Stanislaw Ulam: 105th birthday

I didn't have enough time in the morning but the 95-vs-105 numerical error is still very painful because my late maternal grandfather was born in 1909, too

Stanislaw Ulam was born in Lviv, Galicia, on April 13th, 1909. His broader family, the Ulams, was a very wealthy one in the region. His immediate family was doing fine but not great. He would study at the Lviv Polytechnic Institute which was a Polish school. It's useful to keep these nationalities in mind when you think about Western Ukraine – where Lviv belongs today. Achievers like Ulam would be Polish Jews for quite some time. But Galicia didn't belong to "Poland" at that time; it was a part of "my country", Austria-Hungary.

He was invited to the U.S. by Hans Bethe and has been affiliated with the IAS at Princeton, Harvard, U. of Wisconsin, U. of Colorado, U. of Florida, and Los Alamos National Laboratory at various moments. He became a U.S. citizen in 1941 – before he began to work on the Manhattan Project. He did quite some important calculations over there – both on hydrodynamic calculations of implosions, and the statistics of multiplicative processes. He was the boss of a group of female computers. Female computers are constructed out of women; at that time, they contained less silicon than they contain today. ;-)




Along with Edward Teller, he is the father of the Ulam-Teller design (primary fission explosive; secondary fusion bomb). All thermonuclear bombs that nations possess today are based on this design.

Ulam has done a surprising amount of "real mathematics" whose relationships to the bombs are not self-evident – in disciplines including set theory (also measurable cardinals and abstract measures), topology, transformation theory, ergodic theory, group theory, projective algebra, number theory, combinatorics, and graph theory. The Borsuk-Ulam theorem claims that maps from Euclidean spheres to themselves map at least one pair antipodal points to a pair of antipodal points. A simpler Mazur-Ulam theorem says that surjective isommetries between normed spaces have to be affine. The Kuratowski-Ulam theorem talks about the meager and comeager subsets of products of Polish spaces – a counterpart of the Fubini theorem (on doing multi-dimensional integrals by steps). His name also appears in the Hyers–Ulam–Rassias stability.




But I want to mention two results that are exceptionally important.

The Fermi-Pasta-Ulam problem (this "Pasta" isn't just some spaghetti sandwiched in between two mathematicians; it was an actual John Pasta) was the question why seemingly complicated systems of nonlinear differential equations tend to have periodic rather than ergodic (chaotic) behavior. The answer is that many problems are exactly integrable.

Finally, Stanislaw Ulam developed the modern version of what we currently call the Monte Carlo method. Complicated systems that can't be calculated accurately may be "approximately calculated" by looking at (and statistically analyzing) the properties of random configurations (chosen by a random generators). This is called after "Monte Carlo" because it resembles the way how a good player is "learning from the random experience" in the casinos.

Monte Carlo is an important method used across sciences and applied sciences. On the other hand, it is not hard to invent the method. I am surely among those who would claim that they invented it independently of Ulam – some decades later. (That's the first method I used to calculate the volumes of n-dimensional balls when I was a basic school student; the real discovery for me was that one can calculate the volume in a non-Monte-Carlo way of discretizing the integral of the lower-dimensional balls LOL.)

Stanislaw Ulam died in Santa Fe, New Mexico, in 1984.