In the grand scheme of proving the bounded L^2 curvature conjecture for the Einstein vacuum equations, bilinear estimates play a very important role. To prove them, one needs a good strategy, in particular, the right concept of an approximate solution to the wave equation on a curved background with limited regularity. September 5, 2003

This is the  third paper in the series with I. Rodnianski on  null hypersurfaces  on Einstein vacuum backgrounds  verifying the finite curvature flux condition.. We prove the results of the main lemma in the first paper of the series, see above. concerning sharp trace theorems in Besov type spaces.  This is the most technical of the series.   Now available,   August 21 2003.

 Together with I. Rodnianski we develop a  geometric version,of  LP-theory on  Riemannian manifolds by a heat flow approach.  We show how to recover the usual properties of the classical LP calculus by using only very limited information concerning the reglarity of the manifold. These results are developed  for application to  control spacelike hypersurfaces on Einstien vacuum backgrounds which satisfy the  bounded currvature flux condition of  the above paper.    September  21, 2003.

One of the central difficulties in connection to the L^2 bounded curvature conjecture in general relativity is to be able to control the causal structure of spacetimes with this very limited regularity. This is the first, and main, in a sequence of three papers, written in collaboration with I. Rodnianski , in which we  circumvent this difficulty by showing that the geometry of null hypersurfaces of Einstein vacuum spacetimes can be controlled  by the total curvature flux through the hypersurface.    August 8,  2003

We  show that under stronger asymptotic decay properties than those used in the Christodoulou -Kl. and Kl-Nicolo  global stabilty results, asymptotically flat initial data sets lead to solutions of the Einstein vacuum equations which have strong peeling properties  consistent with the predictions  of the conformal compactification approach of Penrose.  Even the existence of spacetimes with such strong peeling properties has remained open until   recently.  Our methods, based on Kl-Nicolo,  give a systematic picture of the relationship between  asymptotic propertie of the data and the peeling properties of the corresponding developments .  To appear in Quantum and Classic Gravity

The third paper in the series with I. Rodnianski on rough solutions to the Einstein vacuum equations. We prove an important result which was needed in our second paper.

The second paper written in collaboration with I. Rodnianski on solutions with minimal regularity for the Einstein equations.. In the second paper we concentrate on the crucial new estimates for the Eikonal equation.

The first in a sequence of three papers written in collaboration with I. Rodnianski on solutions with minimal regularity for the Einstein equations.. Taking advantage oif the  special structure of the equations, in wave coordinates, we get the optimal regularity result , $H^{2+\epsilon}$, which can be derived by Strichartz type estimates.

A new book written in collaboration with F. Nicolo on the initial value problem in General Relativity. The main goal of the book is to revisit the proof of glaobal nonlinear stability of the Minkowski space by Christodoulou-Klainerman. We provide a new self contained proof of the main part of that result  concerrning the full solution of the radiation problem in vacuum, for arbitrary assymptotically flat initial data. The proof, which is a significant modification of the arguments developed in Ch-Kl is based on a double null foliation rather than the mixed null-maximal foliation used in Ch-Kl.

A paper written in collaboration with I. Rodnianski and T. Tao in which we expereiment with new proofs of bilinear estimates which have the potential to generalize to quasilinear equations.

This paper, in collaboration with G. Staffilani, provides a new proof an old conditional regularity result of Glassey-Strauss

The set of lectures delivered at the IPAM workshop in Oscillatory Integrals and PDE' March 19-25 2001, UCLA.

This paper, in collaboration with I. Rodnianski, is an extension of the recent higher dimensional critical regularity result of T. Tao to general bounded parallelizable Riemannian target manifolds.

This paper, in collaboration with I. Rodnianski, improves Tataru's well known recent regularity result for quasilinear wave equations. The paper relies on a significant improvement of the geometric technique developed earlier in a commuting vectorfield approach paper.

This is a survey paper in collaboration with S. Selberg concerning optimal well posedness results for nonlinear wave equations such as Wave Maps, Maxwell-Klein-Gordon and Yang Mills. The survey provides a unified treatment and simplified proofs for some old results of Klainerman-Machedon, Klainerman-Selberg, Klainerman-Tataru.

This is another philosophical lecture delivered at the AMS Millenium Conference in Los Angeles, August, 2000.

Geometric and Fourier Methods in Nonlinear Wave Equations(Lecture in Tel-Aviv, Aug, 1999).

This is a philosophical essay reflecting my personal views about PDE's. Published in the proceedings of the conference `` Visions in Mathematics'' Tel Aviv 1999.

This paper marks my wholehearted return to quasilinear wave equations, after a detour of more than ten years. I show that one can recover some of the recent results of Chemin-Bahouri and Tataru by reducing the crucial dispersive inequality ( the key ingredient in the Strichartz type estimates) to L^2--L^\infty decay estimates based on energy estimates and an adaptation of the commuting vectorfields method.