
SCIENTIFIC PROGRAMS AND ACTIVITIES 

January 19, 2018  
Working Seminar on Nonlinear Evolution EquationFIELDS ANALYSIS WORKING GROUP

Thursdays  
10
May 2007, 11:10 PM 
Aaron Smith, Queen's University Ricci Flow of the Cigar Manifold Towards the Standard Cigar 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 timedependent biLipschitz maps between the manifolds with both constants approaching one. We derive the optimal rate of convergence of these maps. 
12
April 2007, 11:10 PM 
Susan Friedlander, University of Illinois at Chicago 
12
April 2007, 12:55 PM 
Robert Jerrard, University of Toronto Weak solutions of a degenerate MongeAmpere equation. 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 ), is not convex, and indeed does not satisfy any kind of condition related to convexity. 
5 April 2007,
11:10 PM 
Ben Stephens, University of Toronto Curve shortening and optimal transport of curvature 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 $L^p$ 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 GaussBonnet. This allows us 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 $p=1+\epsilon$ flows behave like the nonlocal $p=1$ flow. 
5 April 2007, 12:55 PM 
Wilfrid Gangbo, Georgia Institute of Technology Hamiltonian dynamics, Wasserstein distance and fluids mechanics. 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 VlasovPoisson systems and the Euler incompressible. (This talk is based on joint works by the author with L. Ambrosio and T. Pacini) 
29 March 2007, 11:10 PM  Thomas A. Ivey, College of Charleston Closed solutions of the vortex filament flow 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 finitegap solutions of this flow, certain geometric and topological features of the evolving curves appear to be correlated with the algebrogeometric data used to construct them. In this talk, I will briefly discuss this construction, and
some lowgenus 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 finitegap 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) 
29 March 2007, 12:55 PM 
Lars Nesheim, University College, London Secondly, we provide a general existence result for the models under consideration. Our results apply to multidimensional problems; in addition, they do not require single crossing conditions à la SpenceMirrlees. We also give a generalization of the SpenceMirrlees 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 onetoone matching; and we discuss an example of the consequences of relaxing the condition. 
22 March 2007, 11:10 PM  Almut Burchard (University of Toronto) Solitary waves on an elastic curve I will discuss the stability of solitary waves on an infinite elastic curve moving in threedimensional space. While slowmoving waves are unstable, higher wave speeds stabilize the solitary wave; I will try to explain the mechanism. This is joint work with Erin Valenti. 
22 March 2007, 12:55 PM  YoungHeon Kim (University of Toronto) Curvature and the continuity of optimal transportation maps II 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). 
15 March 2007, 11:10 PM  Daniel Spirn (University of Minnesota)
Asymptotics of the ChernSimonsHiggs and GinzburgLandau energies I will discuss some new asymptotic regimes of the ChernSimonsHiggs and GinzburgLandau energies, both models for high temperature superconductivity. This is joint work with Matthias Kurzke of Humboldt University. 
15 March 2007, 12:55 PM  Alberto Montero Zarate (University of Toronto) Some remarks on a model for Bose Einstein condensation 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. 
8 March 2007, 11:10 PM  Alan Michael Hammond (Courant Institute) Moment bounds and massconservation in PDE modelling coaslescence We examine the behaviour of solutions to a system of PDE (the Smoluchowski PDE), that model the aggregation of massbearing 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. 
8 March 2007, 12:55 PM  YoungHeon Kim (University of Toronto) Curvature and the continuity of optimal transportation maps We will discuss the continuity of optimal transport maps, in view of a semiRiemannian structure which we have formulated recently. A necessary condition for the continuity is given as some nonnegativity condition on the curvature of this semiRiemannian 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).

1 March 2007, 12:55PM 
Dorian Goldman, University of Toronto

22 Feb.  Reading week 
15 Feb.  
8 Feb. 2007, 11:10 a.m.  Juhi Jang, Brown University Dynamics of Gaseous Stars. In this talk, I will discuss the existence theory and the stability theory of LaneEmden star configurations in EulerPoisson system. 
8 Feb. 2007, 12:55 p.m. 
Dorian Goldman, University of Toronto 
1 Feb, 2007 12:55  1:55 
Maria Sosio, University of Toronto Harnack Estimate for a Doubly Nonlinear Parabolic Equation Harnack estimates for quasilinear 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 measuretheoretical arguments, bypassing any notion of maximum principle and potentials. The main steps to prove an intrinsic Harnack inequality for a nonnegative weak solution of a doubly nonlinear parabolic equation will be presented. [1] E. DiBenedetto, U. Gianazza, V. Vespri, Harnack Estimates for QuasiLinear Degenerate Parabolic Differential Equation, preprint IMATI 4PV 2006, 127, to appear in Acta Mathematica. 
25 Jan,
2007 12:55  1:55 
Paul Lee, University of Toronto 
18 Jan,
2007 11:10  12:10 
Catherine Sulem, University of Toronto Hamiltonian expansions for water waves over a random bottom 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. 
18 Jan, 2007 
Jeremy Quastel, University of Toronto White Noise and KdV 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.) 
11
Jan, 2007 11:10  12:10 
Ben Stephens, University of Toronto 
14
Dec, 2006 11:10  12:10 
Gang Zhou On the formation of singularities in solutions of the critical nonlinear Schroedinger equation 
7 Dec, 2006 
John Lott, University of Michigan 
7 Dec, 2006 
Izabella Laba, University of British Columbia 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 measuretheoretic 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
Nov. 2006 11:1012:10 
Mary Pugh, University of Toronto Entropy Methods for FourthOrder Equations I will present an introduction to entropy methods as used for a class of fourthorder 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). 
23
Nov. 2006 11:1012:10 
Ben Stephens, University of Toronto 
16
Nov. 2006 11:1012:10 
Alex Bloemendal The Regularity of Planar Maps of Bounded Compression and Rotation. 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. 
16
Nov. 2006 12:5013:50 
Jim Colliander,
University of Toronto Maximalintime issues for the nonlinear Schroedinger equation 
9 Nov. 2006  Location change Room WI 2006 (Wilson Hall, New College), the University of Toronto 
2
Nov. 2006, 11:10PM 
YoungHeon Kim,
University of Toronto Regularity results for optimal transport problems with general costs (CONTINUED). 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 cconvexity and cexponential maps, and will describe a geometric interpretation of the A3 condition of cost functions. 
2 Nov. 2006, 12:50PM 
Nikos Tzirakis, University of Toronto 
26
Oct. 2006 11:10AM 
YoungHeon Kim Regularity results for optimal transport problems with general costs In the following couple of talks, I will discuss the work byG. 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 Rober 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 cconvexity and cexponential maps, and will describe a geometric interpretation of the A3 condition of cost functions. 
26 Oct. 2006 12:50PM  Nikos Tzirakis Global wellposedenss and scattering for the defocusing energycritical nonlinear Schrodinger equation in . I will present the global wellposedness result of M. Visan and E. Ryckman for the energycritical defocusing NLS in . The main step of the proof is the establishment of a global bound for the solution. The result is based on previous results for the problem by Bourgain (radial assumption on the initial data) and Colliander, Keel, Staffilani, Takaoka and Tao (no assumptions on the data). 
19 Oct. 2006, 11:10AM 
Robert McCann,
University of Toronto Regularity and counterexamples in optimal transportation. 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, and counterexamples to continuity due to Gregoire Loeper. 
19 Oct. 2006, 12:50PM  Ian Zwiers, University of Toronto On the MerleRaphael results for quintic focusing nonlinear Schrodinger on R. I'll discuss the L2critical 1D problem; the geometric decomposition, orthogonality conditions, and local virial identity and why the Q_b profiles are needed to get the loglog bound. 