PAST SEMINARS

July 28th, 2009
(Tuesday)
Fields
Room 210
12:00  1:00 pm

Filomena Feo (Universita di Napoli "Parthenope")
Comparison results for equations related to Gauss measure
Abstract

August 11th
(Tuesday)
Fields
Room 210
12:00  1:00 pm 
MaxKonstantin von Renesse
(Technische Universitaet Berlin)
An Optimal Transport Perspective on
the Schroedinger Equation
We show that the Schroedinger equation is a lift of Newton's
2nd law of motion to the space of probability measures, on
which derivatives are taken w.r.t. the Wasserstein Riemannian
metric.Here the potential is the is sum of the total classical
potential energy of the extended system, plus its Fisher information.
The precise relationship is established via a well known (`Madelung')
transform which is shown to be a symplectic submersion of
the standard symplectic structure of complex valued functions
into the canonical symplectic space over the Wasserstein Riemannian
manifold.

August 25th
(Tuesday)
Fields
*** STEWART LIBRARY***
12:00  1:00 pm 
Emanuel Milman (University of Toronto)
A Generalization of Caffarelli's Contraction Theorem via
(reverse) Heat Flow
A theorem of Caffarelli asserts that the optimal transport
map $T$ (for the quadratic cost in Euclidean space) between
the Gaussian probability measure $\gamma$ and any probability
measure of the form $\gamma \exp(V)$, where $V$ is a convex
function, is necessarily a contraction: $T(x)  T(y) \leq
xy$. We generalize this result for some more general measures
$\gamma$, but less general convex functions $V$, using a different
map $T$. Contrary to the optimal transport map, our map is
constructive, in the sense that its inverse is obtained by
running along a specific heat flow having $\gamma$ as its
invariant measure. Some applications and further insights
on the optimal map will also be discussed.

September 24th
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Robert McCann (University of Toronto)
An optimal multidimensional price strategy facing informational
asymmetry
The monopolist's problem of deciding what types of products
to manufacture and how much to charge for each of them, knowing
only statistical information about the preferences of an anonymous
field of potential buyers, is one of the basic problems analyzed
in economic theory. The solution to this problem when space
of products and of buyers can each be parameterized by a single
variable (say quality X, and income Y) garnered Mirrlees (1971)
and Spence (1974) their Nobel prizes in 1996 and 2001, respectively.
The multidimensional version of this question is a largely
open problem
in the calculus of variations described in Basov's book "Multidimensional
Screening". I plan to give a couple of lectures explain
recent progress with A Figalli and YH Kim, which identifies
structural conditions on the value b(X,Y) of product X to
buyer Y, which reduce this problem to a convex program in
a Banach space leading to uniqueness and stability results
for its solution, confirming robustness of certain economic
phenomena observed by Armstrong (1996) such as the desirability
for the monopolist to raise prices enough to drive a positive
fraction of buyers out of the
market, and yielding conjectures about the robustness of other
phenomena observed Rochet and Chone (1998), such as the clumping
together of products marketed into subsets of various dimension.
Ideas from differential geometry / general relativity and
optimal transportation are relevant to passage to several
dimensions.

October 1st
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Robert McCann (University of Toronto)
An optimal multidimensional price strategy facing informational
asymmetry  Part II
The monopolist's problem of deciding what types of products
to manufacture and how much to charge for each of them, knowing
only statistical information about the preferences of an anonymous
field of potential buyers, is one of the basic problems analyzed
in economic theory. The solution to this problem when space
of products and of buyers can each be parameterized by a single
variable (say quality X, and income Y) garnered Mirrlees (1971)
and Spence (1974) their Nobel prizes in 1996 and 2001, respectively.
The multidimensional version of this question is a largely
open problem
in the calculus of variations described in Basov's book "Multidimensional
Screening". I plan to give a couple of lectures explain
recent progress with A Figalli and YH Kim, which identifies
structural conditions on the value b(X,Y) of product X to
buyer Y, which reduce this problem to a convex program in
a Banach space leading to uniqueness and stability results
for its solution, confirming robustness of certain economic
phenomena observed by Armstrong (1996) such as the desirability
for the monopolist to raise prices enough to drive a positive
fraction of buyers out of the
market, and yielding conjectures about the robustness of other
phenomena observed Rochet and Chone (1998), such as the clumping
together of products marketed into subsets of various dimension.
Ideas from differential geometry / general relativity and
optimal transportation are relevant to passage to several
dimensions.

October 8th
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Colin Decker (University of Toronto)
Uniqueness of matching in the marriage market
Economists are interested in studying marriage behaviour
because it provides insight into a basic economic unit, the
household, and because changes in marital behaviour offer
insight into other social and economic variables of interest.
Given agents described by multi dimensional discrete types,
and their preferences, a competitive model of the marriage
market describes how individuals will arrange themselves in
marriage. Whether this described arrangement is unique is
a key question and one that recurs in the study of matching
markets.
Choo and Siow (2006) introduced a competitive model of the
marriage market that incorporates several important features
from economic theory. It is not known whether the ChooSiow
model predicts a unique marital arrangement given the preferences
of agents. I will identify sufficient conditions on the preferences
of agents to guarantee the existence of a unique marital arrangement.
To achieve this result, Robert McCann, Ben Stephens and I
adapted the continuity method, commonly used in the study
of elliptic PDE, to the setting of isolating the positive
roots of a system of polynomial equations.

October 15th
(Thursday)
Fields
Room 210
12:00  1:00 pm 
R.J. McCann (University of Toronto)
An optimal multidimensional price strategy facing informational
asymmetry, Part III
The monopolist's problem of deciding what types of products
to manufacture and how much to charge for each of them, knowing
only statistical information about the preferences of an anonymous
field of potential buyers, is one of the basic problems analyzed
in economic theory. The solution to this problem when space
of products and of buyers can each be parameterized by a single
variable (say quality X, and income Y) garnered Mirrlees (1971)
and Spence (1974) their Nobel prizes in 1996 and 2001, respectively.
The multidimensional version of this question is a largely
open problem in the calculus of variations described in Basov's
book "Multidimensional Screening". I plan to give
a couple of lectures explain recent progress with A Figalli
and YH Kim, which identifies structural conditions on the
value b(X,Y) of product X to buyer Y, which reduce this problem
to a convex program in a Banach space leading to uniqueness
and stability results for its solution, confirming robustness
of certain economic phenomena observed by Armstrong (1996)
such as the desirability for the monopolist to raise prices
enough to drive a positive fraction of buyers out of the market,
and yielding conjectures about the robustness of other phenomena
observed Rochet and Chone (1998),such as the clumping together
of products marketed into subsets of various dimension. Ideas
from differential geometry / general relativity and optimal
transportation are relevant to passage to several dimensions.

October 22nd
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Paul Lee (University of California Berkeley)
The MaTrudingerWang conditions for natural mechanical
actions.
The MaTrudingerWang conditions are important necessary
conditions for the regularity theory of optimal transportation
problems. In this talk, we will discuss new costs arising
from natural mechanical actions which satisfy this condition.
This is a joint work with R. McCann.

October 29th
(Thursday)
Fields
Room 210
12:00  1:00 pm 
J. Colliander (University of Toronto)
Towards partial regularity for nonlinear Schrödinger?
What happens at the end of life of an exploding solution of
a nonlinear Schrödinger equation? This talk will describe
ideas toward a description of the set of points where the
solution becomes singular. In particular, a heuristic argument
suggesting Hausdorff dimension upper bounds on the singular
set will be presented. These upper bounds are saturated by
recent examples of blowup solutions with thick singular sets.
Comparisons with corresponding results for NavierStokes and
other equations will also be discussed.

November 5th
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Elliott Lieb (Princeton University)
Mathematics of the Bose Gas: A truly quantummechanical
manybody problem
The peculiar quantummechanical properties of the lowest
energy states of Bose gases that were predicted in the early
days of quantummechanics have finally been verified experimentally
recently. The mathematical derivation of these properties
from Schroedinger's equation has also been difficult, but
much progress has been made in the last few years and some
of this will be reviewed in this talk. For the low density
gas with finite range interactions these properties include
the leading order terms for the lowest state energy, the validity
of the GrossPitaevskii equation in traps (including rapidly
rotating traps), BoseEinstein condensation and superfluidity,
and the transition from 3dimensional behavior to 1dimensional
behavior as the crosssection of the trap decreases. The phenomena
described are highly quantummechanical, without a classical
physics explanation, and it is very satisfying that reality
and these mathematical predictions agree.

November 19th
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Robert Jerrard (University of Toronto)
Partial regularity for hypersurfaces minimizing elliptic
parametric integrands
I will give one or two expository talks on an old paper
of Schoen, Simon, and Almgren in which they prove that hypersurfaces
in R^{n+1} that solve certain geometric variational problems
are smooth away from a closed set of n2 dimensional Hausdorff
measure 0. The geometric variational problems in question
 the "parametric elliptic integrands" mentioned
in the title  should be thought of as generalizations of
the minimal surface problem.

November 26th
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Robert Jerrard (University of Toronto)
Partial regularity for hypersurfaces minimizing elliptic
parametric integrands  Cont.
I will give one or two expository talks on an old paper
of Schoen, Simon, and Almgren in which they prove that hypersurfaces
in R^{n+1} that solve certain geometric variational problems
are smooth away from a closed set of n2 dimensional Hausdorff
measure 0. The geometric variational problems in question
 the "parametric elliptic integrands" mentioned
in the title  should be thought of as generalizations of
the minimal surface problem.

December 3rd
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Laszlo Erdös (Munich)
Dynamical formation of correlations
in a BoseEinstein condensate
We consider the evolution of N bosons interacting with a
repulsive short range pair potential in three dimensions.
The potential is scaled according to the GrossPitaevskii
scaling, i.e. it is given by N2V(N(xi ? xj)). We monitor the
behavior of the solution to the Nparticle Schrödinger
equation in a spatial window where two particles are close
to each other. We prove that within this window a short scale
interparticle structure emerges dynamically. The local correlation
between the particles is given by the twobody zero energy
scattering mode. This is the characteristic structure that
was expected to form within a very short initial time layer
and to persist for all later times, on the basis of the validity
of the GrossPitaevskii equation for the evolution of the
BoseEinstein condensate. The zero energy scattering mode
emerges after an initial time layer where all higher energy
modes disperse out of the spatial window.
This is a joint work with A. Michelangeli and B. Schlein.

***POSTPONED***
November 26th
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Almut Burchard (University of Toronto)
Competing Symmetries and convergence of sequences of random
symmetrizations
Rearrangements change the shape of a function while preserving
its size. The symmetric decreasing rearrangement, which is
used for finding extremals of functionals that involve gradients
or convolutions, replaces a given function f with a radially
decreasing function f^*.
The symmetric decreasing rearrangement can be approximated
by sequences of simpler rearrangements, such as Steiner symmetrizations
or polarization. In this talk, I will discuss the convergence
of random Steiner symmetrizations to the symmetric decreasing
rearrangement. The Competing Symmetries technique of Carlen
and Loss will be explained in detail.
References:
A. Volcic, Random Steiner symmetrization of measurable sets.
http://arxiv.org/abs/0902.0462
A. Burchard, Short course on rearrangements (Section 3.2).
http://www.math.utoronto.ca/almut/rearrange.pdf
A. Burchard, Steiner Symmetrization is continous in W^{1,p}
(Theorem 3 and Sections 67). GAFA 7 (1997), 823860.
http://www.math.utoronto.ca/almut/preprints/steiner.ps

January 7, 2010
(Tuesday)
Fields
Room 210
12:00  1:00 pm 
Oana Pocovnicu (Orsay)
Traveling waves for the cubic Szeg equation on the real line
We consider the cubic Szeg equation on the real line. This equation
was introduced by Grard and Grellier on the circle as a toy
model for nondispersive evolution equations in studying the
nonlinear Schrdinger equation on a subRiemannian manifold.
It turns out that this equation is completely integrable. i.e.,
it has a Lax pair and there is an infinite sequence of conserved
quantities. In this talk, after discussing its wellposedness
in the Hardy space on the upper halfplane, we show that the
only traveling waves are of the form C=(xp) , with Im p<0
. Moreover, they are shown to be stable, in contrast to the
situation on the circle where some traveling waves were shown
to be unstable.

January 14th,
2010
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Almut Burchard (University of Toronto)
Competing Symmetries and convergence of sequences of random
symmetrizations
Rearrangements change the shape of a function while preserving
its size. The symmetric decreasing rearrangement, which is
used for finding extremals of functionals that involve gradients
or convolutions, replaces a given function f with a radially
decreasing function f^*.
The symmetric decreasing rearrangement can be approximated
by sequences of simpler rearrangements, such as Steiner symmetrizations
or polarization. In this talk, I will discuss the convergence
of random Steiner symmetrizations to the symmetric decreasing
rearrangement. The Competing Symmetries technique of Carlen
and Loss will be explained in detail.
References:
A. Volcic, Random Steiner symmetrization of measurable sets.
http://arxiv.org/abs/0902.0462
A. Burchard, Short course on rearrangements (Section 3.2).
http://www.math.utoronto.ca/almut/rearrange.pdf
A. Burchard, Steiner Symmetrization is continous in W^{1,p}
(Theorem 3 and Sections 67). GAFA 7 (1997), 823860.
http://www.math.utoronto.ca/almut/preprints/steiner.ps

January 21st
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Ian Zwiers (University of Toronto)
http://www.math.utoronto.ca/izwiers/
Minimal NavierStokes Singularities
Suppose that in three dimensions there exists a solution
to NavierStokes that forms a singularity in finite time (for
data in H21_ ). Then there exists such data of minimal norm.
This is a recent result of Rusin and ?verák, building
on LemariéRieusset's development of local Leray solutions.

January 28
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Jaiyong Li (University of Toronto)
New examples on spaces of negative sectional curvature
satisfying MaTrudinger Wang conditions
When the domain of the optimal transportation problem is
a Riemannian manifold, an interesting problem is to analyze
the regularity of the optimal map, with the transport cost
related to the Riemannian distance. There have been extensive
studies about the Riemannian distance squared on the sphere
and the quotient of the sphere. In this talk, we discuss the
regularity of the optimal map on a manifold with constant
sectional curvature, with the transport cost given by a realvalued
function composed with the Riemannian distance. We will show
the relation between the Jacobi vector field and the MaTrudingerWang
tensor, which is an important quantity for the regularity
of the optimal map. As a consequence of this relation, we
give new examples of cost functions satisfying the MaTrudingerWang
conditions, and a perturbative result of the distance squared
on the Euclidean space. This is joint work with P. Lee.

February 25, 2010
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Brendan
Pass (University of Toronto)
The multimarginal optimal transportation problem
I consider an optimal transportation problem with more than
two marginals. I will discuss how the signature of a certain
pseudoRiemannian form provides an upper bound for the dimension
of the support of the optimal measure. Time permitting, I
will also discuss conditions on the cost function that ensure
existence and uniqueness of an optimal map.

March 4, 2010
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Brendan
Pass (University of Toronto)
Rectifiability of optimal transportation plans
I will prove the following result (which represents joint
work with Robert McCann and Micah Warren): any solution to
a Kantorovich optimal transportation problem on two smooth
ndimensional manifolds X and Y is supported on an ndimensional
Lipschitz submanifold of the product X \times Y, provided
the cost is C^2 and nondegenerate. If time permits, I will
discuss how this generalizes to the multimarginal problem.

March 11, 2010
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Geordie Richards (University of Toronto)
Almost Sure Local Wellposedness for the Stochastic KdVBurgers
Equation
We consider the stochastic KdVBurgers equation on the 1d
torus as a toy model for a stochastic Burgers equation. The
stochastic Burgers equation we model is obtained by differentiating
the wellknown KardarParisiZhang (KPZ) equation in space.
We present almost sure local wellposedness in H1=2() for
the stochastic KdVBurgers equation. Time permitting, we will
discuss the issue of global existence in time. This is a joint
work with Tadahiro Oh and Jeremy Quastel.

April 1, 2010
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Marina
Chugunova (University of Toronto)
On the speed of the propagation of the thin film interface
The equation $u_t+[u^n(u_{xxx}+\alpha^2 u_x  sin(x))]_x=0$
with periodic boundary conditions is a model of the evolution
of a thin liquid film on the outer surface of a horizontal
cylinder in the presence of gravity field. We use energyentropy
methods to study different properties of generalized weak
solutions of this equation. For example: finite speed of the
compact support propagation for n(13) is proved by application
of local energyentropy estimates. Joint work with A. Burchard,
M. Pugh, B. Stephens, and R. Taranets

April 8, 2010
(Thursday)
Fields
Room 210
12:00  1:00 pm 
Frederic
Rochon (University of Toronto)
Ricci flow and the determinant of the Laplacian on noncompact
manifolds
After introducing the notion of determinant of the Laplacian
on a noncompact surface with ends asymptotically isometric
to a cusp or a funnel, we will show that in a given conformal
class (with 'renormalized area' fixed), this determinant is
maximal for the metric of constant scalar curvature, generalizing
a wellknown result of Osgood, Phillips and Sarnak in the
compact case. This will be achieved by combining a corresponding
Polyakov formula with some long time existence result for
the Ricci flow for such metrics. This is a joint work with
P. Albin and C.L. Aldana.

April 15, 2010
(Thursday)
Fields
Room 210
1:00  2:00 pm
**PLEASE NOTE TIME CHANGE** 
Frederic
Rochon (University of Toronto)
Ricci flow and the determinant of the Laplacian on noncompact
manifolds Part 2
After introducing the notion of determinant of the Laplacian
on a noncompact surface with ends asymptotically isometric
to a cusp or a funnel, we will show that in a given conformal
class (with 'renormalized area' fixed), this determinant is
maximal for the metric of constant scalar curvature, generalizing
a wellknown result of Osgood, Phillips and Sarnak in the
compact case. This will be achieved by combining a corresponding
Polyakov formula with some long time existence result for
the Ricci flow for such metrics. This is a joint work with
P. Albin and C.L. Aldana.

April 29th
(Tuesday)
Fields
Room 210
12:00  1:00 pm 
Dominic
Dotterrer (University of Toronto)
The probabilistic method in geometry: Bourgain's Theorem
For quite some time now, the probabilistic method has been yielding
geometric fruit. By studying "typical" geometric structures,
we can sometimes understand the "extremal" ones. This
is exemplified in the proof of Bourgain's embedding theorem,
which states that any finite metric space on n points can be
embedded in Euclidean space with less than O(log n) metric distortion.
I will give a careful, annotated proof of this theorem and show
that the estimate is sharp.
