## Tuesday, July 26/11:15 AM/Grande Ballroom

# The John von Neumann Lecture

This prize, established in 1959, is in the form of an honorarium for an invited lecture called The John von Neumann Lecture. The lecturer will survey and evaluate a significant and useful contribution to mathematics and its applications. It may be awarded to a mathematician or to a scientist in another field, but in either case, the recipient should be one who has made distinguished contributions to pure and/or applied mathematics.
The 1994 John von Neumann lecturer is Martin D. Kruskal, Rutgers University.

**Chair: Avner Friedman, University of Minnesota, Minneapolis**

## Surreal Numbers

The surreal numbers are a fascinatingly rich system discovered by John H. Conway a couple of decades ago but not yet widely appreciated. In a compellingly natural way, they simultaneously generalize the usual real numbers and Cantor's ordinal numbers, encompassing also a vast slew of infinite and infinitesimal numbers and combinations thereof.
These numbers and their operations and relations are defined very simply and explicitly, and their basic arithmetic properties have strikingly short proofs, so that even restricted to the real numbers their construction is a great improvement on the classical treatment. The theory proceeds from a new fundamental arithmetic concept and leads to unsuspected insights also about simple functions.

At present the only substantial application of the surreal numbers is to the game theory whence they sprang, but there is hope for significant and deep applications to analysis and mathematical physics via the emerging field of asymptotics beyond all orders (exponential asymptotics).

**Martin D. Kruskal, David Hilbert Professor of Mathematics, Rutgers University**

Martin Kruskal grew up in New Rochelle, NY, where as a child he met Richard Courant. He studied at the University of Chicago and then at New York University, where he was much influenced by the ambience Courant had created. Professor Kruskal spent many years at what is now the Princeton Plasma Physics Laboratory working on the mathematical physics and engineering problems of thermonuclear fusion. This led to one of his main lifelong interests, asymptotics in the broadest sense. A subsidiary interest in mathematical foundations made it natural to study surreal numbers when he first learned of them in the early '70s, and he was later galvanized on becoming aware of their still unrealized promise of highly relevant application to asymptotics. He received his B.S. degree from the University of Chicago; his M.S. and Ph.D. degrees from New York University. Martin Kruskal is currently the David Hilbert Professor of Mathematics at Rutgers University, and professor emeritus of astrophysics and mathematics at Princeton University. In 1993, he was awarded the President's National Medal of Science.