Adventures of a Mathematician

A Note on S. M. Ulam’s Mathematics by Jan Mycielski

Stanislaw Ulam belonged to a group of mathematicians who came to the United States from Poland before and during the Second World War and played a very important role in the life of mathematics in this country. Among them one can name Natan Aronszajn, Stefan Bergman, Samuel Eilenberg, Witold Hurewicz, Mark Kac, Otton Nikodym, Alfred Tarski, and Antoni Zygmund. They escaped the fate of many millions killed during the war, among whom were the important Polish mathematicians Jozef Marcinkiewicz, Stanislaw Saks, and Juliusz Schauder. The history of mathematics would have been very different if the war had not decimated the Polish youth and almost exterminated the Jewish youth of Europe.

Ulam wrote over 150 technical papers and three books: A Collection of Mathematical Problems, published in 1960, Mathematics and Logic (with Mark Kac), published in 1968, and this autobiography published in 1976. Selections of his papers were published in Sets, Numbers, Universes[1] (referred to as SNU), Science, Computers, and People[2] and Analogies Between Analogies[3] (referred to as ABA). A collection of articles about his contributions appeared in From Cardinals to Chaos.[4]

Françoise, his wife, had a great influence upon his life. She managed their various homes with incredible efficiency and helped in the organization of his many travels. In addition she edited this autobiography from tapes and was a coeditor of the collections of his papers published posthumously.

Endowed with exceptional charm and intelligence, Ulam had an unusual facility for establishing contacts with people. His most pleasant quality was his openness and spontaneity. Without hesitation he shared his ideas with all who were interested. Moreover he had a remarkable memory and a broad humanistic culture acquired in an excellent secondary school in Lwów.

Ulam had a unique ability to raise important unsolved problems. These problems exerted an exceptional influence upon the work of many scientists. In the years of our acquaintance which began in 1969, he preferred to invent open questions, especially those at the boundary of mathematics, physics, and biology, rather than to go into the details of mathematical work. He became a pioneer of computer applications to the heuristic study of dynamical systems. Although he encouraged younger mathematicians and liked to see the work of people from its best side, he looked at mathematics from the point of view of a scientist whose purpose is to study nature. He criticized these mathematical problems which do not appear to have a direct or natural motivation, calling them “Chinese puzzles.” He preferred to think about physics and biology, and the mathematical problems that derive from them.

This gift is best illustrated by one of his conjectures which was soon proved by K. Borsuk. It is called “the antipodal theorem.” To each continuous function f from the spherical surface to the plane, there exist two antipodal points on the sphere, x and — x, such that f(x) = f(-x).

A well-known consequence of the antipodal theorem was formulated by Steinhaus and is called “the ham and cheese sandwich theorem.” It says that if you have a sandwich composed of bread, ham, and cheese, there is one plane which evenly divides the bread, the ham, and the cheese. In more formal language: For any three measurable sets in three-dimensional space, there exists a plane which halves each of them.

Ulam’s work pertains to so many areas of mathematics and other sciences connected with it, and is so manifold, that only his most important contributions can be mentioned here.

Ulam’s doctoral dissertation, written in 1931 (SNU, pp. 9–19), pertaining to the size of certain infinite cardinal numbers, contains results which are very important for the foundations of modern mathematics.

Like geometry in the times of Euclid, modern mathematics can be synthesized within a compact list of axioms. The concepts used in that axiomatization belong to Cantor’s set theory. Strictly speaking we have an infinite hierarchy of stronger and stronger set theories. The first of them, called Zermelo-Fraenkel set theory (and denoted ZF) suffices for almost all mathematics. But an intriguing part is still missing. The theorems of that part require some stronger theories of this hierarchy. (Those theorems belong to descriptive set theory, problems of measurability, capacitability, property of Baire, determinacy, the theory of ideals of sets, etc.) The most natural way to gauge the strength of those theories is to look at the size of infinite cardinal numbers whose existence can be proved in them.

Ulam’s work constitues the first deeper investigation of the size of a few such gauge-cardinals. Since 1931 the theory has undergone substantial development. But it was not until 1960, almost thirty years after his work, that W. Hanf and A. Tarski, using a theorem of J. Los, made the next step on that road. Since then many mathematicians (Keisler, Martin, Reinhardt, Solovay, and Woodin, to name only a few), have developed this topic.

Ulam was also the first to define in The Scottish Book[5] the binary infinite game of perfect information which was used later by H. Steinhaus and this writer to express the axiom of determinacy. (See commentary in The Scottish Book, pp. 113–116). Ulam’s contributions to set theory are reproduced and their subsequent influence is outlined in SNU.

Before the war Schreier and Ulam were studying the group of homeomorphisms of the n-dimensional sphere S (see SNU). Later Oxtoby and Ulam (SNU) proved several fundamental results about this group.

The most important of these can be explained as follows (omitting mathematical precision): If we gently mix a glass of water, we get a transformation which is continuous (because water has some viscosity) and preserves volumes (because water is nearly incompressible). Now their theorem tells us that almost all such transformations have the property that no portion of the water which has a positive volume, except the whole glass, will occupy the same location before and after the mixing.

Transformations with the above properties are called ergodic, and before the work of Oxtoby and Ulam, it was not even known that such transformations exist.

Ulam had a very geometrical way of thinking and he was interested in problems of geometric topology (e.g., the antipodal theorem mentioned above). But he also contributed to general topology.

With Kuratowski he extended Fubini’s classical theorem from the paradigm of measure to the paradigm of the Baire category. With Schreier he showed that every homeomorphism of the sphere S

Everett and Ulam wrote important papers about processes of the Galton-Watson type (ABA). They studied the probability connected with the cascades of elementary particles caused by collisions of energetic particles. Their chief results pertained to the relative number of particles of different kinds such as neutrons and uranium nuclei. Since that time, the theory of branching processes has undergone a great deal of further development.

The theory of nonlinear systems was initiated by the work of Fermi, Pasta, and Ulam (SNU, ABA). Experimenting with a computer, they discovered that a vibrating string whose classical equation is perturbed by a certain nonlinear term almost returns to its original position, and much earlier than expected from statistical considerations. Thus, in spite of the existence of the nonlinear term, energy does not disperse over higher harmonics but remains concentrated over the first few and often returns to the very first. This work initiated a great number of studies by physicists and mathematicians; in particular it led to the theory of solitons.

Stein and Ulam obtained interesting results on nonlinear transformations, their stable and non-stable fixed points, periodic points, and others (see SNU or ABA). The subject of their studies is connected with the classical paper of Volterra about the fluctuation of relative population numbers of different species of fishes in ponds. The relative number of individuals of each of the n species is represented by baricentric coordinates of a point in an n-1 dimensional simplex. The simplex is subject to a transformation which describes the law of the evolution of the population. Stein and Ulam studied the movement of points in the simplex under various simple transformations of that kind and they obtained very strange trajectories of these points. Their experiments suggested the existence of complicated sets, so-called strange attractors, and others. This became a very lively field of study.

Papers with von Neumann and Richtmyer (ABA), and with Metropolis (SNU), propose the application of computers to statistical studies by the method of random samples. This became known as the Monte Carlo Method. Multidimensional integrals can often be evaluated by this method even when all other methods fail. In this way the efficiency of shields of nuclear reactors, so-called neutron transport problems, and many other problems were solved. By means of computers Ulam also studied the evolution of populations of stars under the law of gravitational attraction and the evolution of genes in the genetic pool of a species, taking into account mutations and sexual reproduction. He also played with simple rules leading to the interesting evolution of some discrete dynamic systems (ABA).

His work on fusion and the hydrogen bomb remains classified. (See article by Hirsch and Mathews.)

The life and work of Stanislaw Ulam teaches us that one can make important contributions to science by letting one’s imagination roam freely upon unexplored topics and by taking the fullest advantage of the universality of the language of mathematics.

JAN MYCIELSKI,

SANTA FE, 1990