SCIENTIFIC PROGRAMS AND ACTIVITIES

2011-12 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.
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2:30 - 3:30 p.m.
Aljoša Volcic (Università della Calabria, Consenza, Italy)
Geometric Tomography
*This event is cross-listed 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 X-ray pictures taken are so sharp that the darkness at each point determines the length of a chord along an X-ray line. (No diusion, please.) How many pictures must be taken to permit exact reconstruction of the body if:

a. The X-rays issue from a nite point source?
b. The X-rays 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 X-ray 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 X-ray problem relevant for the present paper.
The earliest papers concern Hammers question (b). The (parallel) X-ray 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 X-rays 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 X-rays 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 cross-ratio 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 X-rays in these directions determine planar convex bodies.
The (point) X-ray 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 X-rays 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 X-ray 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 X-rays 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 two-dimensional Rayleigh-Bténard convection
Rayleigh-Benard convection is the buoyancy-driven 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 Rayleigh-Benard convection we describe recent results for mathematically rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-Benard convection between stress-free isothermal boundaries derived from the Boussinesq approximation of the Navier-Stokes 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 well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion
I will present global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with viscosity only in the horizontal direction. This work improves the global well-posedness 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 Littlewood-Paley 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

2:10-3:00
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 re-formulation 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 non-canonical 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 Lie-Poisson 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.

3:10-4:00
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 time-periodic 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 self-dual polar decomposition for vector fields
I shall explain how any non-degenerate 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