PDE Seminar, Fall 2007

Wednesdays from 4 to 5 PM at MIT (room 2-135).

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Next seminar:

2008

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Full schedule: (Some slides/notes available below)

• September 12: Andras Vasy (Stanford)
  ``Asymptotics of solutions of the wave equation on De Sitter and De Sitter-Schwarzschild spaces.''
  

• September 19: No seminar

• September 26: Hans Christianson (MIT)
  ``Dispersive Equations and Hyperbolic Orbits.'' notes available
  

  Abstract: I will discuss several related PDE problems which all fall loosely into the class of "dispersive equations". These problems have all be studied extensively in Euclidean space and other non-trapping geometries, but a hyperbolic periodic orbit changes things. I will discuss the similarities and differences, and explain a microlocal estimate near the periodic orbit which can be "glued" into non-trapping results to study these problems in the case of one trapped hyperbolic orbit.

• October 3: No seminar

• October 10: Gigliola Staffilani (MIT)
  Weak turbulence for 2D periodic cubic NLS
  

• October 17: No seminar on account of Wiener lectures

• November 7: Alex Ionescu (UW - Madison)
  "A conditional "no hair" theorem for smooth manifolds"
  

  Abstract: In joint work with S. Klainerman, we prove that if E is the exterior region of a smooth, regular, stationary Einstein vacuum spacetime M of dimension 4, and if certain technical conditions are satisfied, then E is locally isometric to the domain of outer communication of a Kerr spacetime. The main technical condition we assume is an identity relating the Ernst potential and the Killing scalar on the bifurcate sphere of the spacetime M.

• November 14: Jeremy Marzuola (Columbia)
  "A stable class of perturbations for minimal mass soliton solutions of saturated Nonlinear Schrödinger Equations in 3D"
  

• November 21: Mihalis Dafermos (Cambridge)
  ``The wave equation on Schwarzschild and Schwarzschild-de Sitter spacetimes.''
  

• November 28: Carlos Kenig (Chicago)
  "Global well-posedness and scattering for the energy critical, focusing non linear wave equation"
  

• December 5: Alvaro Pelayo (MIT)
  "Symplectic actions of two-tori on four-manifolds''
  

  Abstract: Some results are joint with J.J. Duistermaat. I will present a classification of symplectic actions of two-tori on compact, connected symplectic four-manifolds. The classification is in terms of a collection of invariants of the topology of the manifold, of the torus action and of the symplectic form. I will present explicit models of such symplectic manifolds with torus actions, defined in terms of these invariants. A stepping stone for this classification is the previous classification of symplectic torus actions with coisotropic principal orbits due to J.J. Duistermaat and the speaker. One important ingredient of the proof of the classification is the construction of certain connections for some principal torus bundle determined by the 4-manifold.

• December 12: Colin Guillarmou (Nice) (Note special time and location: 3 PM in 4-163)
  "Strichartz estimate with no loss in hyperbolic trapping"
  

  Abstract: (with Burq and Hassell) we show that Strichartz estimate with no loss holds in some cases with hyperbolic trapped set.

• December 12: Lydia Bieri (Harvard)
  "An Extension of the Stability Theorem of the Minkowski Space in General Relativity"
  

  Abstract: The talk addresses the global, nonlinear stability of solutions of the Einstein equations in General Relativity. In particular, it deals with the initial value problem for the Einstein vacuum equations, generalizing the results of D. Christodoulou and S. Klainerman in 'The global nonlinear stability of the Minkowski space'. Every strongly asymptotically flat, maximal, initial data which is globally close to the trivial data gives rise to a solution which is a complete spacetime tending to the Minkowski spacetime at infinity along any geodesic. We consider the Cauchy problem with more general, asymptotically flat initial data. This yields a spacetime curvature which is not bounded in L^∞ any more. The main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime.