
Fields Institute Colloquium/Seminar in Applied Mathematics
20102011
Organizing Committee 

Jim Colliander (Toronto)
Walter Craig (McMaster)
Catherine Sulem (Toronto) 
Robert McCann (Toronto)
Adrian Nachman (Toronto)
Mary Pugh (Toronto)

The Fields Institute Colloquium/Seminar in Applied Mathematics
is a monthly colloquium series for mathematicians in the areas of
applied mathematics and analysis. The series alternates between
colloquium talks by internationally recognized experts in the field,
and less formal, more specialized seminars.In recent years, the
series has featured applications to diverse areas of science and
technology; examples include superconductivity, nonlinear wave
propagation, optical fiber communications, and financial modeling.
The intent of the series is to bring together the applied mathematics
community on a regular basis, to present current results in the
field, and to strengthen the potential for communication and collaboration
between researchers with common interests. We meet for one session
per month during the academic year. The organizers welcome suggestions
for speakers and topics.
201011
Schedule  Future talks to be held at the Fields Institute

Wednesday, June 1, 2011
3:10PM
Fields Institute, Room 230

Erwan Faou (University of Rennes)
Resonances and long time integration of nonlinear Schroedinger
equations
In this talk, we will review some recent advances in long
time simulation of Hamiltonian PDE, by focusing on the special
case of the nonlinear Schrödinger equation with cubic
nonlinearity and without potential (the resonant case). After
discussing some results concerning the long time behavior
of the exact solution (preservation of the actions in dimension
1, energy cascade in dimension 2), we will study the persistence
of such qualitative behaviors by fully discrete splitting
schemes. In particular, we will show how the choice of the
number of grid points or the stepsize can lead to numerical
instabilities, and on the other hand how implicit schemes
are in general unable to reproduce correctly the energy exchanges
in such a resonant situation. The main tool to analyze these
phenomena is the use of backward error analysis for splitting
methods under CFL applied to Hamiltonian PDEs, as stated in
a recent common work with B. Grébert.

Past Talks

Wednesday,
April 20, 2011
3:10PM
Fields Institute, Room 230 
Nicolas Perinet, UOIT
Numerical simulation of Faraday waves
When two superposed fluids are vertically shaken the initially
plane interface forms patterns if the oscil lation is of
suffcient amplitude. This phenomenon, called the Faraday instability,
constitutes an amazingly rich macroscopic model for the study
of pattern formation. In addition to the classically observed
crystalline patterns (stripes, squares or hexagons) very singular
structures have been seen: quasipatterns, oscillons and superlattices.
In an attempt to survey the patterns supplied by the Faraday
experiment, we solve the complete nonlinear Navier–Stokes
equations by a finitedifference projection method coupled
to a FrontTracking technique for the calculation of the surface
tension forces and advection of the interface.
In the linear regime the instability thresholds and temporal
eigenmodes calculated numerically are com pared to those
obtained from Floquet theory. In the nonlinear regime we compare
with experimental work which provides quantitative features
of squares and hexagons arising at saturation for several
forcing amplitudes. The evolution of the nonlinear spatial
modes and the spatiotemporal spectra are in good agreement
with experimental results.
However, experiments and early numerical simulations highlight
that the hexagonal symmetry may only be transient. The alternate
emergence of hexagonal structures and patterns with other
symmetries suggests that the hexagonal regime may be a fixed
point belonging to a homoclinic orbit. We have developed an
algorithm which forces the hexagonal symmetry in order to
calculate the fixed state that will be the starting point
for the exploration of this orbit.
Finally, we have carried out a numerical study of the drift
instability in the Faraday experiment in an annular configuration.
An azimuthal displacement of initially stationary patterns
has been experimentally observed when the oscillation amplitude
exceeds a secondary threshold. Our numerical simulations have
con?rmed this result. Bifurcation diagrams displaying additional
instabilities have been constructed, as well as a complementary
spatiotemporal spectral analysis.
This is joint work with Damir Juric (Laboratoire d’Informatique
pour la Mecanique et les Sciences de l’Ingenieur, CNRS,
Orsay) and Laurette Tuckerman (Laboratoire de Physique et
Mecanique des Milieux Heterogenes, ESPCICNRS, Paris)

Wednesday, April 20, 2011
2:10PM
Fields Institute, Room 230

Greg Lewis, UOIT
Mixedmode solutions in the differentially heated rotating
annulus
The differentially heated rotating annulus experiment has
long been regarded as a useful tool for studying baroclinic
waves. These waves can be generated in rotating fluids with
an imposed horizontal temperature gradient, and thus, may
play an important role in the poleward transport of heat and
momentum in the atmosphere.
We present a bifurcation analysis of a mathematical model
that uses the (threedimensional) NavierStokes equations
in the Boussinesq approximation to describe the flow of a
near unity Prandtl number fluid (i.e. air) in the differentially
heated rotating annulus. In particular, we study the Hopf
bifurcations that correspond to the transition from axisymmetric
to nonaxisymmetric flow, where the axisymmetric flow loses
stability to an azimuthal mode of integer wave number, and
rotating waves may be observed. Of particular interest are
the double Hopf (HopfHopf) bifurcations that occur along
the transition, where thereis an interaction of two modes
with azimuthal wave numbers differing by one.
The analysis shows that in certain regions in parameter space,
stable quasiperiodic mixedazimuthal mode solutions result
from the modeinteraction. These flows have been called wave
dispersion and interference vacillation. The results differ
from similar studies of the annulus with a higher Prandtl
number fluid (e.g.water). In particular, we show that a decrease
in Prandtl number can stabilize these mixedmode solutions.
We also discuss the modeinteraction with 1:2 spatial resonance,
which indicates another mechanism by which a mixedmode solution
may arise.

Wednesday, March 16, 2011
3:10PM
BA6183, Bahen Center, 40 St. George St.

Professor Charles Fefferman, Princeton University
Breakdown of Smoothness in the Muskat Problem
The problem concerns the evolution of the interfaces between
two or more fluids in a porous medium. The talk presents new
phenomena arising when at least three fluids are present.
(Joint work with several coauthors.)

December
8, 2010
2:10 p.m
Room 230 
Abdelmalek
Abdesselam (U of Virginia)
Introduction to the renormalization group as a rigorous tool
in probability theory
Ever since its introduction by Kenneth G. Wilson in the seventies,
the renormalization group has been the main conceptual tool
used by physicists in order to make meaningful calculations
with functional integrals. The latter are, largely conjectural,
infinitedimensional probability measures over spaces of functions
which one can try to construct rigorously using a scaling
limit of similar measures where the continuum is discretized
by finer and finer grids.
The renormalization group is a dynamical system corresponding
to averaging over the short distance fluctuations of the random
function and zooming out by a fixed scale ratio. Fixed points
of this dynamical system correspond to the possible scaling
limits one can achieve. The renormalization group provides
a far reaching generalization of the familiar central limit
theorem, in a situation where the random variables are dependent,
in a way which is subordinated to the geometry of the space
labeling these variables. In this nontechnical presentation,
we will provide an introduction to the basic ideas of the
renormalization group.

November
10, 2010
2:10 p.m.
Room 230 
Boris
Khesin (University of Toronto)
Optimal transport and geodesics for H1 metrics on diffeomorphism
groups
We describe the Wasserstein space for the homogeneous H1
metric which turns out to be isometric to (a piece of) an
infinitedimensional sphere. The corresponding geodesic flow
turns out to be integrable, and it is a generalization of
the HunterSaxton equation. The corresponding optimal transport
can be used for the "sizerecognition", as opposed
to the "shape recognition". This is a joint work
with J. Lenells, G. Misiolek, and S. Preston.

November
10, 2010
3:10 p.m.
Room 230 
CANCELLED
Tony Gomis (NBI)
Adomian Decomposition Method, Cherruault Transformations,
Homotopy Perturbation Method, and Nonlinear Dynamics: Theories
and Comparative Applications to Frontier problems
New global methods for solving complex, nonlinear, continuous
and discrete,deterministic and stochastic,differential or
integral, and combined functional equations, will be presented
and compared. This talk will outline the Adomian Decomposition
Method,and the Homotopy Perturbation Techniques, all offering
solutions as convergent infinite functional series.In this
talk , the Cherruault Alienor transformations based on a generalization
of the spacefilling curves theory(for quasilossless dimensionality
compression, and for functional Global Optimization ) will
be outlined and applied to realworld problems.

August
20, 2010
2:10 pm
Stewart Library 
Apala
Majumdar (University of Oxford)
The Landaude Gennes theory of nematic liquid crystals: Uniaxiality
versus Biaxiality
In this talk, we review two widelyused continuum theories for
nematic liquid crystals: the OseenFrank theory, which is restricted
to uniaxial nematics, and the more general Landaude Gennes
theory. We also discuss the analogies between the Landaude
Gennes theory and the celebrated GinzburgLandau theory for
superconductors in three dimensions. We treat uniaxial and biaxial
cases separately. The uniaxial case can be viewed as a generalized
GinzburgLandau theory from a threedimensional source into
a threedimensional target manifold although there are important
technical differences arising from the nonlinearities in the
governing equations. The biaxial case deals with maps from a
threedimensional source space into a fivedimensional target
manifold and presents a whole host of new complexities. We use
a combination of GinzburgLandau techniques and methods from
singular perturbation theory and harmonic map theory to prove
qualitative results on the structure, stability and dimension
of defect sets in equilibrium configurations on threedimensional
domains and describe the equilibrium behaviour away from the
defect set in terms of a limiting harmonic map. We also show
that biaxiality is inevitable in certain model situations, particularly
in the vicinity of defects. A part of this talk is joint work
with Arghir Zarnescu.

July
28
2pm
Room 230

Chris
King (Northeastern University)
The classical capacity of a quantum channel
Quantum channels describe the dynamical evolution of open
quantum systems. From the point of view of information theory,
a quantum channel is also the simplest quantum analog of the
discrete memoryless channel whose capacity for information
transmission was analyzed by Shannon more than sixty years
ago. This analogy has led to many interesting questions and
conjectures concerning the capacities of quantum channels.
In this talk I will describe recent progress regarding the
transmission of classical information through a quantum channel,
and in particular discuss recent counterexamples to the additivity
conjecture.

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