SCIENTIFIC PROGRAMS AND ACTIVITIES

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

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.

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: $|T(x) - T(y)| \leq |x-y|$. 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.

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 Y-H 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.

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 Y-H 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.

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 Choo-Siow 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.

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 Y-H 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.

Paul Lee (University of California Berkeley)
The Ma-Trudinger-Wang conditions for natural mechanical actions.

The Ma-Trudinger-Wang 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.

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 Navier-Stokes and other equations will also be discussed.

Elliott Lieb (Princeton University)
Mathematics of the Bose Gas: A truly quantum-mechanical many-body problem

The peculiar quantum-mechanical properties of the lowest energy states of Bose gases that were predicted in the early days of quantum-mechanics 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 Gross-Pitaevskii equation in traps (including rapidly rotating traps), Bose-Einstein condensation and superfluidity, and the transition from 3-dimensional behavior to 1-dimensional behavior as the cross-section of the trap decreases. The phenomena described are highly quantum-mechanical, without a classical physics explanation, and it is very satisfying that reality and these mathematical predictions agree.

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 n-2 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.

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 n-2 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.

Dynamical formation of correlations in a Bose-Einstein 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 Gross-Pitaevskii scaling, i.e. it is given by N2V(N(xi ? xj)). We monitor the behavior of the solution to the N-particle 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 two-body 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 Gross-Pitaevskii equation for the evolution of the Bose-Einstein 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.

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 6-7). GAFA 7 (1997), 823-860.
http://www.math.utoronto.ca/almut/preprints/steiner.ps

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 6-7). GAFA 7 (1997), 823-860.
http://www.math.utoronto.ca/almut/preprints/steiner.ps

Ian Zwiers (University of Toronto)
http://www.math.utoronto.ca/izwiers/
Minimal Navier-Stokes Singularities

Suppose that in three dimensions there exists a solution to Navier-Stokes 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 Lemarié-Rieusset's development of local Leray solutions.

Jaiyong Li (University of Toronto)
New examples on spaces of negative sectional curvature satisfying Ma-Trudinger 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 real-valued function composed with the Riemannian distance. We will show the relation between the Jacobi vector field and the Ma-Trudinger-Wang 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 Ma-Trudinger-Wang conditions, and a perturbative result of the distance squared on the Euclidean space. This is joint work with P. Lee.

Brendan Pass (University of Toronto)
The multi-marginal optimal transportation problem

I consider an optimal transportation problem with more than two marginals. I will discuss how the signature of a certain pseudo-Riemannian 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.

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 n-dimensional manifolds X and Y is supported on an n-dimensional 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 multi-marginal problem.

Geordie Richards (University of Toronto)
Almost Sure Local Well-posedness for the Stochastic KdV-Burgers Equation

We consider the stochastic KdV-Burgers equation on the 1-d torus as a toy model for a stochastic Burgers equation. The stochastic Burgers equation we model is obtained by differentiating the well-known Kardar-Parisi-Zhang (KPZ) equation in space. We present almost sure local well-posedness in H1=2() for the stochastic KdV-Burgers equation. Time permitting, we will discuss the issue of global existence in time. This is a joint work with Tadahiro Oh and Jeremy Quastel.

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 energy-entropy 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 energy-entropy estimates. Joint work with A. Burchard, M. Pugh, B. Stephens, and R. Taranets

Frederic Rochon (University of Toronto)
Ricci flow and the determinant of the Laplacian on non-compact manifolds

After introducing the notion of determinant of the Laplacian on a non-compact 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 well-known 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.

Frederic Rochon (University of Toronto)
Ricci flow and the determinant of the Laplacian on non-compact manifolds Part 2

After introducing the notion of determinant of the Laplacian on a non-compact 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 well-known 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.