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Fields Institute Colloquium/Seminar in Applied Mathematics
2007-2008
| Organizing Committee |
|
Jim Colliander (Toronto)
Walter Craig (McMaster)
Barbara Keyfitz (Fields) |
Robert McCann (Toronto)
Adrian Nachman (Toronto)
Mary Pugh (Toronto)
Catherine Sulem (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 super-conductivity,
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.
Schedule - Future talks to be
held at the Fields Institute
|
Tuesday
June 24
11:10am
room 230 |
James
Hill (School of Mathematics and Applied Statistics,
University of Wollongong)
Geometry and mechanics of carbon nanotubes and gigahertz nano-oscillators.Fullerenes
and carbon nanotubes are of considerable interest due to their
unique properties, such as low weight, high strength, flexibility,
high thermal conductivity and chemical stability and they
have many potential applications in nano-devices. In this
talk we present some recent new results on the geometric structure
of carbon nanotubes and related nanostructures. One concept
that has attracted much attention is the creation of nano-oscillators,
to produce frequencies in the gigahertz range, for applications
such as ultra-fast optical filters and nano-antennae. The
sliding of an inner shell inside an outer shell of a multi-walled
carbon nanotube can generate oscillatory frequencies up to
several gigahertz, and the shorter the inner tube the higher
the frequency. A C60-nanotube oscillator generates high frequencies
by oscillating a C60 fullerene inside a single-walled carbon
nanotube. Here we discuss the underlying mechanisms of nano-oscillators
and some recent results using the Lennard-Jones potential
together with the continuum approach to mathematically model
three different types of nano oscillators including double-walled
carbon nanotube, C60-nanotube and C60-nanotorus oscillators.
|
| PAST
TALKS 2007-08 |
April 2
3:10 p.m. |
Yuri
A. Kordyukov (Russian Academy of Sciences, Ufa, Russia)
Slides of talk
Spectral gaps for periodic Schroedinger operators with magnetic
wells
Consider a periodic Schroedinger operator with magnetic wells
on a noncompact, simply connected, Riemannian manifold equipped
with a properly disconnected, cocompact action of a finitely
generated, discrete group of isometries. We will discuss sufficient
conditions on the magnetic field, which ensure the existence
of a gap (or, even more, an arbitrarily large number of gaps)
in the spectrum of such an operator in the semi-classical
limit. The proofs are based on the study of the tunneling
effect in the corresponding quantum system. This is joint
work with B. Helffer.
|
March 19
3:10 p.m. |
Govind
Menon, Brown University
Min-driven clustering
The study of domain coarsening in the Allen-Cahn equation
has several interesting dynamical aspects such as metastability
and connections with a hierarchy of reduced models for clustering.
Motivated by this problem, we consider a process (`min-driven
clustering') that may be described informally as follows: at
each step a random integer $k$ is chosen with probability $p_k$
and the smallest cluster merges with $k$ randomly chosen clusters.
We study a mean-field model of this process. We prove optimal
results on well-posedness, the approach to self-similarity,
and the classification of eternal solutions. The analysis
relies on an explicit solution formula discovered by Gallay
and Mielke, and a careful choice of time scale.
This is work with Barbara Niethammer (Oxford) and Bob Pego
(Carnegie Mellon).
|
March 12
3:00 p.m.**
New time |
Horng-Tzer
Yau, Harvard University Slides of
talk
Dynamics of Bose-Einstein Condensates
Consider a system of $N$ bosons interacting via a repulsive
short range pair potential. Let $\psi_{N,t}$ be the
solution to the Schrödinger equation of the N-particle
dynamics. We prove that the one-particle density matrix of
$\psi_{N,t}$ solves the time-dependent Gross-Pitaevskii equation,
a cubic non- linear Schrödinger equation. We shall also
review general problems related to quantum dynamics of N particle
systems.
|
Feb 27,
2008
2:10 p.m. |
Jerry Bona (University
of Illinois at Chicago)
Recent results in nonlinear wave theory |
|
Feb. 6, 2008
3:10 p.m.
|
Kehinde Ladipo (University of Houston)
Finite Element Analysis of Fluid motion in Conical Diffusers
- Part I
A finite element analysis of the flow of an incompressible
Newtonian fluid through a conical diffuser is presented. Time
discretization of the equations of motion by three-operator
splitting is combined with the wave-like-equation method of
treating advection. The effect of the diffuser-included angle
on the fluid motion is investigated. The objective of this
work is to develop an efficient finite element model for conical
diffusers and use the model to determine the optimal diffuser-included
angle that will eliminate (or reduce to a negligible level),
the re-circulation region that usually develops behind the
smaller diameter pipe. The re-circulation is as a result of
flow separation which also translates to pressure losses across
the diffuser. Results are presented for the numerical simulation
using diffuser-included angles q = 28.08 degrees and 22.60
degrees, and diffuser-diameter ratio 1.5. Plots of the streamlines
and velocity contours, as well as the horizontal velocity
profile revealed the expected re-circulation region when the
included diffuser angle is large. The length of the re-circulation
region, determined from the streamlines and contour plots,
provided a prediction of the appropriate range of included
angles that can eventually be applied to model a diffuser
that will be re-circulation free.
|
Jan
23, 2008
3:10 p.m. |
Reinhard Illner,
University of Victoria
From Fokker-Planck type kinetic traffic models tostop-and-go
waves in dense traffic
We discuss kinetic models of Fokker-Planck type for multilane
traffic flow and compare them with models of conservation law
type from conceptual and practical points of view. The kinetic
models allow calculations of fundamental diagrams (density-flux
diagrams) in equilibrated traffic and offer in particular an
explanation why such diagrams appear to be multi-valued when
lane changing is included. The modeling suggests that lane-changing
is necessary for this phenomenon to occur, and allow to predict
fluxes as functions of density with or without lane changes.
If, in dense traffic, "diffusive' effects in driver behaviour
becomes small, the Fokker-Planck models degenerate into a Vlasov-type
kinetic equation with spatial nonlocality (nonlocality is a
hallmark of all these models). An ansatz $f(x,v,t)= \rho(x,t)
\delta(v-u(x,t))$ leads to macroscopic equations for $(\rho,u).$
Eliminating the nonlocality by Taylor approximations leads to
the pressureless gas dynamics equations at the zeroth order,
to PDEs of conservation type (more precisely, of Hamilton-Jacobi
type) like the Aw-Rascle model at the first order, and to a
system of equations of Hamilton-Jacobi equations with diffusive
corrections at second order. This latter case looks complicated,
but a search for traveling wave solutions produces traveling
waves that emulate the phenomenon of stop-and-go wave formation
on freeways. For each wave speed there appears to be a velocity
domain where traveling waves of that speed will not form because
the constant state $(\rho,u)$ is stable.
The latter work is a recent and ongoing collaboration with M.
Herty. The models were inspired by traffic observations made
by B. Kerner on the German autobahn, and our results are consistent
with these observations, at least from a qualitative point of
view.
|
November
13,
2:10 pm |
Isom
Herron , Rensselaer Polytechnic Institute. |
A new look at the principle of exchange of stabilities
In the classic work of Chandrasekhar, Hydrodynamic and Hydromagnetic
Stability, one of the most referenced ideas is this principle,
which is now described as "In the linearized stability
problem, the first unstable eigenvalue has imaginary part equal
to zero". For some problems, this situation is clear, when
the underlying operator is self adjoint. For other problems,
this principle has defied suitable verification. We have developed
techniques based on the analyzing the resolvent structure such
as, among other things, positive Green's functions as oscillation
kernels, which verifies this in diverse contexts: Taylor-Couette
flow, convection problems and others. |
|
November 13, 3:10pm.
|
Laurette
Tuckerman, (PMMH-ESPCI) University of Pierre and
Marie Curie
Patterns in Turbulence
The greatest mystery in fluid dynamics, and perhaps in all
of physics, is transition to turbulence. The simplest shear
flow, plane Couette flow -- the flow between parallel plates
moving at different velocities -- is linearly stable for all
Reynolds numbers (nondimensionalized velocity gradients), but
nevertheless undergoes sudden transition to 3D turbulence at
Reynolds numbers near 325. At precisely these Reynolds numbers,
it was recently discovered experimentally that there appears
a steady and regular pattern of alternating wide turbulent and
laminar bands, tilted at an angle with respect to the direction
of motion of the bounding plates. We report on numerical simulations
of this remarkable flow. |
Oct
31, 2007
Fields Institute
3:10pm |
Joint
Fields/Physics Colloquium
Jun Zhang,
NYU
Free-moving boundaries interacting with thermal convective
fluids
Thermal convection has come to be regarded as one of the
most important prototypical systems of dynamical systems. It
has been extensively studied over the past 3 decades or so.
An experimental system often consists of a fluid confined within
a rigid box that is heated at the bottom and cooled
at the top.
Our experimental studies explore the intriguing phenomena when
its rigid boundary is partly replaced either by a freely moving,
thermally opaque (which reduces local heat transport) "floater"
or by a collection of free-rolling spheres (a deformable mass).
We identify from our table-top experiments several dynamical
states, ranging from oscillation to localization to intermittency.
A phenomenological, low-dimensional model seems to reproduce
most of the experimental results. Through our on-going experiments,
we further seek their possible implications in geophysical processes
such as continental drift.
This colloquium is jointly sponsored by the Department of Physics
and the Fields Institute. |
Nov
1, 2007 4:10pm, **McLennan Physics
MP 102 **
Note location |
Joint Fields/Physics Colloquium
Jun Zhang,
NYU
The unidirectional flight of flapping wings
The locomotion of most fish and birds is realized by flapping
their wings or fins transverse to the direction of travel.
Here, we study experimentally the dynamics of a symmetric
wing that is "flapped" up and down but is free to
move in the horizontal direction. In this table-top prototypical
experiment, we show that flapping flight occurs abruptly at
a critical flapping frequency as a symmetry-breaking bifurcation.
We then investigate the separate effects of the flapping frequency,
the flapping amplitude, the wing geometry and the influence
from the solid boundaries nearby. Through dimensional analysis,
we found that there are two dimensionless parameters well
describe this intriguing problem that deals with fluid-solid
interaction. The first one is the dynamical aspect ratio that
combines four length scales, which includes the wing geometry
and the flapping amplitude. The second parameter, the Strouhal
number, relates the flapping efforts in the vertical direction
to the resultant forward flight speed. We also investigated
the effect of flexibility and passive pitching of the wings.
We find that these help to increase the flight speed significantly,
as observed in our experiments.
This colloquium is jointly sponsored by the Fields Institute
and the Department of Physics.
|
Oct
17, 2007
3:10 p.m. |
AUDIO OF TALK
Weinan
E, Princeton
Mathematical theory of solids: From atomic to macroscopic
scales
I will give an overview of a program on building a mathematical
theory of crystalline solids, starting from atomistic models.
I will discuss what the crucial issues are. I will start by
reviewing the geometry of crystal lattices, the quantum as
well as classical atomistic models of solids. I will then
focus on a few selected problems:
(1) The crystallization problem -- why the ground states of
solids are crystals and which crystal structure do they select?
(2) stability of crystals;
(3) instability of crystals;
(4) the generalized Peierls-Nabarro model for defects in crsytals.
|
Oct
10, 2007
3:10 p.m. |
AUDIO
OF TALK
Robert
MacPherson, IAS, Princeton
The Geometry of Grains
A metal or ceramic is naturally decomposed into cells
called "grains". The geometry of this cell complex
influences the properties of the material. Some interesting
mathematical problems arise in trying to understand the time
evolution of these grains. In 1952, von Neumann gave a simple
formula for the growth rate of a grain in 2 dimensions, which
has been used as the basis for much of the work on grain evolution.
This formula will be generalized to 3 (and higher) dimensions
(joint work with David Srolovitz). The generalization relies
on a good notion of the linear dimension of a 3 dimensional
grain called the "mean width", which should be useful
in other contexts.
|
Oct.
3, 2007
3:10 p.m. |
Michel
Chipot, University of Zurich
Exponential rate of convergence for
the solution of elliptic problems in strips |
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