# 2009-2010 FAWG Seminar

The Fields Analysis Working Group Seminar takes place (usually) at the Fields Institute Room 210 on Thursdays at 12h10. The seminar is co-organized by J. Colliander, Colin Decker, R.J. McCann, and L. Guth.

The seminar also has a page at the Fields Institute.

Of related interest in Toronto (and sometimes cross-listed):

## July 23, Denzler, Friday, 15h10, Fields Institute Room 210 *Please Note the Unusual Time*

Title: Asymptotics of fast diffusion via dynamical systems.
Abstract: 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 with ideas coming from linearized stability of dynamical systems. Joint work with Robert McCann and Herbert Koch.

## April 29, Dotterrer, 12:00-1:00 @Fields Institute Room 210

 Dominic Dotterrer homepage (U. Toronto) Fields Analysis Working Group Thursday April 29 12:00-1:00 Fields Institute Room 210 Title: The probabilistic method in geometry: Bourgain's Theorem Abstract: 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. [ arXiv] 2010_04_29_Dotterrer_Notes 2010_04_29

## April 15, Rochon, Thursday 1:10-2:00, Fields Institute Room 210 *Please Note the Unusual Time*

Title: Ricci flow and the determinant of the Laplacian on non-compact manifolds, Part 2.
Abstract: 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.
2010_04_08_FAWG_Rochon_Notes, arXiv

## April 8, Rochon, Thursday 12:10-1:00, Fields Institute Room 210

Title: Ricci flow and the determinant of the Laplacian on non-compact manifolds
Abstract: 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.
2010_04_08_FAWG_Rochon_Notes, arXiv

## April 1, Chugunova, Thursday 12:10-1:00, Fields Institute Room 230

Title: On the speed of the propagation of the thin film interface
Abstract: 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

## March 11, Richards, Thursday 12:10-1:00, Fields Institute Room 210

• Geordie Richards (University of Toronto)
Title: Almost Sure Local Well-posedness for the Stochastic KdV-Burgers Equation.
Abstract: 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 $H^{-1/2-}(\mathbb{T})$ 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.

## March 4, Pass, Thursday 12:10-1:00, Fields Institute Room 210

• Brendan Pass (University of Toronto)
Title: Rectifiability of optimal transportation plans.
Abstract: 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.

## February 25, Pass, Thursday 12:10-1:00, Fields Institute Room 230

• Brendan Pass (University of Toronto)
Title: The multi-marginal optimal transportation problem.
Abstract: 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.

## January 28, Li, Thursday 12:10-1:00, Fields Institute Room 230

• Jiayong Li (University of Toronto)
Title: New examples on spaces of negative sectional curvature satisfying Ma-Trudinger-Wang conditions.
Abstract: 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.

## January 21, Zwiers, Thursday 12:10-1:00, Fields Institute Room 230

Title: Minimal Navier-Stokes Singularities
Abstract: Suppose that in three dimensions there exists a solution to Navier-Stokes that forms a singularity in finite time (for data in $\dot{H}^\frac{1}{2}$). Then there exists such data of minimal norm. This is a recent result of Rusin and Šverák, building on Lemarié-Rieusset's development of local Leray solutions.
References:
W. Rusin and V. Šverák, Minimal initial data for potential Navier-Stokes singularities.
http://arxiv.org/abs/0911.0500
P.G. Lemarié-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall / CRC, 2002
(Math Library)

## January 14, Burchard, Thursday 12:10-1:00. Fields Institute Room 210

Title: Competing Symmetries and convergence of sequences of random symmetrizations
Abstract: 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

## January 7, Pocovnicu, Thursday 12:10-1:00, Fields Institute Room 230

• Oana Pocovnicu (Orsay)
Title: Traveling waves for the cubic Szegö equation on the real line
Abstract: We consider the cubic Szegö equation on the real line. This equation was introduced by Gérard and Grellier on the circle as a toy model for non-dispersive evolution equations in studying the nonlinear Schrödinger equation on a sub-Riemannian 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 well-posedness in the Hardy space on the upper half-plane, 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.
(Some background information is available in the paper of Gérard-Grellier.)

## December 3, Erdös, Thursday 12:10-1:00. Fields Institute Room 210

Title: Dynamical formation of correlations in a Bose-Einstein condensate
Abstract: 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(xixj)). 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.
The paper of the same title is linked here.
Slides from the talk.
Numerical simulation of correlations.

## November 26, Jerrard, Thursday 12:10-1:00, Fields Institute Room 210

Title: Partial regularity for hypersurfaces minimizing elliptic parametric integrands Part II
Abstract: 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.
1977_Schoen_Simon_Almgren_Acta
mathscinet

## November 19, Jerrard, Thursday 12:10-1:00, Fields Institute Room 210

Title: Partial regularity for hypersurfaces minimizing elliptic parametric integrands
Abstract: 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.
1977_Schoen_Simon_Almgren_Acta, :mathscinet

## November 5, Lieb, Thursday 12:10-1:00. Fields Institute Room 210

Title: Mathematics of the Bose Gas: A truly quantum-mechanical many-body problem
Abstract: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.

## October 29, Colliander, Thursday 12:10-1:00. Fields Institute Room 210

Title: Towards partial regularity for nonlinear Schrödinger?
Abstract: 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.

## October 22, Lee, Thursday 12:10-1:00. Fields Institute Room 210

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

## October 15, McCann, Thursday 12:10-1:00. Fields Institute Room 210

Title: An optimal multidimensional price strategy facing informational asymmetry, Part III
Abstract: 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.

## October 8, Decker, Thursday 12:10-1:00. Fields Institute Room 230

• Colin Decker (University of Toronto)
Title: Uniqueness of matching in the marriage market.
Abstract: 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.

## October 1, McCann, Thursday 12:10-1:00. Fields Institute Room 230

Title: An optimal multidimensional price strategy facing informational asymmetry, Part II.
Abstract: See below.

## September 24, McCann, Thursday 12:10-1:00. Fields Institute Room 210

Title: An optimal multidimensional price strategy facing informational asymmetry.
Abstract: 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.
Organizational discussion and lunch to follow.