Carlos E. Kenig will deliver the Howard Rowlee Lecture at 4:00pm on Friday, April 16, 2010, in 115 Avery Hall.

To be preceded by a reception in 348 Avery Hall from 3:15-3:50pm.

Publicity poster

In addition to the Rowlee Lecture, there will be a conference on Harmonic Anaysis and PDE's held April 16-18, 2010.

## The Global Behavior of Solutions to Critical Nonlinear Dispersive and Wave Equations

### Abstract

We describe a method (which I call the concentration-compactness/rigidity theorem method) which Frank Merle and I have developed to study global well-posedness and scattering for critical non-linear dispersive and wave equations. Such problems are natural extensions of non-linear elliptic problems which were studied earlier, for instance in the context of the Yamabe problem and of harmonic maps. We illustrate the method with some concrete examples and also mention other applications of these ideas.

### About the Speaker

Professor Carlos Kenig currently holds the position of Louis Block Distinguished Service Professor in the Department of Mathematics of the University of Chicago. His area of expertise is in the fields of harmonic analysis and partial differential equations (PDE's). The term harmonic analysis classically refers to the decomposition of a given function (or oscillation) into the sum of pure sine oscillations, plus possibly a constant part. Research problems in this branch of analysis have been pursued since the eighteenth century, by some of the greatest minds in the history of mathematics; e.g., Joseph Fourier, Jean Le Rond d'Alembert, Joseph Louis Lagrange, Daniel Bernoulli, Pierre-Simon de Laplace and Alberto Calderón. By dint of the breadth and sheer fundamental importance of his work in harmonic and PDE analysis, one can readily argue that Professor Kenig is a worthy heir to a very rich legacy.

In fact, Professor Kenig currently has around 200 publications, which in the aggregate has been cited 3213 times, according to MathSciNet. (MathSciNet is a citation database operated by the American Mathematical Society, and so is accorded to be the more reliable in the mathematical sciences.) The unifying theme of his widespread research has been the application and creation of harmonic analytical techniques in the solution of those concrete PDE's which come directly from the physical sciences and engineering disciplines. Particular examples of Professor Kenig's work include his seminal paper with G. Ponce and L. Vega, Well-posedness and scattering results for generalized Korteweg-de Vries equations via the contraction principle, *Comm. Pure Appl. Math., Vol. 46* (1993), 527-620; his remarkable work with A. Ionescu, Global well-posedness of the Benjamin-Ono equation in low regularity spaces, *J. Amer. Math. Soc. 20* (2007), 753-798; and his outstanding work with F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation, *Acta Math. 201* (2008), 147-212. Aside from the intrinsic worth of his papers, many of them have served as a roadmap for subsequent researchers attempting to invoke harmonic analysis in order to answer deep questions in PDE's. Because of Professor Kenig's research accomplishments he was recently honored with the 2008 AMS Maxime Bôcher Memorial Prize. Presented every three years by the American Mathematical Society, the Bôcher Prize is one of the highest distinctions in the field of analysis. In the past, he has also held Sloan and Guggenheim Fellowships; in 1986 and 2002 he was an invited speaker at the *International Congress of Mathematicians*, at Berkeley and Beijing, respectively. Moreover, since 2002 he has been a fellow of the American Academy of Arts and Sciences.

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