Joel Smoller, Professor and Lamberto Cesari chair in Mathematics at the University of Michigan, has done groundbreaking research in a number of different areas connected with nonlinear partial differential equations and their applications, from population dynamics to fluid mechanics to, most recently, general relativity and cosmology. He is perhaps best known for his work on shock wave theory, having written the standard textbook on the subject (Shock Waves and Reaction Diffusion Equations, Springer 1983, 2nd edition 1994) and almost single-handedly populated the field with his students and their own advisees. He is well known for his dynamic and engaging lecture style, and his gift for producing outstanding students. In all, he has successfully guided twenty-three students to a Ph.D., many of whom are now leaders in their (various) fields of mathematics.

Appearing in the top journals of the filed, Smoller has published over 120 research papers, of which seventeen date since 1997. Most recently, his paper, "The Coupling of Gravity to Spin and Electromagnetism," with F. Finster and S.T. Yau received an honorable mention in the Gravity Research Foundation competition in 1999; it is rare and quite significant that a mathematical paper be recognized in this way by a physical body. He has received several prestigious awards, including a Guggenheim Fellowship (1980-1981), the Margaret and Herman Sokol award for excellence in teaching and research (The University of Michigan, 1992), and the aforementioned Cesari chair. He has served as editor of the first-rate journals Transactions of the American Mathematical Society (1981-1985) and Journal for Applicable Analysis (currently). He has held long-term visiting positions at such esteemed institutions as the Courant Institute (New York), Harvard University, the University of Paris, the University of Warwick, Ecole Normale Superieure (Paris), Ecole Polytechnique (Paris), the University of Wisconsin, and the University of California at Davis (Regents visiting professor, 1996). He has given numerous invited lectures around the world: for example, an hour-long invited address at the 500th anniversary of the founding of the University of Heidelberg (1986), a highly prestigious Harvard-MIT-Brandeis colloquium at Harvard (1992), invited addresses at Academica Sinica in the People's Republic of China, Fudan and Bejiing (the two top mathematics institutions), and recent plenary invitations for the International Conference on Hyperbolic Problems, Magdeburg, Germany (2000), and at Los Alamos laboratories (2000).

In his first lecture, Smoller will speak on his recent investigations in gravitation and cosmology, relating to black holes and the big bang theory: specifically on new cosmological models discovered with his collaborator Blake Temple, which seem to contradict existing mathematical theory. In particular, these have the feature that the details of the early universe cannot be known, as information is continually lost; that is, unlike standard cosmological models in which the history of the universe can be traced backwards to its origins in an initial big bang, the models of Smoller and Temple are time-irreversible. Rather than emerging whole from a black-hole singularity, these solutions have two quite different parts, which collide violently along a sort of blast front or shock wave, where the solution loses its smoothness. As is familiar to practitioners of shock and detonation wave theory, such a loss of smoothness entails also loss of information, entropy, and irreversibility: features that to modern schemes of thought are perhaps more palatable than determinism back to the beginning of time. These unexpected new solutions, and their implications, have rapidly caused a buzz of new activity in the field of cosmology.

A second and thematically-related topic explored by Smoller and various collaborators concerns the effects of coupled Einstein and Yang-Mills, Dirac, or Dirac-Maxwell equations, that is, the effects of coupling between gravitational and quantum mechanical and/or electrodynamic effects. A striking result is that the relatively weak force of gravitation can still serve as a regularizing effect, allowing the existence of localized, particle-like solutions; indeed, there exists an infinite family of such solutions. (The nonexistence of particle-like solutions is well known in the absence of gravity.) A second result is that when one couples quantum mechanics (Diracs equation) to gravity and electromagnetism or to the weak nuclear force, stationary black-hole solutions cease to exist. This is in sharp contrast to the situation in the absence of quantum mechanical effects. These issues will be discussed in Smoller's second lecture.

In addition to the Patten Foundation lectures, Professor Smoller will be presenting a Department of Mathematics colloquium April 20 in Swain Hall East, room 240, at 4:00 pm. Following his Patten visit he will be in attendance at the Institute for Advanced Study for an additional week.