# SCIENTIFIC PROGRAMS AND ACTIVITIES

October 23, 2016

## Fields Analysis Working Group 2010-11

A working group seminar and brown bag lunch devoted to nonlinear dynamics and the calculus of variations meeting once a week for three hours at the Fields Institute. The focus will be on working through some key papers from the current literature with graduate students and postdocs, particularly related to optimal transportation and nonlinear waves, and to provide a forum for presenting research in progress.
The format will consist of two presentations by different speakers, separated by a brown bag lunch.

Interested persons are welcome to attend either or both talks and to propose talks to the organizers Robert McCann and James Colliander <colliand@math.toronto.edu> and <mccann@math.toronto.edu>.

Seminars will be held on Thursdays at 12 noon in Room 210 unless stated otherwise. Please check the website regularly for updates.

 Fields Visitor Resources 2009-10 Working Seminar on Nonlinear Evolution Equation and Fields Analysis Working Group Directions

PAST SEMINARS

Apr. 7, 2011
12:10 pm
Room 210
Brendan Pass (University of Toronto)
Structural results on optimal transportation plans
This presentation consitutes the departmental thesis defense of a University of Toronto PhD student. In this thesis we prove several results on the structure of solutions to optimal transportation problems.
The second chapter represents joint work with Robert McCann and Micah Warren; the main result is that, under a non-degeneracy condition on the cost function, the optimal is concentrated on a n-dimensional Lipschitz submanifold of the product space. As a consequence, we provide a simple, new proof that the optimal map satisfies a Jacobian equation almost everywhere. In the third chapter, we prove an analogous result for the multi-marginal optimal transportation problem; in this context, the dimension of the support of the solution depends on the signatures of a family of semi-Riemannian metrics on the product space. In the fourth chapter, we identify sufficient conditions under which the solution to the multi-marginal problem is concentrated on the graph of a function over one of the marginals. In the fifth chapter, we investigate the regularity of the optimal map when the dimensions of the two spaces fail to coincide. We prove that a regularity theory can be developed only for very special cost functions, in which case a quotient construction can be used to reduce the problem to an optimal transport problem between spaces of equal dimension. The final chapter applies the results of chapter 5 to the principal-agent problem in mathematical economics when the space of types and the space of products differ. When the dimension of the space of types exceeds the dimension of the space of products, we show if the problem can be formulated as a maximization over a convex set, a quotient procedure can reduce the problem to one where the two dimensions coincide. Analogous conditions are investigated when the dimension of the space of products exceeds that of the space of types.

The thesis can be viewed at http://www.math.utoronto.ca/bpass/ut-thesis.pdf.
Everyone welcome. Refreshments will be served in the Bahen Math Lounge before the exam.

Mar. 31, 2011
12:10 pm
Room 210

Jordan Bell
Global well-posedness for KdV in L2
In this talk I will outline the proof of the bilinear estimate of Kenig, Ponce and Vega, and show how global well-posedness of KdV with initial data in L2 follows from it.

Mar. 24, 2011
12:10 pm
Room 210
Ivana Alexandrova (SUNY Albany)
Aharonov--Bohm Effect in Resonances of Magnetic Schrodinger Operators with Potentials with Supports at Large Separation

Vector potentials are known to have a direct significance to quantum particles moving in the magnetic field. This is called the Aharonov--Bohm effect and is known as one of the most remarkable quantum phenomena. Here we study this quantum effect through the resonance problem. We consider the scattering system consisting of two scalar potentials and one magnetic field with supports at large separation in two dimensions. The system has trajectories oscillating between these supports. We give a sharp lower bound on the resonance widths as the distances between the three supports go to infinity. The bound is described in terms of the backward amplitude for scattering by each of the scalar potentials and by the magnetic field, and it also depends heavily on the magnetic flux of the field. This is joint work with Hideo Tamura.

Feb. 17, 2011
12:10 pm
Room 210
Shibing Chen (University of Toronto)
Sharp new conformally invariant integral inequalities.

There is a well known inequality for harmonic functions by Carleman: $\int_{B_{2}}e^{2u}dx\leq\frac{1}{4\pi}(\int_{\partial B_{2}}e^{u}d\theta)^{2}$, for all harmonic functions in the unit disc $B_{2}$. This inequality was used by Carleman to prove the isoperimetric inequality on minimal surfaces, which was the first isoperimetric inequality on surfaces with variable Gaussian curvature. We give a new proof for this inequality, and furthermore from this new proof we can obtain some generalizations of this inequality to higher dimensions. However, we don't yet know whether the higher dimensional inequalities admit geometric explanations or interesting consequences.

Feb. 3, 2011
12:10 pm
Room 210

Jing Wang (University of Toronto)
An introduction to the Concentration-Compactness-Rigidity method at critical regularity

In this talk, firstly I will give a short introduction about Kenig and Merle' Concentration-Compactness-Rigidity (CCR) method, which is used to prove global wellposedness at critical regularity. The CCR method reduces the problem to the existence of some special solution called almost periodic solution. Secondly I will decribe some new ideas to preclude the almost periodic solution. This part mainly comes from Visan's paper, where she presented a new proof of global wellposenss of defocusing energy critical Schrodinger equation, based on the new idea introduced by Dodson at L2 critical.

Nov. 18, 2010
12:10 pm
Room 210

James Colliander (Toronto)
An expository talk about critical regularity for Navier-Stokes (following Koch-Kenig)

In this talk, I will describe recent work of G. Koch and C. Kenig. Beginning with breakthrough work of J. Bourgain, new ideas have been introduced to prove global well-posedness and understand the maximal-in-time behavior of wave equations at critical regularity. These ideas have been streamlined into a robust roadmap in recent works by Kenig and F. Merle. The paper I will describe in this talk applies this approach to a parabolic equation and reproduces (essentially) the state-of-the-art understanding of the Cauchy problem for the 3d Navier-Stokes system.

Nov. 11, 2010
12:10 pm
Room 210
TBA
Oct. 28, 2010
12:10 pm
Room 210
Jiakun Liu (Princeton University)
Global regularity of the reflector problem

In this talk we study a reflector system which consists of a point light source, a reflecting surface and an object to be illuminated. Due to its practical applications in optics, electro-magnetics, and acoustic, it has been extensively studied during the last half century. This problem involves a fully nonlinear partial differential equation of Monge-Ampere type, subject to a nonlinear second boundary condition. In the far field case, it is related to the reflector antenna design problem. By a duality, namely a Legendre type transform, Xu-Jia Wang has proved that it is indeed an optimal transportation problem. Therefore, the regularity results of optimal transportation can be applied. However, in the general case, the reflector problem is not an optimal transportation problem and the regularity is an extremely complicated issue. In this talk, we give necessary and sufficient conditions for the global regularity and briefly discuss their connection with the Ma-Trudinger-Wang condition in optimal transportation.

Oct. 21, 2010
12:10 pm
Room 210
Quentin Merigot (University of Toronto)
A Generalized Euler-Lagrange Equation for the Principal-Agent Problem
The principal-agent problem in economics can be modelized as a minimization problem for a certain functional L over the cone K of positive convex functions. This convexity constraints forbids the use of classical techniques of the calculus of variations: because the set K is very small (compared to, say, W^{1,2}), the solutions of the principal agent problem do not follow in general the classical Euler-Lagrange equation associated to L. Following Rochet and Chon we will show how to use the theory of dilatations to get a necessary and sufficient condition for optimality for the principal-agent problem, in the form of a generalized Euler-Lagrange equation.
Oct. 7, 2010
12:10 pm
Room 210

Shibing Chen (University of Toronto)
Regularity of solutions for a variational problem related to the Rochet-Chone model.
I will explain Caffarelli and Lions' proof of C11 regularity of solutions to the variational problem related to an economic model introduced by Rochet and Chone. Roughly speaking, letting u minimize an elliptic functional J(u) among all convex functions defined on a convex domain, they proved that the minimizer is a C11 function. The proof is based on a delicate perturbation argument.

Sept. 16, 2010
12:10 pm
Room 210
Colin Decker (University of Toronto)
Existence and uniquness of equilibrium in a noisy model of the marriage market

Economists model the marriage market with the aim of gaining insight into household behaviour and predicting population growth. Classical results in matching predict that agents of high quality in a certain characteristic will marry agents with high quality in that characteristic. However, real marriage data fails to have this structure, as agents of differing qualities are often seen to be paired in marriage. Recently (in Choo-Siow 2004), a more subtle model was created that uses random preferences to smear the predicted marriage distribution, so that agents are permitted to marry away from their own type. The predicted marriage distribution is an implicit function of parameters, and as a result it is not a priori clear whether it exists as a single valued function of them. Joint with Lieb, McCann, and Stephens, we answer this question in the affirmative and are able to qualitatively characterize how changes in parameters affect changes in the distribution.

July 23, 2010
3 pm
Room 210
Jochen Denzler (University of Tennessee at Knoxville)
Asymptotics of fast diffusion via dynamical systems
Joint work with Robert McCann and Herbert Koch
This gives an outline of how to implement rigorously a formalism that gives the convergence rate of fast diffusion to the (self-similar) Barenblatt solution along the ideas coming from linearized stability of dynamical systems.
July 14th, 2010
12:10 pm
Room 210
Young-Heon Kim (University of British Columbia)
Regularity of optimal transportation maps on multiple products of spheres

Optimal transportation seeks for a map which transports a given mass distribution to another, while minimizing the transportation cost. Existence and uniqueness of optimal transportation maps is well known on Riemannian manifolds where the transportation cost of moving a unit mass is given by the distance squared function. However, regularity (such as continuity and smoothness) of such maps is much less known, especially beyond the case of the round sphere and its small perturbations. Moreover, if the manifold has a negative curvature somewhere, there are discontinuous optimal maps even between smooth mass distributions. In this talk, we explain a regularity result of optimal maps on products of multiple round spheres of arbitrary dimension and size. This is a first such result given on non-flat Riemannian manifolds whose curvature is not strictly positive. This is joint work with Alessio Figalli and Robert McCann.