Carolina Meeting on Harmonic Analysis and PDE

Friday, January 30 - Sunday, February 1, 2009

University of North Carolina, Chapel Hill


Two one hour lectures each by:

  • Carlos Kenig, University of Chicago

    • Lecture 1: The concentration-compactness/rigidity theorem method for critical nonlinear dispersive and wave equations
      • We will describe a method, developed in a series of joint works with Merle to study global well-posedness and scattering for critical nonlinear dispersive and wave equations in focusing and defocusing settings
    • Lecture 2: Compact radial solutions for energy supercritical nonlinear wave equations in 3d, with applications
      • We will discuss recent results with Merle establishing pointwise decay estimates for radial solutions of nonlinear wave equations in the energy supercritical range in 3d. Applications to global existence and scattering in the defocusing case will be mentioned.

  • James Colliander, University of Toronto

    • Lecture 1: Rough Blowup Solutions to Cubic Focusing Nonlinear Schrödinger on R^2.
      • Qualitative properties of so-called log-log blowup solutions of L^2 critical nonlinear Schrödinger equations have been studied in a remarkable series of papers by Merle and Raphaël. This talk will describe recent work (with Pierre Raphaël) establishing stability of the log-log regime of blowup solutions under small rough perturbations. The techniques involve a blending of the log-log toolbox with a the I-method of almost conservation laws in the setting of a big bootstrap.
    • Lecture 2: Recent Progress on Nonlinear Schrödinger-type Equations.
      • This talk will survey some recent results on NLS-type equations. I will describe work (with Tristan Roy) which proves global well-posedness for the cubic defocusing NLS for general rough data. I will also describe a result by Ian Zwiers which constructs a solution of cubic focusing NLS in three space dimensions which explodes precisely on a circle. I'll also discuss new qualitative blowup propertiesfor the elliptic-elliptic Davey-Stewartson system obtained by Geordie Richards.

  • Hart Smith, University of Washington

    • Spectral cluster and Strichartz estimates on manifolds with boundary
      • We will discuss recent work on establishing Lp bounds on spectral clusters (approximate eigenfunctions) on compact manifolds with boundary, and
        closely related work on Strichartz estimates for the wave equation.  In the case of manifolds with convex boundary, for example convex domains in Euclidean space, the presence of multiply reflected geodesics and gliding rays significantly complicates the study of these and other dispersive estimates. Our joint work with Sogge, and Sogge-Blair, nevertheless establishes a range of useful estimates that in some cases are sharp. We will also discuss a striking example of Ivanovici which places a limit on the range of Strichartz estimates that are valid inside such domains.

One hour lectures each by:

  • Ioan Bejenaru, University of Chicago

    • Global Schrodinger maps in dimensions d\ge 2: small data in the critical Sobolev spaces

  • Jeremie Szeftel, Princeton University

    • Around the bounded L^2 curvature conjecture in general relativity
      • We will talk about recent results obtained jointly with Sergiu Klainerman and Igor Rodnianski on the construction and the control of a parametrix to the homogeneous wave equation \square_g\phi=0, where g is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix when one only assumes L^2 bounds on the curvature tensor R of g is a major step towards the proof of the bounded L^2 curvature conjecture.

  • Nikolaos Tzirakis, University of Illinois

    • Asymptotically linear solutions in $H^1$ of the 2-d defocusing nonlinear Schr\"odinger and Hartree equations
      • In this talk I will show how in 2d, given an $H^1$ solution to the linear Schr\"odinger equation one can prove the existence (but not the uniqueness) of an $H^1$ solution to the defocusing nonlinear Schr\"odinger (NLS) equation in the short range case, and the existence of an $H^1$ solution to the defocusing Hartree equation for certain interaction powers, such that their difference in the $H^1$ norm tends to zero as time tends to infinity. This is a partial result towards the existence of well-defined continuous wave operators from $H^1 to H^1$ for these equations. This is joint work with Justin Holmer.


Schedule:  All talks will be in Phillips 332.  The registration and teas will take place in the adjacent lounge, Phillips 330.

Friday -
  1:30    Arrival/Registration
  2:00    H. Smith
  3:00    Coffee/Tea
  3:30    J. Szeftel		 Saturday -
  9:00    Coffee/Tea
  9:30    J. Colliander
 10:45   N. Tzirakis
 11:45   Lunch
  1:15    C. Kenig
  2:30    I. Bejenaru
  3:30    Coffee/Tea
  4:00    H. Smith

  6:00   Banquet		 Sunday -
  9:00   Coffee/Tea
  9:30   J. Colliander
  10:45  C. Kenig

Banquet:  There will be a banquet for speakers and participants on Saturday, January 31 beginning at 6:00pm at the Carolina Inn.

Lodging:  There are a block of rooms available at a group rate at the Holiday Inn in Chapel Hill.  Please call 1-888-452-5765.  The rooms are $89.95/night for either king size bed or two double beds.  To get the group rate, reservations need to be made by January 15, and you should ask for the "Harmonic Analysis and PDE" rooms.  The hotel has a shuttle service that will bring guests from the hotel to campus.  Please inquire at the front desk.

Registration:  In order to get a head count for the teas and banquet, please register by emailing.

Support:  There are some funds available for those without other funding available to assist participants with the cost of their travel and lodging .  To apply, please email.

Local Organizers:  Joseph Cima, Jason Metcalfe, Michael Taylor, Mark Williams, Warren Wogen

Some local information:  Campus maps, Visiting the UNC math