains a Lip(1/2) driving term. We prove that if the slit is
first-order tangential,
then the driving term is Lip(1/3).
WINTER 2007
Friday, January 12, 2-3pm
Burnside 920
Michael Levitin (Heriot-Watt)
Calculating eigenvalues and resonances in domains
with regular ends
Monday, January 29, 2:30-3:30pm
Burnside 920
Jason Metcalfe (Berkeley)
Global Strichartz and smoothing estimates for rough Schrodinger
equations
Abstract I will talk about recent work with J. Marzuola and D.
Tataru. We prove some frequency localized smoothing estimates for
Schrodinger equations with C^2 asymptotically flat coefficients. In
particular, we make no assumptions on the trapping and prove the said
estimates outside of the region where the trapping may occur, modulo lower
order error terms. We then use a long time parametrix construction of
Tataru to prove global-in-time Strichartz estimates on this same region.
Friday, February 2, 2:00-3:00pm (Note time change!)
Burnside 920
Dimiter Vassilev (UC Riverside)
Conformal geometry and function theory on quaternionic contact
manifolds
Abstract
We shall present some results concerning Einstein structures
on quaternionic contact manifolds and their conformal deformations.
From the analytical point of view the study concerns the best constant
in the Folland-Stein L2 embedding theorem and geometric function theory
on quaternionic contact manifolds.
Joint Applied Mathematics/Analysis seminar
Friday, February 9, 1:30-2:30pm
Room TBA
Olaf Steinbach (Graz)
Boundary Integral Equations: Analysis and Applications
Abstract For a long time boundary integral equations were
used to prove the
unique solvability of second order boundary value problems. Using
indirect boundary integral formulations with single and double layer
potentials those considerations led to first and second kind
boundary integral equations to be solved. In particular for the
latter Neumann series are an appropriate choice. However, there was
and there is still an ongoing discussion about the convergence
estimates in suitable function spaces.
In this talk we will present some results to prove the contraction
property of the double layer potential in fractional Sobolev spaces.
These estimates have an deep impact for the design of efficient
boundary element methods, i.e. preconditioning and adaptivity.
Moreover, there are strong relations with domain decomposition methods
and the Dirichlet to Neumann map involved.
Monday, February 12, 2:30-3:30pm
Burnside 920
Rajesh Pereira (Univ. of Saskatchewan)
Majorization, Matrices and the Geometry of Polynomials
Abstract In this talk, we explore the connections between the
majorization order, matrix theory and the geometry of polynomial root
sets. We will show how majorization can be used to prove results about
polynomials (such as the solution of the De Bruijn-Springer conjecture and
an improvement to Mahler's inequality) and will discuss an application of
majorization to the fastest mixing Markov chain problem of Boyd, Diaconis and
Xiao. We will also explore a possible Hilbert space approach to the
Bombieri norm on polynomials.
Friday, March 16, 14:30-15:30
Burnside 920
Julian Edward (Florida International U.)
Ingham-type inequalities for complex frequencies and
applications to control theory
Abstract For complex valued sequences $\{
\omega_n\}_{n=1}^{\infty}$ of the form $\om_n=a_n+ib_n$ with $a_n\in
{\bf R}$ and $b_n\geq 0$, we prove inequalities of the form
$\int_0^T|\sum_{n=p}^{\infty}x_ne^{it\om_n}|^2dt\geq C
\sum_{n=p}^{\infty}|x_n|^2/(1+b_n)$, for all sequences $\{ x_n\}$
with $\sum_{n=1}^{\infty}|x_n|^2/(1+b_n)<\infty.$ We apply these to
prove exact null controllability for a class of hinged beam
equations with mild internal damping.
Friday, March 23, 14:30-15:30
Burnside 920
Malabika Pramanik (UBC)
$L^2$ decay estimates for oscillatory integral operators in
several variables with homogeneous polynomial phases
Abstract Oscillatory integral operators mapping $L^2(\mathbb R^{n_1})$
to $L^2(\mathbb R^{n_2})$ play an important role in many problems in
harmonic analysis and partial differential equations. We will briefly
discuss the applicability of these operators in various contexts and give
an overview of the current literature.
We also mention recent results (joint with Allan Greenleaf and Wan Tang)
where, extending earlier work of Phong and Stein in the case $n_1 = n_2 =
1$, we obtain optimal decay rates for the $L^2$ operator norm of
oscillatory integral operators in $2+2$ variables with generic phases.
Some other higher dimensional situations are also addressed.
Friday, March 30, 14:30-15:30
Burnside 920
Alexander Koldobsky (Missouri)
Intersection bodies and L_p-spaces
AbstractIntersection bodies have become one of the central objects in
convex geometry. We discuss a recently discovered connection between this
class of bodies and $L_p$-spaces with $p<0.$ This connection allows to get
new geometric results by extending to negative values of $p$ different
results from the classical theory of $L_p$-spaces.
Thursday, April 5, 17:00-18:00
Concordia, LB 921-4 (Library building)
D. Dryanov (Concordia)
Kolmogorov Entropy for Classes of Convex Functions
Abstract Kolmogorov (or $\epsilon$-) entropy of a compact set in
a metric space measures "how massive is it" and thus replaces
its dimension (which is usually infinite). This notion is widely
applied in the approximation
theory. However, recently the need to estimate the Kolmogorov entropy
emerged in the Fluid Dynamics, where some nontrivial compact sets appear as
final states of the evolution of ideal incompressible flows. An example of
such set is the set of plane-parallel flows in a periodic strip with
parallel walls having convex velocity profile.
In our talk we answer the following question asked by A.Shnirelman: What is
the exact asymptotic of the $\epsilon$-entropy for uniformly bounded
classes of convex function in the $L^p$-metric? We show that for
$1<=p<=\infty$, the Kolmogorov $\epsilon$-entropy of the metric space of
convex and uniformly bounded functions, equipped with $L^p$-metric has the
precise asymptotic $\epsilon^{-1/2}$.
Friday, April 13, 14:30-15:30
Burnside 920
Yuri Safarov (King's College, London)
Asymptotic formulae for the spectral function
Abstract The talk will discuss asymptotic behaviour of the spectral
function of the Laplace operator on a manifold without boundary
for large values of the spectral parameter. It will give an overview
of known results obtained with the use the wave equation technique and
Fourier Tauberian theorems.
Friday, April 20, 14:30-15:30
Burnside 920
C.J. Xu (Univ. de Rouen)
Hypoellipticity of Boltzmann equation
Abstract
In this talk, we prove the regularity(in Sobolev space and in Gevrey
class) effet of Cauchy problem for Boltzmann equation without Grad's
angular cutoff.
We study full nonlinear Boltzmann eqution for spatially homogeneous space,
and linear Boltzmann eqution in inhomogeneous case. We use the nonlinear
microlocal calculus and Fefferman-Phong's uncertainty principle to get the
regularity of weaks solutions. So it is quite different with classical
H-theorem for Boltzmann equation.
Monday, April 30, 14:30-15:30
Burnside 920
G. Staffilani (MIT)
Nonlinear Schrodinger equation in Hyperbolic spaces
Tuesday, May 1, 11:00-13:00
Burnside 920
F. Germinet (Univ. Cergy-Pontoise, Paris)
1. Random Schrodinger operators and Anderson localization
2. A concentration inequality and applications
The third lecture
3. Single energy multiscale analysis and Anderson localization
will be given from 14:40-15:40 on May 2, during the Analysis Day at
CRM (see below)
Abstract In these lectures, I will review standard and new results
on Anderson
localization for random Schrodinger operators. Such models describe
the motion of an electron in disordered media. As discussed by
Anderson in 1958, a disordered medium should have a trapping effect,
at least at low energies, and one then refers to ``Anderson
localization". In the first talk, I will present different models as
well as standard results obtained over the last 30 years. In
the second talk, I will state and prove a new concentration bound for
particular functions of independant random variables. As an application,
it will yield Anderson localization for an arbritrary (non degenerate)
underlying probability measure. In the third talk, I will show how to
extract exponentially decaying eigenfunctions out of the basic and
standard single energy multiscale analysis introduced by Frohlich
and Spencer in 1983.
ANALYSIS DAY: Wednesday, May 2, 11:00-16:00
CRM, Room 5340
11:00-12:00
Francois Germinet (Univ. Cergy-Pontoise, Paris)
Single energy multiscale analysis and Anderson localization
Abstract
In the talk, I will show how to extract exponentially
decaying eigenfunctions out of the basic and standard single energy
multiscale analysis introduced by Frohlich and Spencer in 1983.
12:00-13:30 Lunch
13:30-14:30
Francis Clarke (Institut universitaire de France et Universite
de Lyon)
Regularity of solutions in the calculus of variations
Abstract An overview of some classical and recent results on the
regularity of solutions in the (single and multiple integral) calculus
of variations. Along the way are reviewed the celebrated theorems of
Hilbert-Haar and De Giorgi.
14:40-15:40
Igor Wigman (CRM and McGill)
Nodal lines for random eigenfunctions of the Laplacian on the torus
Abstract We study the volume of nodal sets for eigenfunctions of the
Laplacian on the standard torus in two or more dimensions.
We consider a sequence of eigenvalues $4\pi^2\lambda$ with growing
multiplicity $N\to\infty$, and compute the expectation and variance
of the volume of the nodal set with respect to a Gaussian probability
measure on the eigenspaces. We show that the expected volume of the
nodal set is $const\sqrt{\lambda}$.
Our main result is that the variance of the volume normalized by
$\sqrt{\lambda}$ is bounded by $O(1/\sqrt{N})$, so that the normalized
volume has vanishing fluctuations as we increase the
dimension of the eigenspace. This is joint work with Z. Rudnick
(Tel Aviv university).
Thursday, May 3, 15:30-16:30
Burnside 920
J. Schenker (IAS, Princeton and Michigan State Univ.)
Estimating the real part of complex eigenvalues of
non-self adjoint Schroedinger operators via complex dilations
Abstract This talk will focus on a lower bound for the real part of
eigenvalues of dissipative operators L = H + i g F with H the 1D Harmonic
oscillator, F an analytic function on the real line, and g the coupling
parameter. When the coupling g is large the eigenvalues
lie in a half plane {z: Re z > g^v} with v an exponent that depends on F.
This cannot be seen directly by variational arguments since the
perturbation i g F is skew-adjoint, but can be seen using complex dilations.
As a result of the bound, one sees that the dissipation of the semi-group
e^{-t L} is greatly enhanced compared to e^{-t H} although the difference
of the generators is skew-adjoint. This is a linear manifestation of the
phenomenon ``hypercoercetivity'' identified by C. Villani.
Monday, May 7, 14:30-15:30
Burnside 920
S. Roy (Laval)
Extreme Jensen measures. Abstract.
FALL 2006
Friday, September 29, 2:30pm
Burnside 920
Dan Mangoubi (CRM/McGill)
On inner radius of nodal domains
Friday, October 6, 2pm
Burnside 920
Simone Warzel (Princeton)
The Canopy Graph and Level Statistics for Random Operators
on Trees
Abstract
For random operators with homogeneous disorder, it is
generally believed that the spectral characteristics of the
infinite operators, which may vary over different energy ranges,
are reflected also in the distributions of the energy gaps in
finite volume versions of the operators. Whereas pure point
spectrum of the infinite operator goes along with Poisson level
statistics, it is expected that purely absolutely continuous
spectrum would be associated with gap distributions resembling the
corresponding random matrix ensemble. We prove that on regular
rooted trees, which is the only example where both spectral types
have been established, the level statistics is always Poisson.
However, we also find that this does not contradict the common
belief if that is carefully reviewed, as the relevant limit of
finite trees is not the infinite tree graph but rather what is
termed here the canopy graph. For this tree graph, the random
Schoedinger operator is proven to have only pure-point spectrum at
any strength of the disorder.
Friday, October 20, 2pm
Burnside 920
Steve Zelditch (Johns Hopkins)
Inverse spectral problem for analytic plane domains with one
symmetry
Abstract
We sketch the proof that simply connected real analytic plane domains
with one symmetry are determined by their Dirichlet or Neumann spectral
among other such domains.
Monday, October 23, 2:30pm
Burnside 920
Alexander Strohmaier (Bonn)
High Energy Limits of Laplace-type and Dirac-type Eigenfunctions and
Frame Flows
Abstract We relate high-energy
limits of Laplace-type and Dirac-type operators to frame flows on the
corresponding manifolds, and show that the ergodicity of frame flows
implies quantum ergodicity in an appropriate sense for those operators.
Observables for the corresponding quantum systems are matrix-valued
pseudodifferential operators and therefore the system remains
non-commutative in the high-energy limit. We discuss to what extent
the space of stationary high-energy states behaves classically.
This is about a joint work with Dmitry Jakobson.
Friday, October 27, 2pm
Burnside 920
Y. Pautrat(Paris-Sud)
Quantum Central Limit Theorem
Abstract Following a joint project with Jaksic, Ogata and Pillet we
study the linear response of coupled fermionic systems close to
thermodynamic equilibrium. The cornerstones of linear response
theory are the Kubo formula, the Onsager reciprocity relations
and the fluctuation-dissipation theorem, which relates the
fluctuations of observables around their equilibrium value, to
the transport properties of the system.
In this talk we describe a general central limit theorem, which
allows us to construct an algebra of operators representing
time fluctuations of, e.g. flux observables. Dynamical
properties of the system appear in this algebra; in particular
the fluctuation algebra is commutative if and only if the
system is in equilibrium and in that case we recover the
fluctuation-dissipation thorem.
Friday, November 3, 2pm
Burnside 920
Der-Chen Chang (Georgetown)
Fundamental Solutions for Hermite and Subelliptic Operators
Abstract In this talk, we first introduce a geometric method based on
multipliers to compute heat kernels for Laplacian operators with potentials.
Using the heat kernel, one may compute the fundamental solution for
the Hermite operator with singularity at an arbitrary point on the
Euclidean spaces and Heisenberg groups. As a consequence, one may
obtain the fundamental solutions for the sub-Laplacian $\Box_J$ in
a family of quadratic submanifolds which was introduced by Nagel, Ricci and
Stein.
Friday, November 10, 2pm
Burnside 920
Pavel Bachurin (Toronto)
Ergodicity and the size of a neighbourhood of
singularity manifolds of dispersing billiards.
Abstract Ergodic theory of multi-dimensional dispersing billiards
has been developed in 1980s. An important part of the theory
is the analysis of the structure of the sets, where the billiard
map is discontinuous. They were assumed to be smooth manifolds till
recently, when a new pathological type of behaviour of these
sets was found. Thus a reconsideration of earlier arguments
was needed. I'll show that for a generic configuration
of scatterers these sets behave relatively well and dispersing
billiards are ergodic. The proof is base on an estimate of the
volume of a neighborhood of a level set of a smooth function
(joint with Ch. Fefferman), which has independent interest.
SUMMER 2006
Monday, June 12, 2:30pm
McGill, Burnside 920
Alexander Vedenov (Kurchatov Institute, Moscow)
Bacterial cell from physical point of view
Abstract:
Bacterial cell is considered as system of polymerizing molecular machines of
three types - DNA polymerases, RNA polymerases and ribosomes.
For an exponentially growing population of bacteria, we suppose synchronous
operation of three
subsystems of polymerizing molecular machines. Then it is possible to estimate
the numbers of these machines in bacterial cell.
The result is qualitatively consistent with experimental data for E.coli.
The exponential growth on mediums with a various composition of
deoxynucleotides triphosphates,
ribonucleotide triphosphates and amino acids is discussed.
The results compared with experimental data for cell size and for population
doubling time.
Thursday, June 29, 3:00pm
McGill, Burnside 920
Eric Sawyer (McMaster)
Regularity of certain subelliptic Monge-Ampere equations
Abstract:
Assuming that k is smooth and vanishes only at nondegenerate critical
points, we prove that C^2 solutions u to the Monge-Ampere equation
detD^2u=k are smooth if and only if the subGaussian principal curvature
of u is positive. More general k are treated as well, and the proofs
involve an n-dimensional partial Legendre transform, Calabi's identity
for the square of third order derivatives of solutions, the Campanato
method of Xu and Zuily, the Stein-Rothschild lifting and approximation of
vector fields, Jerison's Poincare inequality, subelliptic DeGeorgi Nash
Moser theory, Guan's commutator lemma and other subelliptic techniques.
Hilbert's 17th problem also plays a role, and we discuss the ways in
which these varied topics enter into the problem of detecting smoothness
in subelliptic Monge-Ampere equations. In two dimensions, Guan had
earlier obtained such results for C^1,1 solutions, but the extension to
C^1,1 solutions in higher dimensions remains a challenging open problem.
2007/2008 Seminars
2005/2006 Analysis Seminar
2004/2005 Seminars
2004/2005 Seminar in Nonlinear Analysis and Dynamical Systems
2003/2004 Seminars
2003/2004
Seminar in Nonlinear Analysis and Dynamical Systems
2003/2004 Working Seminar in Mathematical Physics
2002/2003 Seminars
2001/2002 Seminars
2000/2001 Seminars
1999/2000 Seminars