
THE
FIELDS INSTITUTE
FOR RESEARCH IN MATHEMATICAL SCIENCES 
Fields
Institute Colloquium/Seminar
in Applied Mathematics
20112012

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.
201112
Past Talks

May 2, 2012
12:10 p.m.
Fields Institute
Stewart Library

12:00  1:00 p.m.
Ingrid Carbone
(Università della Calabria, Consenza, Italy)
A new class of low discrepancy sequences of partitions
and points
An abstract of the talk is available here.
Slides of the talk are available here.

2:30  3:30 p.m.
Aljoša
Volcic (Università della Calabria, Consenza,
Italy)
Geometric Tomography
*This event is crosslisted with Thematic
Programme on Inverse Problems and Imaging
In 1963, Hammer posed the following problem.Suppose there
is a convex hole in an otherwise homogeneous solid and that
Xray pictures taken are so sharp that the darkness at each
point determines the length of a chord along an Xray line.
(No diusion, please.) How many pictures must be taken to
permit exact reconstruction of the body if:
a. The Xrays issue from a nite point source?
b. The Xrays are assumed parallel?
From a modern perspective, Hammers questions are clearly
geometrical variants of the sort of problems considered in
computerized tomography, the science behind the CAT scanner
used in most ma jor hospitals. Hammers Xray problem was a
ma jor inspiration for the development of geometric tomography,
the area of mathematics dealing with the retrieval of information
about a geometric ob ject from data concerning its sections,
or projections, or both. A full survey of geometric tomography
is provided in [7], from
Chapters 1 and 5 of which we present the following short summary
of the contributions to Hammers Xray problem relevant for
the present paper.
The earliest papers concern Hammers question (b). The (parallel)
Xray of a convex body K in the direction u is the function
giving the lengths of all the chords of K parallel to u. The
uniqueness aspect of question (b) is equivalent to asking
which nite sets of directions are such that the corresponding
Xrays distinguish between dierent convex bodies. Simple
examples show that there are arbitrarily large sets of directions
that do not have this desirable property and that no set of
three directions does. A complete solution was found by Gardner
and McMullen (1980) who proved that there are sets of four
directions in such that the Xrays of any planar convex body
in these directions determine it uniquely among all planar
convex bodies. Gardner and Gritzmann showed later that suitable
sets of four directions are those whose set of slopes, in
increasing order, have a rational crossratio not equal to
3/2, 4/3, 2, 3, or 4. It follows that if w1 = (1; 0), w2 =
(2; 1), w3 = (0; 1), and w4 = (?1; 2), for example,
then the directions ui = wi=kwik, i = 1; : : : ; 4 are such
that Xrays in these directions determine planar convex bodies.
The (point) Xray of a convex body K at a point p is the function
giving the lengths of all the chords of K lying on lines through
p. The uniqueness aspect of Hammers question (a) is not completely
solved, but it is known that a planar convex body K is determined
uniquely among all planar convex bodies by its Xrays taken
at
(i) two points such that the line through them intersects
K and it is known whether or not K lies between the two
points (Falconer and Gardner, 1983);
(ii) three points such that K lies in the triangle with
these points as vertices (Falconer and Gardner, 1983);
(iii) any set of four collinear points whose cross ratio
is restricted as in the parallel Xray case above (Gardner
1987);
(iv) any set of four points in general position (V., 1986).
Several algorithms have been proposed to provide reconstruction
and not only uniqueness results (Kolzow, Kuba and V. (1989),
Gardner and V.(1995), Lam and Solmon (2001)). More recently
Gardner and Kiderlen (2006) proved almost sure convergence
from noisy data of sequences of certain polygons constructed
using nitely many Xrays when uniqueness is guaranteed.
Even more recently Gritzmann, Langfeld and Wiegelmann (preprint
2012) found new interesting connections between geometric
tomography and discrete tomography.

April 12, 2012
12:10 p.m.
Fields Institute
Stewart Library

12:10  1:00 p.m.
Charles
Doering (University of Michigan)
"Ultimate state'' of twodimensional RayleighBténard
convection
RayleighBenard convection is the buoyancydriven flow
of a fluid heated from below and cooled from above. Heat transport
by convection an important physical process for applications
in engineering, atmosphere and ocean science, and astrophysics,
and it serves as a fundamental paradigm of modern nonlinear
dynamics, pattern formation, chaos, and turbulence theory.
Determining the transport properties of high Rayleigh number
convection turbulent convection remains a grand challenge
for experiment, simulation, theory, and analysis. In this
talk, after a general survey of the theory and applications
of RayleighBenard convection we describe recent results for
mathematically rigorous upper limits on the vertical heat
transport in two dimensional RayleighBenard convection between
stressfree isothermal boundaries derived from the Boussinesq
approximation of the NavierStokes equations. The bounds on
the heat transport scaling challenge some popular theoretical
arguments regarding the asymptotic high Rayleigh number convection.
This is joint work with Jared Whitehead.
2:10  3:00 p.m.
Evelyn
Lunasin (University of Michigan)
Global wellposedness for the 2D Boussinesq system with
anisotropic viscosity and without heat diffusion
I will present global existence and uniqueness theorems for
the twodimensional nondiffusive Boussinesq system with viscosity
only in the horizontal direction. This work improves the global
wellposedness results established recently by R. Danchin
and M. Paicu for the Boussinesq system with anisotropic viscosity
and zero diffusion. We follow some of their ideas, and in
proving the uniqueness result, we have used an alternative
approach by writing the transported temperature (density)
as $\theta = \triangle\xi$ and adapting the techniques of
V. Yudovich (1963) for the 2D incompressible Euler equations.
This new approach allows us to establish uniqueness results
with fewer assumptions on the initial data for the transported
quantity $\theta$. Furthermore, our proof is more elementary
in that we do not need to resort to using LittlewoodPaley
theory or the paraproduct calculus of J. Bony. This is joint
work with Adam Larios and Edriss S. Titi

March 21, 2012
2:10 p.m.
Fields Institute
Stewart Library

2:103:00
Christian
Lessig (Technische Universitaet Berlin and University
of Toronto)
The Geometry of Light Transport
Founded on Lambert's radiometry from the 18th century, light
transport theory describes the propagation of visible light
energy in macroscopic environments. While already in 1939
the theory was characterized as "a case of `arrested
development'", no reformulation has been undertaken
since then. Following recent literature, we develop the geometric
structure of light transport by studying the short wavelength
limit of a lifted representation of electromagnetic theory
on the cotangent bundle. This shows that light transport is
a Hamiltonian system with the transport of the light energy
density, the phase space representation of electromagnetic
energy, described by the canonical Poisson bracket. A noncanonical
Legendre transform relates light transport theory to geometric
optics, and by considering measurements, as did Lambert, we
are able to obtain classical concepts from radiometry. In
idealized environments where the Hamiltonian vector field
is defined globally, we show that light transport is a LiePoisson
system for the group Diff_{can}(T^*Q) of canonical transformations.
The Poisson bracket then describes the infinitesimal coadjoint
action in the Eulerian representation while the momentum map
yields the convective light energy density as Noetherian quantity.
The group structure also unveils a tantalizing similarity
between ideal light transport and the ideal Euler fluid, warranting
to consider the systems as configuration and phase space analogues
of each other.
3:104:00
David
Ambrose (Drexel University)
Two Existence Problems in Interfacial Fluid Dynamics
Much progress has been made in recent years in existence theory
for initial value problems in interfacial fluid dynamics.
We will introduce two other existence problems: the problem
of global weak solutions for interfacial flows with surface
tension, and the problem of timeperiodic interfacial flows.
We will report on progress for these problems, which includes
both analytical and numerical work. This is joint work with
Milton Lopes Filho, Helena Nussenzveig Lopes, Walter Strauss,
and Jon Wilkening.

Nov 23, 2011
2:10 p.m.
Fields Institute
Room 230

Nassif Ghoussoub
(University of British Columbia)
A selfdual polar decomposition for vector fields
I shall explain how any nondegenerate vector field on a bounded
domain of Rn is monotone modulo a measure preserving involution
S (i.e., S2=Identity). This is to be compared to Brenier's
polar decomposition which yields that any such vector field
is the gradient of a convex function (i.e., cyclically monotone)
modulo a measure preserving transformation. Connections to
mass transport which is at the heart of Brenier's decomposition
is elucidated.
This is joint work with A. Momeni


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