SCIENTIFIC PROGRAMS AND ACTIVITIES

Susan Friedlander, University of Illinois at Chicago
Onsager's Conjecture and a Model for Turbulence.
We discuss properties of a shell type model for the inviscid fluid equations. We prove that the forced system has a unique equilibrium which is an exponential global attractor. Every solution blows up in H5 / 6 in finite time . After this time, all solutions stay in Hs,s < 5 / 6, and "turbulent" dissipation occurs. Onsager' conjecture is confirmed for the model system. We discuss how intuition obtained from the model carries over to the 3-D Euler equations where we obtain the "sharp" space for Onsager's conjecture concerning energy conservation.
This is joint work with Alexey Cheskidov, Natasa Pavlovic, Roman Shvydkoy and Peter Constantin.

(This talk is based on joint works by the author with L. Ambrosio and T. Pacini)

In this talk, I will briefly discuss this construction, and some low-genus examples (in particular, Kirchhoff elastic rod centerlines) where this correlation is well understood. I will mainly discuss recent joint work with Annalisa Calini, describing how to generate a family of closed finite-gap solutions of increasingly higher genus via a sequence of deformations of the multiply covered circle. We prove that every step in this sequence corresponds to constructing a cable on previous filament; moreover, the cable's knot type (which is invariant under the evolution)
can be read off from the deformation sequence.

Lars Nesheim, University College, London
Nonlinear hedonic pricing: finding equilibria through linear programming
First, we consider a general framework for studying hedonic prices problems with quasi-linear preferences, and show that it is equivalent to a matching model with transferable utilities. From a mathematical perspective, both problems can in turn be rephrased under the form of a linear program, in fact an optimal transportation problem of
Monge-Kantorovich form.

Secondly, we provide a general existence result for the models under consideration. Our results apply to multi-dimensional problems; in addition, they do not require single crossing conditions à la Spence-Mirrlees. We also give a generalization of the Spence-Mirrlees condition which guarantees uniqueness of the stable match (in the matching model) or of the equilibrium (in the hedonic model) even in the absence of one-to-one matching; and we discuss an example of the consequences of relaxing the condition.

This represents joint work with Robert McCann (University of Toronto).

This is joint work with Fraydoun Rezakhanlou.

This is joint work with Robert McCann (University of Toronto).

Dorian Goldman, University of Toronto
Chaotic dynamics in various models in Meteorology, Part II.
I will introduce 3 well known models used extensively in meteorology that are approximations to the full Navier Stokes equations in a rotating frame of reference. The three models are the 2D Euler equations, the Quasigeostrophic and finally the
Semigeostrophic approximation. I will derive these approximations
and show that when the initial data corresponds to regions of uniform elliptial (ellipsoidal in the case of the 3D Quasigeostrophic model) potential vorticity (which is defined analagously to the vorticity for 2D Euler in the cases of Quasi and
Semigeostrophy), that the dynamics are very similar and the phase
portraits are essentially equivalent. In particular the Semigeostrophic equations, modelled in dual variables take the form of a Monge ampere equation, transforming the problem into an optimal transportation problem. In particular each phase portrait contains a hyperbolic fixed point with homoclinic saddle connection. Extending the result of Bertozzi (1988) who used Melnikov analysis to
demonstrate that the 2D Euler equations responded chaoticically to
vertical stretching of the elliptical columnm I will show the method
extends to the Semigeostrophic equations and most likely to the
Quasigeostrophic equations as well.

Dorian Goldman, University of Toronto
Chaotic dynamics in various models in Meteorology, Part I.
I will introduce 3 well known models used extensively in meteorology that are approximations to the full Navier Stokes equations in a rotating frame of reference. The three models are the 2D Euler equations, the Quasigeostrophic and finally the Semigeostrophic approximation. I will derive these approximations and show that when the initial data corresponds to regions of uniform elliptial (ellipsoidal in the case of the 3D Quasigeostrophic model) potential vorticity (which is defined analagously to the vorticity for 2D Euler in the cases of Quasi and Semigeostrophy), that the dynamics are very similar and the phase portraits are essentially equivalent. In particular the Semigeostrophic equations, modelled in dual variables
take the form of a Monge ampere equation, transforming the problem into an optimal transportation problem. In particular each phase portrait contains a hyperbolic fixed point with homoclinic saddle connection. Extending the result of Bertozzi (1988) who used Melnikov analysis to demonstrate that the 2D Euler equations responded chaoticically to
vertical stretching of the elliptical columnm I will show the method extends to the Semigeostrophic equations and most likely to the Quasigeostrophic equations as well.

[1] E. DiBenedetto, U. Gianazza, V. Vespri, Harnack Estimates for Quasi-Linear Degenerate Parabolic Differential Equation, preprint IMATI 4-PV 2006, 1-27, to appear in Acta Mathematica.

Paul Lee, University of Toronto
Infinite-dimensional geometry of optimal mass transport
In the 60's, Arnold shows that the Euler equation can be thought of as the geodesic flow on the group of all volume preserving diffeomorphism. In a similar spirit, Otto shows that the mass transport problem can be consider as the geodesic problem on the space $W$ of all volume forms with the same total volume. In particular, the space $W$ can be regarded as the quotient of the group of all diffeomophisms by the subgroup of volume preserving ones, while the geodesic flow on the diffeomorphism group, given by the Burger's equation, is closely related to that on the space $W$. It turns out that this relation between diffeomorphism group and the space $W$ can be understood via Hamiltonian reduction.
We also consider the following nonholonomic version of the classical Moser theorem: given a bracket generating distribution on a manifold, two volume forms of equal total volume can be isotoped by the flow of a vector field
tangent to this distribution. We discuss these results from the point of view of an infinite-dimensional non-holonomic distribution on the diffeomorphism groups.

18 Jan, 2007
12:55 - 1:55

Ben Stephens, University of Toronto
Modulational stability of ground states of nonlinear Schroedinger equation
The nonlinear Schroedinger equation has a family of ground states related to the scaling invariances of the equation. I will present Weinstein's paper showing how states starting near this ground state family stay near the family as time increases.

7 Dec, 2006
11:10 - 12:10
Room 230

John Lott, University of Michigan
The work of Grigory Perelman
This will be a exposition, for nonspecialists, of Perelman's work on Ricci flow.

7 Dec, 2006
1:00 - 2:00
Room 230

Izabella Laba, University of British Columbia
Harmonic analysis and incidence geometry
Many problems in Euclidean harmonic analysis, especially restriction theory, lead to purely geometric questions, typically involving arrangements of large families of ``thin" objects such as lines, circles, or surfaces. Conversely, geometric measure-theoretic questions involving projections or intersections of sets are often approached via harmonic analytic methods.

While the connections between harmonic analysis and geometric measure theory go back several decades, it was only in the 1990s that harmonic analysts began to explore links to the area of combinatorics known as discrete geometry, or more specifically incidence geometry. The latter studies many of the same questions that harmonic analysts are interested in - for example, projection sets, distance sets, or arrangements of families of thin objects - but in a discrete setting, as opposed to the geometric measure-theoretic setting encountered in harmonic analysis.

The purpose of this talk is to give a brief survey of this area of research, including both classical results and current directions, with emphasis on the connections between the analysis and the combinatorics.

Ben Stephens, University of Toronto
Modulational stability of ground states of nonlinear Schroedinger equation
The nonlinear Schroedinger equation has a family of ground states related to the scaling invariances of the equation. I will present Weinstein's paper showing how states starting near this ground state family stay near the family as time increases

Nikos Tzirakis, University of Toronto
Global well-posedenss and scattering for the defocusing energy-critical nonlinear Schrodinger equation in \mathbb{R}^{1+4}. (CONTINUED)
I will present the global well-posedness result of M.Visan and E. Ryckman for the energy-critical defocusing NLS in \mathbb{R}^{1+4}. The main step of the proof is the establishment of a global L_t^6 L_x^6 bound for the solution. The result is based on previous results for the \mathbb{R}^{1+3} problem by Bourgain (radial assumption on the initial data) and Colliander, Keel, Staffilani, Takaoka and Tao (no assumptions on the data).

19 Oct. 2006, 11:10AM