Opinion and Los Alamos. It is by gaining a thorough understand- Pure and Applied ing of the problems arising in the subject proper that one develops a feeling for it, and with it, intuition. So, as Mathematics Coifman said, in subjects like biology and informatics we are in a pre-Newtonian age. Toward the end of the recent International Congress of In applied mathematics the emphasis on rigor and Mathematicians in Madrid, there was a panel discussion proof must come at the appropriate stage. Let us consider about whether pure and applied mathematics are drift- an example. Feynman had great intuition but didn’t care ing apart. The panelists were Lennart Carleson, Ronald much for rigor or proofs. He says in one of his autobio- Coifman, Yuri Manin, Helmut Neunzert, and Peter Sarnak. graphical writings that while he was at Cornell he used to John Ball, the president of the International Mathematical talk to William Feller and Mark Kac, the famous probabi- Union, moderated. The following reflections were inspired lists. It is a happy circumstance, for science in general and by that session. mathematics in particular, that Feller and Kac didn’t dis- I studied at the Courant Institute under Harold Grad, miss Feynman as a sloppy crackpot but instead patiently a distinguished applied mathematician. However, I think listened to him. Thus was born the great Feynman-Kac of myself as having one foot in pure mathematics and the formula. The moral, I think, is that pure mathematicians, other in applied. The majority of the audience as well as the panelists while insisting correctly on rigor and proofs, must be were pure mathematicians. So perhaps it would be helpful patient and show some respect toward intuition born to ask, What is applied mathematics? A very good answer out of a deep knowledge of a subject. Attitudes like that was provided by the late Kurt Friedrichs, who distin- of the late Paul Halmos— “Applied mathematics is bad guished himself in both pure and applied mathematics. mathematics” —are shortsighted. For their part, applied He used to say, “Applied mathematics consists in solving mathematicians, while using intuition as their guide, must exact problems approximately and approximate problems recognize the need for and the importance of proofs. On exactly.” Initial and boundary value problems associated the other hand, it is rare that a single individual embodies with the Navier-Stokes equations are an example of prob- all the requisite qualities to a high degree. So often what lems that are extremely difficult to solve exactly and where is needed is a joining of hands of people with disparate we look for approximate solutions. Hence computing is abilities, strengths, and points of view rather than a sepa- an important part of applied mathematics. The Bhatnagar- ration or drifting apart. Gross-Krook equation in kinetic theory and plasma physics It is interesting to note that one doesn’t always know is an example of a solvable problem that approximates an whether one’s work is useful or not. For example, G. H. intractable one. Hardy is considered the quintessential pure mathemati- Apropos of definitions, Peter Lax remarked once, quot- cian. However, Joseph and Maria Mayer (she won the Nobel ing Joe Keller, that pure mathematics is a branch of applied Prize in physics) use, in their classic book on statistical mathematics; this was echoed by one of the ICM panelists. mechanics, a result of Hardy and Ramanujan in the evalu- Some of the greatest mathematicians of the past—Newton, ation of the partition function of an imperfect gas. Some Euler, Lagrange, Gauss, and Riemann—and more recently say that Hardy’s distaste for applications and the pride he Hilbert, Weyl, Wiener, von Neumann, and Kolmogorov did took in the uselessness of his work had roots in his paci- both pure and applied mathematics. We are of this world, fism. The use of poison gas in the First World War, which and nothing comes entirely from within ourselves without he regarded as an application of chemistry, appalled him reference to the external world. and may have had the unfortunate effect of turning him Sarnak said that proofs are essential in pure math, but against applications altogether. One cannot but admire they are essential in applied math too, except that the path such loftiness, however. one takes is rather different. As Carleson said, applied Acknowledgment. It is a pleasure to thank Peter Lax math relies heavily on the inductive method, as opposed to for helpful comments. the deductive method preferred by pure mathematicians. In pure mathematics the emphasis is on rigor. However, —Chelluri C. A. Sastri ideas are far more important. Ideas come from intuition, of Dalhousie University course, which in turn comes from living and breathing the [email protected] subject. Sarnak and his colleague Weinan E are quite right in insisting that even applied mathematicians need basic training in mathematics—roughly what is taught in good North American graduate schools in the first two years of study—but that is not enough. As Grad used to say, one must also immerse oneself completely in the subject to which one wants to apply mathematics. For instance, since he worked in magneto-fluid dynamics, to stay in touch with the experimentalists he regularly visited Oak Ridge APRIL 2007 NOTICES OF THE AMS 471 Letters to the Editor to its adulthood. There were the good Young mathematicians in China need old days when ideas were shared and not just encouragement but a better new frontiers were explored. It was perspective of what the most exciting Retarded Differential Equations during this period that the Ricci flow and promising directions of research and Quantum Mechanics was introduced and investigated by are. My public lecture in Beijing on Hamilton. Thirty years later, geomet- June 20 [2006] was addressed to the G. W. Johnson and I wish to draw ric analysis has reached maturity, and mathematics community and a large attention to the work of C. K. Raju the proof of the Poincaré conjecture group of string theorists. In that talk that is related to some of the ideas is perhaps its most spectacular suc- I focused on the achievements of discussed by Sir Michael Atiyah in cess to date. I expect many more Hamilton and Perelman. Since Cao his talk “The Nature of Space”, which successes to come. and Zhu managed to put together in we reported on in the June/July 2006 The achievements of Hamilton and writing the details of the deep ideas issue of the Notices. Ideas suggesting Perelman in solving the conjecture, of Hamilton and Perelman, I praised a link between retarded differential equations and quantum mechanics especially their major breakthroughs them as well, hoping this would en- were put forward some years ago on singularities of nonlinear para- courage their fellow mathematicians by Raju, and we, along with Atiyah, bolic systems and the structure theo- in China. believe they deserve attention. Inter- rem for 3-dimensional manifolds, are Over the years I have inherited ested readers are encouraged to read, unparalleled. They far exceed the from my teacher S. S. Chern the in particular, the following papers established standards for Fields Med- strong belief that it is the duty of written by Raju: als. I fully support, and have always any mature mathematician to train 1. Time: Towards a Consistent The- said so, the award of the Fields Medal the next generation. Since he and I ory, Kluwer Academic, Dordrecht, to Perelman. (In my view, Hamilton both come from China and there are 1994 (Fundamental Theories of Phys- clearly deserves the Fields Medal many talented young Chinese math- ics, vol. 65), ch. 5b “Electromagnetic also, but he is not eligible at this time ematicians who are not exposed to time” (pp. 116–122), and ch. 6b “Quan- because of the age restriction.) For modern mathematics, we have spent tum mechanical time” (pp. 161–189). anyone to suggest in words or a car- a lot of time helping mathematicians 2. The Eleven Pictures of Time, toon that my position has ever been and students in China. We devoted a Sage, 2003, pp. 298–302. anything but that is both offensive lot of time discussing the challenges 3. “The electrodynamic 2-body and completely untrue. and working together to address problem and the origin of quan- Proving the Poincaré conjecture is them. Thanks to his leadership, there tum mechanics”, Foundations of an intricate and daunting process. In are now many outstanding Chinese- Physics, 34, (June 2004), 937–962. a work of this scale, it is understand- trained mathematicians in Western able that when Perelman released universities. Over the last twenty —Mark E. Walker his manuscripts on arxiv.org, several years, Chern and I have also been University of Nebraska key steps were merely sketched or trying to develop mathematics within [email protected] outlined. These manuscripts posed Chinese universities. Because of the a tremendous challenge for the math Cultural Revolution, the recovery has (Received December 29, 2006) community to digest. For two years been slow. But thanks also to the help many top experts in the field of geo- of many friends from the West, the metric analysis worked hard and situation is improving. The Proof of the Poincaré made steady progress in understand- There have been uninformed re- Conjecture ing and clarifying Perelman’s papers. ports on how the Cao-Zhu paper was In recent months there has been At the end of 2005, Cao and Zhu com- handled by the Asian Journal of Math- considerable attention devoted to pleted a three-hundred-plus-pages ematics, as well as on the joint work the proof of the Poincaré conjecture manuscript that provided a complete by Lian, Liu, and me on the mirror that was started by Hamilton and account of the Hamilton-Perelman symmetry conjecture.