2nd
Thomas Wolff
Memorial Lectures
in Mathematics
April 1, 3,
8, and 10, 2003
4:15 p.m.
Room 151 Sloan
Speaker:
TERENCE TAO
Professor of Mathematics
University of California, Los Angeles
Long-time behavior of semi-linear dispersive equations
There has been much recent progress on the long-time analysis of non-linear dispersive equations such as the Korteweg-de Vries equation, non-linear Schrödinger equations such as the cubic NLS equation, and non-linear wave equations. These equations are model equations for the behavior of various waves in physics. We consider the following general questions about these equations:
I. Local existence. For a specified initial data u(0,x), can one create a unique solution for some non-zero amount of time? What regularity conditions are needed on the initial data? What kind of estimates are available on the solution?
II. Global existence. Can the local solution be extended to a global one, or do singularities form, and if so, how do they form? Do the available bounds on the solution change with time?
III. Long-time behavior of solutions. Given a global solution, what are the asymptotics of this solution? Does the solution eventually approach a solution to the linear equation (i.e., do we have scattering)? Or does the solution resolve into soliton waves (the soliton resolution conjecture)? Or does the energy go from low frequency modes to high frequency modes (turbulence)? If so, how fast (weak turbulence vs. strong turbulence)? Do these infinite-dimensional Hamiltonian equations behave like their finite-dimensional counterparts (e.g., is there “symplectic non-squeezing”)?
The questions in category III are the most interesting for physical applications, but it turns out that one must first develop a sufficiently precise theory for questions I and II before we can obtain good answers to III. In this series of talks, we discuss these questions and some recent results and techniques.
The talks will be arranged as follows:
Talk 1 (April 1). General non-technical overview of results
Talk 2 (April 3). Local and global existence theory
Talk 3 (April 8). Turbulence; scattering
Talk 4 (April 10). Non-squeezing; stability of solitons
TERENCE TAO graduated from Flinders
University of South Australia in 1992 with a B.Sc.(Hons) and M.Sc. in
Mathematics. He earned a Ph.D. in Mathematics from Princeton University in
1996 under the guidance of Elias Stein. He is currently a professor of
mathematics at UCLA and is a recipient of fellowships from the Sloan
Foundation, Packard Foundation, and the Clay Mathematics Institute. He
received the Salem Prize in 2000 for his work in harmonic analysis and for
questions in geometric measure theory and partial differential equations. In
2002, he received the Bôcher Prize for “his recent breakthrough on the
problem of critical regularity in Sobolev spaces of the wave maps equations,
his collaborative papers on global regularity in optimal Sobolev spaces for
KdV, and his contributions to Strichartz and bilinear estimates.”
Tao’s research includes harmonic analysis, partial differential equations (especially non-linear wave and dispersive equations), geometric combinatorics, and representation theory (especially of U(n)).
The Thomas Wolff
Memorial Lectures
In Mathematics
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, mathematics department chair 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
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.