**"Invariant manifolds and dispersive Hamiltonian evolution equations" **

Abstract: By means of certain dispersive PDEs (such as the nonlinear
Klein-Gordon equation) we will exhibit a new family of phenomena
related to the ground state stationary solutions. These ground states are exponentially
unstable, and one can construct stable, (un)stable, and center(-stable) manifolds
associated with them in the sense of hyperbolic dynamics.

In terms of these invariant manifolds one can completely characterize the global dynamics
of solutions whose energy exceeds that of the ground states by at most a
small amount. In particular, we will establish a trichotomy in forward time giving either finite-time
blow up, global forward existence and scattering to zero, or global existence
and scattering to the ground states as all possibilities. It turns out that all
nine sets consisting of all possible combinations of the forward/backward
trichotomies arise.

This extends the classical Payne-Sattinger picture from 1975 which gives such a
characterization at energies below that of the ground state; in the latter case
the aforementioned (un)stable and center(-stable) manifolds do not arise, since they require
larger energy than that of the ground state. Our methods proceed by combining
a perturbative analysis near the ground states with a global and variational
analysis away from them. This work is joint with Kenji Nakanishi of Kyoto University, Japan.

Wilhelm Schlag received his undergraduate education at the University of Vienna and his graduate education at the University of California at Berkeley and the California Institute of Technology. He obtained his Ph.D. degree in mathematics under Thomas Wolff in 1996 at Caltech.
He has held faculty positions at Princeton University, Caltech, and the University of Chicago, as well as visiting positions at the Institute for Advanced Study, Princeton and the Mathematical Sciences Research Institute at Berkeley. He is currently a Professor of Mathematics at the University of Chicago.
He is the author of more than fifty scholarly papers and acts as editor of Geometric and Functional Analysis, the Bulletin of the Mathematical Society of France, and Analysis & PDE. He has been an invited speaker at numerous national and international meetings, including the International Congress of Mathematical Physics in 2004 at Lisbon, Portugal.

* The
Thomas Wolff lectures, sponsored by donations from his widow and his parents, memorialize
Caltech’s great analyst who was tragically killed at age 46 in an automobile accident
in July 2000. Wolff was a specialist in
analysis, particularly harmonic analysis. Professor Wolff made numerous highly original
contributions to the mathematical fields of Fourier analysis, partial differential
equations, and complex analysis. A recurrent theme of his work was the application of
finite combinatorial ideas to infinite, continuous problems.*

*His early work on the Corona theorem, done as a Berkeley graduate student, stunned the mathematical community with its simplicity. Tom never wrote it up himself since several book writers asked for permission to include the proof in their books where it appeared not long after he discovered it. After producing a number of very significant papers between 1980 and 1995, he turned to the Kakeya problem and its significance in harmonic analysis, works whose impact is still being explored.*

*Peter Jones, professor of mathematics at Yale, described Tom’s contributions as follows: “The hallmark of his approach to research was to select a problem where the present tools of harmonic analysis were wholly inadequate for the task. After a period of extreme concentration, he would come up with a new technique, usually of astonishing originality. With this new technique and his well-known ability to handle great technical complications, the problem would be solved. After a few more problems in the area were resolved, the field would be changed forever. Tom would move on to an entirely new domain of research, and the rest of the analysis community would spend years trying to catch up. In the mathematical community, the common and rapid response to these breakthroughs was that they were seen not just as watershed events, but as lightning strikes that permanently altered the landscape.”*

*Tom Wolff was noted for his analytic prowess, the depth of his insights, and the passion with which he nurtured the talents of young mathematicians. We miss him.*