# SCIENTIFIC PROGRAMS AND ACTIVITIES

February 24, 2017

## 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)
 2006-07 Colloquium/ Seminar Series 2002-03 Colloquium/Seminar Series 2005-06 Colloquium/Seminar Series 2001-02 Colloquium/Seminar Series 2003-04 Colloquium/Seminar Series 2000-01 Colloquium/Seminar Series

### Overview

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.

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