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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 nondegeneracy condition on the
cost function, the optimal is concentrated on a ndimensional 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 multimarginal
optimal transportation problem; in this context, the dimension of the
support of the solution depends on the signatures of a family of semiRiemannian
metrics on the product space. In the fourth chapter, we identify sufficient
conditions under which the solution to the multimarginal 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 principalagent 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/utthesis.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 wellposedness 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 wellposedness of KdV with
initial data in L2 follows from it.

Mar. 24, 2011
12:10 pm
Room 210 
Ivana Alexandrova (SUNY Albany)
AharonovBohm 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 AharonovBohm
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 ConcentrationCompactnessRigidity method
at critical regularity
In this talk, firstly I will give a short introduction about Kenig
and Merle' ConcentrationCompactnessRigidity (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 NavierStokes (following
KochKenig)
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 wellposedness and understand the maximalintime
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 stateoftheart
understanding of the Cauchy problem for the 3d NavierStokes 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, electromagnetics, and
acoustic, it has been extensively studied during the last half century.
This problem involves a fully nonlinear partial differential equation
of MongeAmpere 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, XuJia 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 MaTrudingerWang condition
in optimal transportation.

Oct. 21, 2010
12:10 pm
Room 210 
Quentin Merigot (University of Toronto)
A Generalized EulerLagrange Equation for the PrincipalAgent Problem
The principalagent 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 EulerLagrange 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 principalagent
problem, in the form of a generalized EulerLagrange equation.

Oct. 7, 2010
12:10 pm
Room 210 
Shibing Chen (University of Toronto)
Regularity of solutions for a variational problem related to the
RochetChone 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 ChooSiow 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 (selfsimilar) Barenblatt
solution along the ideas coming from linearized stability of dynamical
systems.

July 14th, 2010
12:10 pm
Room 210 
YoungHeon
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 nonflat Riemannian manifolds whose curvature is not
strictly positive. This is joint work with Alessio Figalli and Robert
McCann.
