2007/08 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)
WINTER 2008
CRM-ISM Colloquium, January 4, 4:00-5:00pm
UdeM, Pav. Andre-Aisenstadt, 2920 Chemin de la Tour, Room 6214
M. Jakobson (Univ. of Maryland)
Attractors and Invariant measures in low-dimensional dynamical systems
Abstract. We consider typical behavior of trajectories in dynamical
systems from topological and measure theoretical prospectives. They
coincide for uniformly hyperbolic systems, but can be different for
non-uniformly hyperbolic ones. Sinai-Ruelle-Bowen (SRB) measures are
crucial for understanding ergodic properties of dynamical systems.
In the case when a system has an absolutely continuous SRB measure its
typical trajectories exhibit random behavior and at the same time their
statistical properties can be described quantitatively. We discuss several
types of dynamics which arise in one-parameter families of low-dimensional
dynamical systems.
Monday, January 14, 2:30pm (DATE AND TIME CHANGED!)
Burnside 920
Z. Yan (McGill)
On a Birkhoff mormal form theorem for 1-d Gross-Pitaevskii equations
Abstract. We consider a 1-dimensional Gross-Pitaevskii (GP)
equation and want to
determine if there exits any Birkhoff normal form theorem for it. In doing
so, the following two factors plays important roles: one is to introduce
suitable function spaces; and the other one is to obtain estimates on
integrals of products of Hermite functions, which represent coefficients of
mode coupling. The resulting system is a perturbed system of a completely
resonant system. Then we made an analysis of the impact of the perturbation
on the principal part of the GP system.
Friday, February 8, 2:30pm
Burnside 920
X. Xu (McGill)
$L^\infty$ and gradient estimates of spectral clusters on compact
manifolds with boundary
Abstract.
In this talk, I will discuss $L^\infty$ and gradient estimates
of spectral clusters of both Dirichlet Laplacian and Neumann Laplacian on
compact manifolds with boundary. For interior $L^\infty$ estimates, two
approaches are discussed: one follows from the estimates of the remainder
of Wey's Law (for smooth manifolds), and the other follows Smith's recent
work (for $C^{1,\alpha}$ manifolds). The boundary $L^\infty$ estimates
and gradient estimates are proved via the maximum principle arguments for
Poisson equations.
Friday, February 15, 2:30pm
Burnside 920
A. Girouard (Univ. de Montreal)
Extremal problems for low eigenvalues on planar domains
Abstract. A classical result of G. Szego
states that among all simply-connected
planar domains of fixed area, the first nonzero Neumann eigenvalue is
maximized by a disk. In this talk, I will discuss the maximization
problem for the next eigenvalue. This is joint work with I.
Polterovich and N. Nadirashvili. Along the way, I will survey some
classical and recent results on extremization of small eigenvalues.
Thursday, February 28, 12noon
Burnside 920
M. Nasri (IMPA)
Two iterative methods for equilibrium problem
Abstract. The main purpose of this talk is the study of
finite dimensional
equilibrium problems.
First we guarantee existence of solutions for equilibrium problems under
reasonable
assumptions. We then introduce a proximal point method for finding
solutions of
equilibrium problems, using a regularization technique.
We also discuss an augmented Lagrangian method for solving this kind of
problems whose
feasible sets are defined by convex inequalities, generalizing the
proximal augmented Lagrangian method for constrained optimization.
March 7, 2:00pm
Burnside 934
B. Colbois (Univ. Neuchatel)
Upper bounds on the spectrum of the Laplacian
Abstract. In this talk, I will present a geometric approach of
getting upper bounds on the spectrum of the Laplacian
(with Neumann boundary conditions) on domains of a
Riemannian manifold with Ricci curvature bounded below
(a joint work with D. Maerten) and for
submanifolds in R^n (a work in progress with E. Dryden
and A. El Soufi).
Applied Mathematics/Analysis Seminar,
March 11, 3:30pm
Burnside 1205
Alex Barnett (Dartmouth)
Eigenmodes and quantum chaos: Lost on the frequency axis? Check your
Dirichlet-to-Neumann map!
Abstract:
The Dirichlet eigenmode (or `drum') problem describes vibrations of an
elastic membrane, acoustic cavity, or quantum particle, and is a
paradigm for more complicated applications to electromagnetic and
optical resonators. When the wavelength is much shorter than the
cavity size this becomes a challenging multiscale problem, and
boundary methods are essential. I will explain an accelerated cousin
of the method of particular solutions (MPS, a global basis
approximation method) which allows O(k) modes to be calculated in the
effort usually required for a single mode, k being the wavenumber. It
removes the need for expensive root-searches along the wavenumber (ie
frequency) axis. At very high frequencies and many cavity shapes, it
is the fastest method known, 10^3 times faster than either MPS or
boundary integral methods.
This has enabled large-scale numerical study of the asymptotic
properties of planar eigenmodes. I will present recent data on
`dynamical tunneling' in Bunimovich's mushroom cavity, which has both
chaotic and integrable motion. If time I will mention recent work
with T. Betcke on the Helmholtz BVP using fundamental solutions bases,
which in analytic shapes give spectral accuracy approaching only 2
degrees of freedom per wavelength on the boundary.
March 14, 14:30
Burnside 920
L. Berlyand (Penn. State)
Solutions with Vortices of a Semi-Stiff Boundary Value Problem for
the Ginzburg-Landau Equation
Abstract. We study solutions of the Ginzburg-Landau (GL) equation
for a complex valued order parameter $u$
-\Delta u+(1/\varepsilon^2) u(|u|^2-1)=0, \; x \in A \subset R^2.
This equation is of principal importance in the Ginzburg-Landau theory of
superconductivity and superfluidity.
For a 2D domain $A$ with holes we consider the so-called
"semi-stiff" boundary conditions: the
the Dirichlet condition for the modulus $|u|=1$, and the homogeneous
Neumann condition for
the phase $\arg(u)$. The principal result of this work is that there are
stable solutions with vortices of
this boundary value problem. The vortices are of a novel type: they
approach the boundary
and have bounded energy in the limit of small $\varepsilon$. By contrast,
in the well-studied Dirichlet problem for the GL PDE, the vortices are
distant from boundary and their energy
blows up as $\varepsilon \to 0$. On the other hand there is no stable
solutions with vortices to the homogeneous Neumann ("soft") problem.
In this work we develop a variational method that allows us to construct
local minimizers of the GL energy functional which corresponds to the GL
PDE. We introduce the notion of the approximate bulk degree which is the
key ingredient of our method. We show that, unlike the standard degree
over a curve, the approximate bulk degree is preserved in the weak
$H^1$-limit.
This is a joint work with V. Rybalko.
Monday, March 17, 14:30 (date changed!)
Burnside 920
Y. Pautrat (Paris-Sud, visiting McGill)
A non-commutative Levy-Cramer theorem
Abstract. In classical probability theory, the Levy-Cramer theorem
is the
basic
tool for proving the convergence in distribution of sequences of random
variables. In particular, it shows that the convergence of (joint)
characteristic functions determines the (joint) law of the limiting
random variables.
In non-commutative probability theory, the joint law of random
variables does not exist in general, so joint characteristic functions
may seem useless. In a joint work with Jaksic and Pillet, we proved,
however, that convergence of "quasi characteristic functions" formally
identical to the standard characteristic functions has interesting
consequences for many functionals of the limiting random variables.
We will give a short review of the principles of non-commutative
probability theory, and discuss our result, which is stated in
surprisingly simple and general terms.
March 28, 14:30
Burnside 920
Nir Lev (Tel Aviv)
Span of translates on the real line, and zeros of Fourier transform.
Abstract. A function $f \in L^p(R)$ is called a "generator" if the
set of all translates of $f$ spans the whole space $L^p(R)$. How to
decide whether a given function is a generator or not? We shall
discuss this problem including recent joint work with A. Olevskii.
April 4, 14:00
Burnside 934
T. Shaposhnikova (Ohio State University and Linkoeping University)
Multipliers in spaces of differentiable functions with applications to
PDEs
Abstract. New results on pointwise multipliers between
Besov spaces will be presented. Various applications to differential and
integral operators will be discussed. This is a joint work with V.
Maz'ya.
CRM-ISM colloquium, April 4, 16:00
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
V. Mazya (Ohio State, University of Liverpool and Linkoeping
University)
Unsolved mysteries of solutions to PDEs near the boundary
Abstract. Throughout its long history, specialists in the theory of partial
differential equations gained a deep insight into the boundary
behavior of solutions.Yet despite the apparent progress in this area
achieved during the last century, there are fundamental unsolved
problems and surprising paradoxes related to solvability, spectral,
and asymptotic properties of boundary value problems in domains with
irregular boundaries. I shall formulate some challenging questions
arising naturally when one deals with unrestricted, polyhedral,
Lipschitz graph, fractal and convex domains.
April 7-11
CRM
Workshop on Spectrum and Dynamics
Website
May 2, 10:30-11:30
CRM, Pav. Andre-Aisenstadt Room 5340
Ignacio Uriarte-Tuero (U. Missouri and Fields Institute)
Removability problems for bounded, BMO and Hoelder continuous quasiregular
mappings in the plane.
Abstract.
A classical problem in complex analysis is to characterize the removable
sets for various classes of analytic functions: H\"{o}lder, Lipschitz, BMO,
bounded (this last case gives rise to the analytic capacity and the
Painlev\'{e} problem which has been recently solved by Tolsa.)
One can ask the same questions in the setting of K-quasiregular maps (since
they are a K-quasiconformal map followed by an analytic map.) Most of the
bounded case was dealt with in a joint paper with K. Astala, A. Clop,
J.Mateu and J.Orobitg, [ACMOUT]. The BMO case was dealt with in [ACMOUT]
except for a gap at the critical dimension (Question 4.2 in [ACMOUT].) I
answered the question filling the gap in [UT].
The Lipschitz case was dealt with by A. Clop, as well as most of the
H\"{o}lder case, where again a gap at the critical dimension was left. In a
joint paper with A. Clop [CUT] we closed the gap.
I will summarize the results and give some ideas of the proofs in the above
papers. The talk will be self-contained.
References:
[ACMOUT] Kari Astala, Albert Clop, Joan Mateu, Joan Orobitg and Ignacio
Uriarte-Tuero. Distortion of Hausdorff measures and improved Painlev\'{e}
removability for bounded quasiregular mappings. Duke Math J., to appear.
[CUT] Albert Clop and Ignacio Uriarte-Tuero. Sharp Nonremovability Examples
for H\"{o}lder continuous quasiregular mappings in the plane, submitted.
[UT] Ignacio Uriarte-Tuero. Sharp Examples for Planar Quasiconformal
Distortion of Hausdorff Measures and Removability. IMRN, to appear.
Thursday, May 29, 14:00-15:00
McGill, Room 920
Bernard Helffer (Paris-Sud)
Semi-classical analysis for Schroedinger operators with magnetic fields
Slides
June 2-7
CRM
Workshop "Mathematical Aspects of Quantum Chaos"
Website
FALL 2007
Friday, September 7, 14:00-15:00
Burnside 920
Andrea Malchiodi (ISAS, Trieste)
Concentration phenomena for singularly perturbed elliptic PDEs
Abstract:
We consider a class of nonlinear elliptic equations with a singular
perturbation parameter $\epsilon$, which arise from physical or
biologial models, like the nonlinear Schroedinger equation or the
Gierer-Meinhardt system. In the last twenty years, a lot of attention has
been devoted to study the asymptotics of solutions, and more
recently some results about concentration at sets of positive
dimension have been derived. We discuss the recent progress in
this direction and some open problems.
Friday, September 14, 14:00-15:00
Burnside 920
Dmitry Jakobson (McGill)
On nodal sets of eigenfunctions
Abstract. We describe results about the topology of
nodal sets on S^2 and R^2 (joint work with A. Eremenko and
N. Nadirashvili). We next discuss recent results about
approximation by nodal sets on real-analytic Riemannian manifolds
(joint work with D. Mangoubi). A movie will be shown!
Friday, September 21, 14:00-15:00
Burnside 920
Jan Derezinski (Warsaw)
On the excitation spectrum of homogeneous Bose gas
Abstract.
I will speak about the joint
energy-momentum spectrum of
the Bose gas in thermodynamic limit.
Using rigorous mathematical language I will
formulate some conjectures about its
infimum and describe their relevance for physics
of cold gases.
I will describe a number of beautiful, although
mostly heuristic, arguments going back to Landau
and Bogoliubov, supporting a very interesting
physical and mathematical
picture of such systems.
Friday, September 28, 14:00-15:00
Burnside 920
Vitali Milman (Tel Aviv)
Asymptotic Geometric Analysis; Geometrization of Probability
Abstract. We study the asymptotic behavior of finite-
(but very high-)dimensional
normed spaces and convex bodies when dimension tends to infinity. Contrary
to common intuition, which anticipates enormous diversity and chaotic
behavior, we observe a uniform behavior for the whole family of finite- (but
high-)dimensional spaces. In the first part of our talk we will discuss
different and unexpected phenomena accompanying high dimension. In the
second, the main part of the talk we will explain how the geometric theory
of convexity is extended to a larger category of log-concave measures which
bring inside this class of (probability) measures geometric vision and
approach. This brings inside the theory functional versions for many
geometric inequalities, and also leads to solutions of some central problems
of the theory.
The talk will be understandable to any graduate student in Mathematics.
Friday, October 19, 14:00-15:00
Burnside 920
Jun-Fang Li (CRM and McGill)
Monotonicity formulas under Ricci flow and Normalized Ricci flow
Abstract. We study the steady state of Ricci flow (RF) and
Normalized Ricci flow(NRF). We are interested in finding integral
quantities which have monotonicity properties under RF or NRF, and
the first variation of these functionals vanishes if and only if at a
steady state of RF or NRF. The ideas stem from G. Perelman's
groundbreaking work in the study of Ricci flow. As applications we
classified the steady states of RF and NRF(compact expanding and
steady cases). A byproduct is our results improve some previous work
on eigenvalues monotonicity properties under RF.
Joint Applied Mathematics/Analysis seminar
Monday, October 22, 14:30-15:30
Burnside 1205
George C. Hsiao (Univ. of Delaware)
Boundary Element Methods for the Last 30 Years
Abstract. Variational methods for boundary integral equations
deal with weak
formulations of the equations. Boundary element methods are numerical
schemes for seeking approximate weak solutions of the corresponding boundary
variational equations in finite-dimensional subspaces of the Sobolev spaces
with special basis functions, the so-called boundary elements. This lecture
gives an overview of the method from both theoretical and numerical point of
view. It summarizes the main results obtained by the author and his
collaborators over the last 30 years. Fundamental theory and various
applications will be illustrated through simple examples. Some numerical
experiments in elasticity as well as in fluid mechanics will be included to
demonstrate the efficiency of the methods.
Friday, October 26, 14:00-15:00
Burnside 920
Thierry Daude (CRM and McGill)
Recovering the mass and the charge of a Reissner-Nordstrom black
hole by an inverse scattering experiment
Abstract. In this talk, we shall study inverse scattering of massless
Dirac fields that propagate in the exterior region of a Reissner-Nordstrom
black hole. Using a stationary approach we shall determine precisely the
leading terms of the high energy asymptotic expansion of the associated
scattering matrix which, in turn, will permit us to recover uniquely the
mass and the charge of the black hole.
Friday, November 2, 14:00-15:00
Burnside 920
Cristina Pereyra (Univ. of New Mexico)
Weighted inequalities and Bellman functions
Abstract. We present a modern perspective to the classical problem of
weighted inequalities for singular integrals and dyadic model
operators. The new ingredient is the use of Bellman functions techniques
to track optimal dependence of the operator bounds on weighted Lebesgue
spaces in terms of the $A_p$ or $RH_p$ characteristic of the weights.
This is not just a mathematical exercise, some people do use these
optimal estimates in areas such as quasiconformal mmapping theory and
elliptic differential equations.
Friday, November 9, 14:00-15:00
Burnside 920
Philip Gressman (Yale)
Uniform estimates for cubic oscillatory integrals
Abstract. I will discuss the problem of proving uniform, optimal
asymptotic estimates for
scalar oscillatory integrals with a phase function which satisfies an
appropriate third-order nondegeneracy condition. The proof relies on the
construction of a nontrivial symmetric space structure adapted to the
geometry of the phase.
Friday, November 16, 14:00-15:00
LB 655 (at Concordia)
Hong Yue (Oulu Univ, Finland)
The John-Nirenberg Inequality for $Q_\alpha(R^n)$ and the Related
Fractal Function
Abstract. The John-Nirenberg inequality characterizes functions in the
space BMO in terms of the decay of the distribution function of their
oscillations over a cube. We prove separate necessary and sufficient
John-Nirenberg type inequalities for functions in the space
$Q_\alpha(R^n)$, introduced by Essen, Janson, Peng and Xiao, who
conjectured a version of this inequality. Our results are a modified
version of their conjecture, and we give two counterexamplesto
show the necessity for this modification.
The counterexamples provide us with a borderline
case function for $Q_\alpha(R^n)$ which can be expressed in terms of a
Haar wavelet decomposition. We vary the parameter in this decomposition
and discuss the relation between the function and the spaces. In
addition, we study the fractal properties of the function and determine
the fractal dimensions of its graph. The properties and dimensions
illustrate some form of regularity for functions in $Q_\alpha(R^n)$.
Friday, December 7, 14:00-15:00
Burnside 920
Stephan De Bievre (Univ. Lille)
The Unruh effect revisited
Abstract. This is joint work with M. Merkli. According to Unruh (1976),
a detector accelerated through a relativistic
quantum field in its vacuum state will respond as if bathed in a black
body radiation at a temperature proportional to its acceleration. I will
give a precise mathematical meaning to this statement and show how it can
be understood as a problem of ``return to equilibrium'' in quantum
statistical mechanics.
ANALYSIS TALKS ELSEWHERE
CRM-ISM colloquium
Friday, October 26, 4:00-5:00pm
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
B. Khesin (Toronto)
Pseudo-Riemannian geodesics and billiards
Abstract. In pseudo-Riemannian geometry the spaces of space-like
and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. In the talk I will describe the geometry of these structures, define pseudo-Euclidean billiards and discuss their properties. In particular, I will outline complete integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in pseudo-Euclidean space; these results are pseudo-Euclidean counterparts to the classical theorems of Euclidean geometry that go back to Jacobi and Chasles.
2006/2007 Seminars
2005/2006 Analysis Seminar
2004/2005 Seminars
2004/2005 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