# 2006-2007 FAWG Seminar

This page contains the 2006-2007 calendar for the Fields Analysis Working Group seminar co-organized by James Colliander and Robert McCann.

## June 28, Thursday 13:10 -- 14:00

• Mikhail Gudim (University of Toronto at Scarborough) 13:10--14:00
Abstract:

The Banach-Tarski paradox states that a solid sphere can be decomposed into disjoint subsets, such that by means of rotations and translations these pieces can be reassembled to fill out two spheres of the same radius. The proof uses basic group theory.

## May 10, Thursday 11:10--14:00

• Aaron Smith (Queen's University) 11:10--12:10
Title: Ricci Flow of the Cigar Manifold Towards the Standard Cigar
Abstract:

In this talk, we will briefly introduce Hamilton's Ricci flow. We will then explicitly compute the evolution under Ricci flow of a class of asymmetric conformally flat manifolds, which we call cigar manifolds. One particular subclass of these cigar manifolds converges towards the standard cigar soliton under Ricci flow in the sense that there exist time-dependent bi-Lipschitz maps between the manifolds with both constants approaching one. We derive the optimal rate of convergence of these maps.

## April 12, 2007, Thursday 11:10--14:00

Title: Onsager's Conjecture and a Model for Turbulence.
Abstract:

We discuss properties of a shell type model for the inviscid fluid equations. We prove that the forced system has a unique equilibrium which is an exponential global attractor. Every solution blows up in H5 / 6 in finite time . After this time, all solutions stay in Hs,s < 5 / 6, and "turbulent" dissipation occurs. Onsager's conjecture is confirmed for the model system. We discuss how intuition obtained from the model carries over to the 3-D Euler equations where we obtain the "sharp" space for Onsager's conjecture concerning energy conservation. This is joint work with Alexey Cheskidov, Natasa Pavlovic, Roman Shvydkoy and Peter Constantin.

Title: Weak solutions of a degenerate Monge-Ampere equation.
Abstract: I will discuss some rigidity results for a class of weak solutions of the equation
detD2u = 0

where u is a function of two variables. A large part of the talk will be devoted to the question of how to understand the detD2u for a function u that is not very smooth (say, not in W2,p for any $p\geq 1$), is not convex, and indeed does not satisfy any kind of condition related to convexity.

## April 5, 2007, Thursday 11:10--14:05

• Benjamin Stephens (University of Toronto) 11:10--12:10
Title: Curve shortening and optimal transport of curvature
Abstract:

How does a string in honey behave when you tighten it? In this talk we study gradient flows of curve length in a medium which resists normal movement in an Lp sense. We restrict to planar Frenet curves. For 1 < p < = 2 this corresponds to a normal flow by a power of the curvature. For p = 1 it corresponds to a nonlocal flow and to "gradient flow of length with respect to the flat metric." In all cases, the mass of the curve's curvature function is conserved due to Gauss-Bonnet. We attempt to give a unified treatment of these local and nonlocal flows by deriving the corresponding evolution of their curvature functions as gradient flows of "curvature entropies" with respect to Wasserstein distance. In particular we are interested in studying how local p = 1 + ε flows behave like the nonlocal p = 1 flow.

Title: Hamiltonian dynamics, Wasserstein distance and fluids mechanics.
Abstract:

We briefly review the Riemannian structure on P2(R2d) = M, the set of probability measures on the phase space R2d. We show that M inherits a Poisson structure which allows us to rigorously treat some evolutive systems appearing in fluids mechanics, as Hamiltonian systems on M. The orbits in M become very useful objects for studying these conservative systems governed by a Hamiltonian. Existence of solutions can be established when the Hamiltonian satisfies locally some smoothness properties. The class of equations considered here includes: the semigeostrophic systems and the Vlasov-Poisson systems and the Euler incompressible.

(This talk is based on joint works by the author with L. Ambrosio and T. Pacini)

## March 29, 2007, Thursday 11:10--14:00

Title: Closed solutions of the vortex filament flow
Abstract:

The simplest model of vortex filament motion in an ideal fluid leads to an integrable nonlinear evolution equation, known as the localized induction approximation or the vortex filament flow, closely related to the cubic focussing nonlinear Schroedinger equation. For closed finite-gap solutions of this flow, certain geometric and topological features of the evolving curves appear to be correlated with the algebro-geometric data used to construct them.

In this talk, I will briefly discuss this construction, and some low-genus examples (in particular, Kirchhoff elastic rod centerlines) where this correlation is well understood. I will mainly discuss recent joint work with Annalisa Calini, describing how to generate a family of closed finite-gap solutions of increasingly higher genus via a sequence of deformations of the multiply covered circle. We prove that every step in this sequence corresponds to constructing a cable on previous filament; moreover, the cable's knot type (which is invariant under the evolution) can be read off from the deformation sequence.

• Lars Nesheim (University College, London) 1:02--2:00
Title: Nonlinear hedonic pricing: finding equilibria through linear programming
Abstract:

First, we consider a general framework for studying hedonic prices problems with quasi-linear preferences, and show that it is equivalent to a matching model with transferable utilities. From a mathematical perspective, both problems can in turn be rephrased under the form of a linear program, in fact an optimal transportation problem of Monge-Kantorovich form.

Secondly, we provide a general existence result for the models under consideration. Our results apply to multi-dimensional problems; in addition, they do not require single crossing conditions à la Spence-Mirrlees. We also give a generalization of the Spence-Mirrlees condition which guarantees uniqueness of the stable match (in the matching model) or of the equilibrium (in the hedonic model) even in the absence of one-to-one matching; and we discuss an example of the consequences of relaxing the condition.

## March 22, 2007, Thursday 11:10--14:00

• Almut Burchard (University of Toronto) 11:10--12:10
Title:Solitary waves on an elastic curve
Abstract:

I will discuss the stability of solitary waves on an infinite elastic curve moving in three-dimensional space. While slow-moving waves are unstable, higher wave speeds stabilize the solitary wave; I will try to explain the mechanism. This is joint work with Erin Valenti.

• Young-Heon Kim (University of Toronto) 13:02--14:00
Title: Curvature and the continuity of optimal transportation maps II
Abstract:

This talk is a continutation of the talk on March 8, and we will discuss one key theorem and its proof. We also plan to give a short summary of the previous talk in the beginning.

This represents joint work with Robert McCann (University of Toronto).

## March 15, 2007, Thursday 11:10--14:00

• Daniel Spirn (University of Minnesota) 11:10--12:10
Title: Asymptotics of the Chern-Simons-Higgs and Ginzburg-Landau energies
Abstract:

I will discuss some new asymptotic regimes of the Chern-Simons-Higgs and Ginzburg-Landau energies, both models for high temperature superconductivity. This is joint work with Matthias Kurzke of Humboldt University.

• Alberto Montero Zarate (University of Toronto) 13:02--14:00
Title:Some remarks on a model for Bose Einstein condensation
Abstract:

In this talk I will describe a Gamma convergence result for an energy functional often used by physicists to model Bose Einstein condensates, the Gross Pitaevskii energy. Among oter things, one can use this Gamma convergence result to give a rough initial description of the vortices that appear in some local minimizers of this energy, and that correspond to whirlpools in the condensate according to the physical model. These questions were suggested to me by Bob Jerrard. In some of them I have worked in collaboration with B. Stephens and A. Bourchard.

## March 8, 2007, Thursday 11:10--14:00

• Alan Michael Hammond (Courant Institute) 11:10--12:10
Titile: Moment bounds and mass-conservation in PDE modelling coaslescence
Abstract:

We examine the behaviour of solutions to a system of PDE (the Smoluchowski PDE), that model the aggregation of mass-bearing particles that diffuse and are prone to coagulate in pairs at close range. Conditions under which these solutions conserve mass for all time will be presented, along with stronger estimates, moment bounds that show that heavy particles are rare. Uniqueness of solutions also follows from the moment bounds.

This is joint work with Fraydoun Rezakhanlou.

• Young-Heon Kim (University of Toronto) 12:55--14:00
Title: Curvature and the continuity of optimal transportation maps
Abstract:

We will discuss the continuity of optimal transport maps, in view of a semi-Riemannian structure which we have formulated recently. A necessary condition for the continuity is given as some nonnegativity condition on the curvature of this semi-Riemannian metric, and this result gives a quite general geometric frame work for the regularity theory of Ma, Trudinger, Wang and Loeper on the potential functions of optimal transport.

This is joint work with Robert McCann (University of Toronto).

## March 1, 2007, Thursday 11:10--14:00

• Dorian Goldman (University of Toronto) 12:55--14:00
Title: Chaotic dynamics in various models in Meteorology Part II

## February 8, 2007, Thursday 11:10--14:00

Title: Dynamics of Gaseous Stars
Abstract:

In this talk, I will discuss the existence theory and the stability theory of Lane-Emden star configurations in Euler-Poisson system.

• Dorian Goldman (University of Toronto) 12:55--14:00
Title: Chaotic dynamics in various models in Meteorology Part I
Abstract:

I will introduce 3 well known models used extensively in meteorology that are approximations to the full Navier Stokes equations in a rotating frame of reference. The three models are the 2D Euler equations, the Quasigeostrophic and finally the Semigeostrophic approximation. I will derive these approximations and show that when the initial data corresponds to regions of uniform elliptial (ellipsoidal in the case of the 3D Quasigeostrophic model) potential vorticity (which is defined analagously to the vorticity for 2D Euler in the cases of Quasi and Semigeostrophy), that the dynamics are very similar and the phase portraits are essentially equivalent. In particular the Semigeostrophic equations, modelled in dual variables take the form of a Monge ampere equation, transforming the problem into an optimal transportation problem. In particular each phase portrait contains a hyperbolic fixed point with homoclinic saddle connection. Extending the result of Bertozzi (1988) who used Melnikov analysis to demonstrate that the 2D Euler equations responded chaoticically to vertical stretching of the elliptical columnm I will show the method extends to the Semigeostrophic equations and most likely to the Quasigeostrophic equations as well.

## February 1, 2007, Thursday 11:10--14:00

• Maria Sosio (University of Toronto) 12:55--14:00
Title: HARNACK ESTIMATE FOR A DOUBLY NONLINEAR PARABOLIC EQUATION
Abstract: Harnack estimates for quasi-linear degenerate parabolic equations

obtained by E. DiBenedetto, U. Gianazza and V. Vespri in [1] have been extended to a doubly nonlinear parabolic equation. In all these works the entire procedure is based only on measure-theoretical arguments, bypassing any notion of maximum principle and potentials. The main steps to prove an intrinsic Harnack inequality for a non-negative weak solution of a doubly nonlinear parabolic equation will be presented.

[1] E. DiBenedetto, U. Gianazza, V. Vespri, Harnack Estimates for Quasi-Linear Degenerate Parabolic Differential Equation, preprint IMATI 4-PV 2006, 1-27, to appear in Acta Mathematica.

## January 25, 2007, Thursday 11:10--14:00

• Paul Lee (University of Toronto) 12:55--14:00
Title: Infinite-dimensional geometry of optimal mass transport
Abstract:

In the 60's, Arnold shows that the Euler equation can be thought of as the geodesic flow on the group of all volume preserving diffeomorphism. In a similar spirit, Otto shows that the mass transport problem can be consider as the geodesic problem on the space $W$ of all volume forms with the same total volume. In particular, the space $W$ can be regarded as the quotient of the group of all diffeomophisms by the subgroup of volume preserving ones, while the geodesic flow on the diffeomorphism group, given by the Burger's equation, is closely related to that on the space $W$. It turns out that this relation between diffeomorphism group and the space $W$ can be understood via Hamiltonian reduction. We also consider the following nonholonomic version of the classical Moser theorem: given a bracket generating distribution on a manifold, two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution. We discuss these results from the point of view of an infinite-dimensional non-holonomic distribution on the diffeomorphism groups.

## January 18, 2007, Thursday 11:10-14:00

Title: Hamiltonian expansions for water waves over a random bottom
Abstract: We discuss the motion of the free surface of a perfect fluid with variable bottom.We consider the scaling regime of the Korteweg - deVries equations, the question being to find effective coefficients for the long wave models of free surface water waves in a channel with a variable depth bed. We consider the case of a random bottom, namely situations in which the bottom of the fluid region is described as a stationary ergodic process which exhibits a sufficiently strong property of mixing.We show formally that in this limit, the random effects are governed by a canonical limit process which is equivalent to a white noise through Donsker's invariance principle. The coherent wave motions of the KdV limit are shown to be preserved, while at the same time the random effects on solutions of the KdV are described, and the degree of scattering due to the variable bottom is quantified. This is joint work with Anne de Bouard, Walter Craig and Philippe Guyenne.
Title: White Noise and KdV
Abstract:In joint work with Benedek Valko (Toronto) we found that Gaussian white noise is an invariant measure for KdV on the circle. In this talk we will describe the relevant concepts, what the result means both mathematically and physically, and give some ideas of the proof. (See math.AP/0611152 for the preprint.)

## January 11, 2007, Thursday 11:10-14:00

This is a continuation from the previous talk:

Title: Modulational stability of ground states of nonlinear Schroedinger equation
Abstract: The nonlinear Schroedinger equation has a family of ground states related to the scaling invariances of the equation. I will present Weinstein's paper showing how states starting near this ground state family stay near the family as time increases.

(For Weinstein's paper you may also click the link to pdf file.)

• No seminar at 1pm due to graduate course information session.

## December 14 2006, Thursday 11:10-14:00

• Gang Zhou 13:00-14:00

I will present G. Perelman's paper "on the formation of singularities in solutions of the critical nonlinear Schroedinger equation.

See the article here: Perelman Article

## December 7 2006, Thursday 11:10--14:00, Fields Institute Room 230

• John Lott (University of Michigan) 11:10-12:10
Title: The work of Grigory Perelman
Abstract: This will be a exposition, for nonspecialists, of Perelman's work on Ricci flow.
Title: Harmonic analysis and incidence geometry
Abstract:

Many problems in Euclidean harmonic analysis, especially restriction theory, lead to purely geometric questions, typically involving arrangements of large families of thin" objects such as lines, circles, or surfaces. Conversely, geometric measure-theoretic questions involving projections or intersections of sets are often approached via harmonic analytic methods.

While the connections between harmonic analysis and geometric measure theory go back several decades, it was only in the 1990s that harmonic analysts began to explore links to the area of combinatorics known as discrete geometry, or more specifically incidence geometry. The latter studies many of the same questions that harmonic analysts are interested in - for example, projection sets, distance sets, or arrangements of families of thin objects - but in a discrete setting, as opposed to the geometric measure-theoretic setting encountered in harmonic analysis.

The purpose of this talk is to give a brief survey of this area of research, including both classical results and current directions, with emphasis on the connections between the analysis and the combinatorics.

## 30 November 2006, Thursday 11h10-14h00

Title: Entropy Methods for Fourth-Order Equations
Abstract: I will present an introduction to entropy methods as used for a class of fourth-order evolution equations. The equations in question have degeneracies that are both helpful(the degeneracy ensures solutions are signed) and difficult (the solutions have relatively low regularity).
Title: The Fourier Restriction Problem and the Kakeya Conjecture
Abstract: For a hypersurface $S\subset R^{n}$, when do we have estimates of the form $\| \hat{f}\|_{L^{q}(S)}\leq C\| f\|_{L^{p}(R^{n})}$? The Restriction conjecture proposes a complete answer to this question in the case S = Sn − 1 . I will discuss progress on this conjecture, as well as applications to PDE and to the Kakeya conjecture, which states that a Besicovitch set in Rn must have full Hausdorff dimension.

## 23 November 2006, Thursday 11h10-14h00

Title: Modulational stability of ground states of nonlinear Schroedinger equation
Abstract: The nonlinear Schroedinger equation has a family of ground states related to the scaling invariances of the equation. I will present Weinstein's paper showing how states starting near this ground state family stay near the family as time increases.

(For Weinstein's paper you may also click the link to pdf file.)

Title: Determination of minimal mass blow-ups for the L2 critical nonlinear Schroedinger equation in $\mathbb{R}^{2}$
Abstract: The pseudo-conformal invariance of L2 critical NLS in $\mathbb{R}^{2}$ yields explicit blow-up solutions of minimal L2 norm; solutions are global in time below the mass of the ground state Q. I will present Merle's proof that these are the only minimal mass blow-up solutions for H1 initial data.

## 16 November 2006, Thursday 11h10-14h00. Fields Institute Board Room

Title: The Regularity of Planar Maps of Bounded Compression and Rotation (continued)
Abstract: I will describe a geometric argument of Caffarelli and Milman proving a quanitative injectivity for monotone maps of the plane whose compression and rotation are weakly bounded.
Find references and a discussion of a point from the first part of the talk here.
Title: Maximal-in-time issues for the nonlinear Schroedinger equation (continued)

## 9 November 2006, Thursday 11h10-14h00. *** NB: Wilson Room 2006 at 40 Wilcox St ***

Title: The Regularity of Planar Maps of Bounded Compression and Rotation
Abstract: I will describe a geometric argument of Caffarelli and Milman proving a quanitative injectivity for monotone maps of the plane whose compression and rotation are weakly bounded.
Title: Maximal-in-time issues for the nonlinear Schroedinger equation
Abstract: This talk surveys the known (optimal) local-in-time theory for semilinear monomial Schroedinger equations, our knowledge of this family of equations maximally in time, and the main conjectures about the maximal-in-time behavior of solutions.

## 2 November 2006, Thursday 11:00-14:00. Fields Institute Board Room

Title: Regularity results for optimal transport problems with general costs (Continued)
Title: Global well-posedenss and scattering for the defocusing energy-critical nonlinear Schrodinger equation in $\mathbb{R}^{1+4}$ (Continued)

## 26 October 2006, Thursday 11:00-14:00. Fields Institute Board Room

Title: Regularity results for optimal transport problems with general costs
Abstract: In the following couple of talks, I will discuss the work by G. Loeper for the regularity of optimal transport maps with respect to some general cost functions. This talk is a continuation to the talk given by Robert McCann on Oct. 19, and I will focus more on the geometric view point of the theory. In the first half, I plan to discuss the c-convexity and c-exponential maps, and will describe a geometric interpretation of the A3 condition of cost functions.
Title: Global well-posedenss and scattering for the defocusing energy-critical nonlinear Schrodinger equation in $\mathbb{R}^{1+4}$.
Abstract: I will present the global well-posedness result of M. Visan and E. Ryckman for the energy-critical defocusing NLS in $\mathbb{R}^{1+4}$. The main step of the proof is the establishment of a global $L_t^6 L_x^6$ bound for the solution. The result is based on previous results for the $\mathbb{R}^{1+3}$ problem by Bourgain (radial assumption on the initial data) and Colliander, Keel, Staffilani, Takaoka and Tao (no assumptions on the data).

## 19 October 2006, Thursday 11:00-14:00. Fields Institute Board Room

Title: Regularity and counterexamples in optimal transportation
Abstract: I will give a brief introduction to optimal transportation, leading up to recent results concerning the smoothness of optimal maps by Ma, Trudinger, Wang and Loeper, andcounterexamples to continuity due to Gregoire Loeper.
Title: On the Merle-Raphael Log-log upper bound for 1D quintic focusing nonlinear Schrodinger
Abstract: I'll discuss the L2-critical 1D problem; the geometric decomposition, orthogonality conditions, and local virial identity and why the Qb profiles are needed to get the log-log bound.