Stanislaw Ulam

How to use this course

Stanislaw Ulam described his own gift as a talent for asking questions. He worked in a style formed at the café tables of Lwów: short papers, many problems, quick conversations, and a habit of handing ideas to collaborators. His career split in two. Until 1943 he was a pure mathematician in set theory, topology and measure theory. From 1944 he worked at Los Alamos, where he used the first electronic computers to explore problems no one could solve on paper, and where he contributed the decisive idea of the hydrogen bomb. This course follows both halves and the thread that ties them: probability, iteration and experiment.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04–06 cover the Polish and Harvard years; Lessons 07–09 the Los Alamos work on stochastic processes and computation; Lesson 10 a selection of his problems. The city of his birth was Lemberg in Austria-Hungary, Lwów in interwar Poland, and is Lviv in Ukraine today; this course uses the Polish name of his youth.

Timeline

Lwów, 1909–1927

Stanisław Marcin Ulam was born on 13 April 1909 in Lemberg, the capital of the Austrian crown land of Galicia, which became the Polish city of Lwów in 1918. His family was Jewish, Polish-speaking and wealthy. His father, Józef Ulam, was a lawyer; his mother, Anna Auerbach, came from a family of industrialists in Stryj. He had a younger sister, Stefania, and a younger brother, Adam, who became a historian of the Soviet Union at Harvard. During the First World War the family lived in Vienna, from 1916 to 1918.

As a boy Ulam was drawn first to astronomy. An uncle gave him a telescope, and through popular accounts of relativity he came to mathematics. By his own account he was learning calculus and set theory from books at about fourteen. He attended Gymnasium VII in Lwów and finished in 1927.

The city he grew up in was becoming one of the world’s centres of mathematics. Hugo Steinhaus and Stefan Banach had made Lwów’s Jan Kazimierz University the home of functional analysis, and the Polytechnic had strong mathematicians of its own. Ulam’s family wanted him to study something practical. He chose the Polytechnic, nominally for engineering.

The Polytechnic and the Scottish Café, 1927–1935

In his first year at the Polytechnic, in 1927, Ulam attended the set theory course of Kazimierz Kuratowski, then a young professor there. He solved an open problem from the course and quickly became Kuratowski’s student in all but name. His first paper, Concerning functions of sets, appeared in Fundamenta Mathematicae in 1929. In 1930 he published the paper on measures on general sets that introduced what are now called measurable cardinals (Lesson 04). He received a master’s degree in 1932 and his doctorate from the Polytechnic in 1933, with Kuratowski and Włodzimierz Stożek as advisors. The thesis built on the measure problem that Henri Lebesgue had posed and that Banach had studied.

The Scottish Café

Ulam soon joined the circle around Banach, Stanisław Mazur and Steinhaus that met in the Roma and then the Scottish Café. He enjoyed the style: problems were posed, attacked for hours and sometimes recorded on the marble tables. With Mazur he proved in 1932 that isometries of normed spaces are affine. In conversations with Karol Borsuk he formulated the antipodal theorem that Borsuk proved in 1933. With Józef Schreier he wrote a series of papers on groups of permutations and transformations. Around 1935 the café began keeping its problems in a notebook, the Scottish Book. Ulam contributed about 40 problems by himself, 11 with Banach and Mazur, and 15 with others.

After the doctorate he travelled: to Vienna, Zürich, Paris and Cambridge in England. Abroad he met mathematicians who would shape his career, among them John von Neumann, whom he met in Warsaw in 1935 and who invited him to Princeton.

Princeton, Harvard, Madison, 1935–1943

Ulam arrived in New York on 7 January 1936 for a visit of a few months at the Institute for Advanced Study. Von Neumann became his closest friend and most important collaborator. At Princeton he met George David Birkhoff, who proposed him for the Harvard Society of Fellows. From the autumn of 1936 he spent the academic years at Harvard as a Junior Fellow and the summers in Lwów. At Harvard he worked with John Oxtoby on ergodic theory (Lesson 06).

In August 1939 he sailed from Poland to America with his seventeen-year-old brother Adam. The German invasion of Poland came days later. His father, his sister and most of his family were murdered in the Holocaust. Many of his Lwów colleagues, including Schreier, Juliusz Schauder and Herman Auerbach, were also killed.

With his fellowship ending, Ulam was a lecturer at Harvard in 1939–40 and then became an assistant professor at the University of Wisconsin in Madison. There, in 1940, he asked whether an approximate homomorphism must be close to a true one, the starting point of stability theory for functional equations (Lesson 05). Paul Kelly completed a doctorate under him at Madison in 1942; the graph reconstruction conjecture now bears both their names. On 19 August 1941 Ulam married Françoise Aron, a French exchange student he had met in Cambridge, Massachusetts.

Ulam wanted to contribute to the war effort. In late 1943 von Neumann told him, without saying where, of interesting work at a place in New Mexico. Hans Bethe sent the formal invitation. Ulam asked the Madison library for books about New Mexico and found that the few borrowers before him were colleagues who had quietly disappeared.

Measurable cardinals

For the general reader. Lebesgue measure assigns a length to many sets of real numbers, but not to all. Could some other notion of size, still additive over countably many pieces, measure every set? Ulam showed in 1930 that the answer depends on the size of the infinite set itself. Only an enormous set, far beyond any that ordinary mathematics needs, could carry such a measure. His theorem began the theory of large cardinals.

Let \(S\) be a set. A measure on \(S\) in Ulam’s sense is a function \(\mu:\mathcal P(S)\to[0,1]\) with \(\mu(S)=1\), \(\mu(\{s\})=0\) for each point, and \(\mu\big(\bigcup_nA_n\big)=\sum_n\mu(A_n)\) for disjoint \(A_n\). It is two-valued if it takes only the values 0 and 1. A cardinal \(\kappa\) is real-valued measurable if a set of size \(\kappa\) carries a measure that is \(\kappa\)-additive, and measurable if it carries a \(\kappa\)-additive two-valued one.

In particular, no measure exists on a set of size \(\aleph_1\), or of size \(\aleph_n\), or of any successor cardinal below the first weakly inaccessible. Applied to \(\mathbb R\): if Lebesgue measure on \([0,1]\) extends to a countably additive measure on all subsets, then \(2^{\aleph_0}\) is at least as large as the least weakly inaccessible cardinal. Stefan Banach and Kuratowski had shown in 1929 that the continuum hypothesis rules this out; Ulam’s theorem removed the hypothesis.

The Ulam matrix

Proof. For each \(\alpha<\omega_1\), \(\alpha\) is countable, so fix an injection \(f_\alpha:\alpha\to\omega\). For \(\xi<\omega_1\) and \(n\in\omega\), let

(i) For fixed \(n\), the sets \(A_{\xi,n}\) (\(\xi<\omega_1\)) are pairwise disjoint, because \(f_\alpha\) is injective. (ii) For fixed \(\xi\), \(\bigcup_nA_{\xi,n}=\{\alpha:\alpha>\xi\}\), whose complement is countable and so has measure 0; hence some \(n=n(\xi)\) has \(\mu(A_{\xi,n(\xi)})>0\). Uncountably many \(\xi\) share the same \(n(\xi)=n\), giving uncountably many disjoint sets \(A_{\xi,n}\) of positive measure. In a probability space at most \(k\) disjoint sets have measure \(\ge1/k\), so at most countably many have positive measure. Contradiction.

The same matrix, with \(\kappa\) in place of \(\omega\), shows that a successor cardinal is never the least cardinal carrying a measure, which is the heart of part (a). Two-valued measures led, through Jerzy Łoś’s ultraproducts and the work of Alfred Tarski, H. Jerome Keisler and Dana Scott around 1960, to the modern theory of large cardinals and elementary embeddings.

Antipodes, isometries, stability

For the general reader. At any moment there are two opposite points on the Earth with exactly the same temperature and air pressure. That is the Borsuk–Ulam theorem, which Ulam conjectured and Borsuk proved. Two other theorems from the same years have the same flavour of a simple question with a sharp answer. A map between spaces of vectors that preserves distances must be essentially linear. A map that is nearly additive must lie close to one that is exactly additive.

Borsuk attributed the question to Ulam. Lazar Lyusternik and Lev Schnirelmann had stated a closely related result in 1930. For \(n=1\), set \(g(x)=f(x)-f(-x)\), which is odd; if \(g(x_0)\ne0\), then \(g\) changes sign on a half circle from \(x_0\) to \(-x_0\) and vanishes in between. In higher dimensions one shows that an odd map \(S^n\to S^{n-1}\) would induce on real projective spaces a map that is impossible in cohomology, or one uses the combinatorial Tucker lemma.

Corollaries include the ham sandwich theorem (Steinhaus, Stone and Tukey): any \(n\) bounded measurable sets in \(\mathbb R^n\) can be simultaneously bisected by one hyperplane. Lyusternik–Schnirelmann: if \(S^n\) is covered by \(n+1\) closed sets, one of them contains a pair of antipodes. László Lovász used the theorem in 1978 to compute the chromatic number of Kneser graphs, starting topological combinatorics.

The key step shows that \(T\) maps midpoints to midpoints; additivity and continuity then give real-linearity of \(x\mapsto T(x)-T(0)\). Midpoints are characterized metrically through a decreasing sequence of sets, or by reflections in the proof of Jussi Väisälä (2003). Surjectivity matters: with the max norm on the plane, \(t\mapsto(t,\sin t)\) is an isometric embedding of \(\mathbb R\) that is not affine (Exercise 5.3).

Ulam’s stability question

In a 1940 talk at Madison, Ulam asked: if a map between groups satisfies the homomorphism equation approximately, is it close to a homomorphism? Donald Hyers answered for Banach spaces the following year.

Proof. With \(y=x\), \(\|f(2x)/2-f(x)\|\le\delta/2\). Replacing \(x\) by \(2^kx\) and dividing by \(2^k\), \(\|2^{-k-1}f(2^{k+1}x)-2^{-k}f(2^kx)\|\le\delta2^{-k-1}\). The sequence \(2^{-n}f(2^nx)\) is therefore Cauchy, and summing the geometric series gives \(\|2^{-n}f(2^nx)-f(x)\|\le\delta(1-2^{-n})\), hence \(\|A(x)-f(x)\|\le\delta\). Additivity: \(\|2^{-n}\big(f(2^n(x+y))-f(2^nx)-f(2^ny)\big)\|\le2^{-n}\delta\to0\). Uniqueness is Exercise 5.2.

Ergodicity and chaotic maps

For the general reader. Stir a cup of coffee and the cream spreads evenly. The ergodic hypothesis of statistical physics says that a typical motion explores its whole space in this way. With John Oxtoby, Ulam showed in 1941 that among all continuous, volume-preserving motions of a cube, the ones that mix space thoroughly are the typical ones. Later, with von Neumann, he studied the simplest example of deterministic chaos, a quadratic map whose iterates behave like random numbers.

Let \(T\) be a measure-preserving transformation of a probability space \((X,\mu)\). \(T\) is ergodic (Ulam and Oxtoby said metrically transitive) if every measurable \(A\) with \(T^{-1}A=A\) has \(\mu(A)\in\{0,1\}\). By Birkhoff’s ergodic theorem (1931), time averages of an ergodic map equal space averages: \(\frac1N\sum_{k<N}\varphi(T^kx)\to\int\varphi\,d\mu\) for almost every \(x\).

The proof combines a Baire category argument, of the kind Banach had used for nowhere differentiable functions, with a delicate construction of homeomorphisms that move small cubes around in a prescribed cyclic way. The result was extended to manifolds and remains the model for “typical” statements in dynamics. It does not hold for smooth maps: Kolmogorov–Arnold–Moser theory shows that smooth area-preserving maps often have invariant regions.

The logistic map at parameter 4

In a 1947 abstract, Ulam and von Neumann studied \(f(x)=4x(1-x)\) on \([0,1]\) as a possible source of random numbers. The substitution \(x=\sin^2(\pi y/2)\) conjugates \(f\) to the tent map \(\tau(y)=1-|2y-1|\):

The tent map preserves Lebesgue measure and is ergodic. Transporting the measure gives the invariant density of \(f\):

Ulam returned to such iterations repeatedly, with Paul Stein in the 1960s, using the Los Alamos computers to draw orbits of quadratic maps in one and two dimensions. This experimental work anticipated the study of chaotic dynamics in the 1970s.

Branching processes

For the general reader. A neutron hits a uranium nucleus and releases two or three new neutrons, each of which may do the same. Will the chain die out or explode? In 1944 Ulam and the philosopher David Hawkins worked out the mathematics of such multiplying populations at Los Alamos. With Cornelius Everett, Ulam then extended it to several kinds of particles. The same equations describe family surnames, epidemics and cell populations.

Let \(p_k\) be the probability that a particle has \(k\) offspring, with generating function \(f(s)=\sum_{k\ge0}p_ks^k\) and mean \(m=f'(1)\). Start with \(Z_0=1\) particle; each particle in generation \(n\) independently produces offspring forming generation \(n+1\). Francis Galton and Henry Watson had studied this model in 1874 for the extinction of surnames. Hawkins and Ulam’s report Theory of multiplicative processes (1944) developed it for neutron chains, independently of the earlier work.

Proof. Conditioning on the first generation, \(\mathbb E[s^{Z_{n+1}}]=f\big(\mathbb E[s^{Z_n}]\big)\), so \(f_{n+1}=f\circ f_n\). Differentiating at \(s=1\) gives the mean. The events \(\{Z_n=0\}\) increase to extinction, so \(q=\lim f_n(0)\). Since \(f\) is continuous and increasing, \(q=f(q)\), and if \(r=f(r)\) with \(r\ge0\), induction gives \(f_n(0)\le r\), so \(q\le r\). Finally \(f\) is convex on \([0,1]\) with \(f(1)=1\): if \(m=f'(1)\le1\), the graph lies above the diagonal on \([0,1)\) and the only fixed point is 1; if \(m>1\), the graph crosses the diagonal once below 1.

Several types

Everett and Ulam (1948) treated \(d\) types of particle, as in a reactor with neutrons at different energies or in different regions. Let \(M_{ij}\) be the mean number of type-\(j\) offspring of a type-\(i\) particle. Then \(\mathbb E[Z_n]=Z_0M^n\), and by the Perron–Frobenius theorem the growth is governed by the largest eigenvalue \(\lambda\) of \(M\): for an irreducible process, extinction is certain if \(\lambda\le1\) and has probability less than 1 if \(\lambda>1\). For a reactor, \(\lambda\) is the multiplication factor, and criticality is \(\lambda=1\).

The Monte Carlo method

For the general reader. Recovering from brain surgery in 1946, Ulam played solitaire and wondered how often a game comes out. The combinatorics were hopeless, but he could just play a hundred games and count. He realized that the new electronic computers could do the same for the paths of neutrons in a bomb or a reactor: simulate many random histories and average. With von Neumann and Nicholas Metropolis this became the Monte Carlo method, now used in every quantitative science.

To compute \(I=\mathbb E[g(X)]=\int g\,dP\), draw independent samples \(X_1,\dots,X_N\) from \(P\) and form \(\hat I_N=\frac1N\sum_{i=1}^Ng(X_i)\).

Dimension-independence is the point. A grid-based quadrature rule with \(k\) points per axis needs \(k^d\) evaluations in dimension \(d\), while the histories of a neutron through many collisions live in a space of very high dimension. Variance reduction techniques, such as importance sampling and splitting, which von Neumann and Ulam also proposed, reduce \(\sigma\) rather than increase \(N\).

History

Ulam described the idea to von Neumann in 1946, and in a letter of March 1947 to Robert Richtmyer von Neumann outlined a statistical calculation of neutron diffusion in a fission device. Metropolis suggested the code name Monte Carlo, after the casino where Ulam’s uncle liked to gamble. In spring 1948 a team including von Neumann, his wife Klára Dán von Neumann and Metropolis ran the first fully automated Monte Carlo calculations on the ENIAC. Metropolis and Ulam published The Monte Carlo method in the Journal of the American Statistical Association in 1949. Earlier, Enrico Fermi had used similar sampling by hand in the 1930s without publishing it, and the idea of estimating \(\pi\) by random needles goes back to Buffon in the eighteenth century. The Metropolis algorithm of 1953 extended the method to sampling from Boltzmann distributions.

Sampling

To sample \(X\) with continuous distribution function \(F\), take \(U\) uniform on \((0,1)\) and set \(X=F^{-1}(U)\): then \(P(X\le x)=P(U\le F(x))=F(x)\). For neutron transport, the distance to the next collision in a medium with mean free path \(\ell\) is exponential, \(X=-\ell\ln U\).

The Fermi–Pasta–Ulam–Tsingou problem

For the general reader. Pluck a string whose springs are slightly nonlinear, and physicists in 1953 expected the energy to spread evenly over all its vibration modes, as heat spreads in a solid. Enrico Fermi, John Pasta and Ulam asked the MANIAC computer at Los Alamos to check, with the programming done by Mary Tsingou. The energy did not spread. After wandering through a few modes it came back, almost entirely, to where it started. The surprise led to the discovery of solitons and helped start the modern study of nonlinear dynamics.

The model

Take \(N-1\) moving masses with displacements \(q_1,\dots,q_{N-1}\) and fixed ends \(q_0=q_N=0\), joined by nonlinear springs. In the \(\alpha\)-model, in units with mass and linear spring constant 1,

The \(\beta\)-model replaces the quadratic term by \(\beta\big[(q_{j+1}-q_j)^3-(q_j-q_{j-1})^3\big]\). For \(\alpha=0\) the chain is linear, with normal modes

each of which keeps its energy forever. Fermi expected the small nonlinearity to couple the modes so that, starting with all the energy in mode 1, the time-averaged energies would approach equal shares, as statistical mechanics predicts.

What the computer showed

The runs, in the summer of 1953, used chains of up to 64 particles. The energy did flow from mode 1 into modes 2, 3, 4 and a few others, but it never spread over all modes. After a time much longer than the period of mode 1, almost all the energy returned to mode 1: a near-recurrence. The report Studies of nonlinear problems (Los Alamos report LA-1940, 1955) appeared after Fermi’s death in 1954. It credited Tsingou for the programming but did not list her as an author; since 2008 many physicists have added her name to the problem.

Solitons and chaos

In the continuum limit of long waves, the \(\alpha\)-chain reduces to the Korteweg–de Vries equation \(u_t+uu_x+\delta^2u_{xxx}=0\) (after rescaling). Norman Zabusky and Martin Kruskal integrated it numerically in 1965 and found localized waves that pass through each other unchanged; they named them solitons and explained the FPUT recurrence through them. This led to the inverse scattering transform and the theory of integrable systems. In the other direction, Felix Izrailev and Boris Chirikov argued in 1966 that above an energy threshold the modes overlap resonantly and the chain does thermalize. Both pictures are part of the modern account; the FPUT chain is close to the integrable Toda lattice, which explains its very long-lived near-regularity.

Problems: primes, liars, sequences

For the general reader. Ulam’s A Collection of Mathematical Problems (1960) gathered dozens of questions, many from the Scottish Café, and his later lectures added more. Some looked like puzzles and turned into research fields. This lesson takes four: a spiral of prime numbers, a sequence defined by unique sums, a guessing game with a liar, and the longest increasing run in a random shuffle.

The prime spiral

In 1963, during a long talk at a meeting, Ulam doodled the integers in a square spiral and circled the primes. They clustered along diagonals. With Myron Stein and Mark Wells he extended the picture on the MANIAC II, and Martin Gardner put it on the cover of Scientific American in March 1964. Diagonals of the spiral are values of quadratic polynomials \(4n^2+bn+c\); the prime-rich ones are those whose values avoid small prime divisors. The conjecture F of G. H. Hardy and John Littlewood predicts the density of primes on each line, and is still unproved. The hero illustration starts the spiral at 41, so that the main diagonal carries Leonhard Euler’s polynomial \(n^2+n+41\).

Ulam numbers

Define \(u_1=1\), \(u_2=2\), and let each later term be the smallest integer greater than the previous one that is the sum of two distinct earlier terms in exactly one way. The sequence begins \(1,2,3,4,6,8,11,13,16,18,26,28,36,38,47,\dots\) Ulam asked in 1964 about its density; computations suggest a density near 0.074 and a strange periodicity, both unexplained.

Searching with a liar

In Adventures of a Mathematician (1976) Ulam asked: someone thinks of a number between one and a million and answers yes-or-no questions, but may lie once. How many questions are needed? Write \(q_1(n)\) for the minimum worst-case number for \(n\) possibilities. Alfréd Rényi had posed a similar game in 1961.

The lower bound is a volume argument from coding theory (Exercise 10.2); Andrzej Pelc’s strategy shows it is achieved. The question is equivalent to designing error-correcting codes with feedback.

Increasing subsequences

Let \(\ell_n\) be the length of the longest increasing subsequence of a uniformly random permutation of \(\{1,\dots,n\}\). In 1961 Ulam, using Monte Carlo simulation, observed that \(\mathbb E\ell_n\) grows like \(c\sqrt n\) and asked for \(c\). John Hammersley proved the limit exists (1972). Benjamin Logan and Larry Shepp, and independently Anatoly Vershik and Sergei Kerov, proved \(c=2\) in 1977. In 1999 Jinho Baik, Percy Deift and Kurt Johansson showed that the fluctuations, of order \(n^{1/6}\), follow the Tracy–Widom law of random matrix theory, a result that connected Ulam’s problem with random matrices and growth models.

Los Alamos, the Super and Orion, 1944–1965

Ulam arrived at Los Alamos in February 1944 and joined the theoretical division under Hans Bethe, working first in Edward Teller’s group on the “Super”, the hypothetical hydrogen bomb, and then on the hydrodynamics of implosion with von Neumann. His daughter Claire was born in Santa Fe in July 1944. After the war he took a position at the University of Southern California in September 1945. In January 1946 he fell ill with viral encephalitis and underwent emergency brain surgery. He recovered fully, and while convalescing conceived the Monte Carlo method. By the spring of 1946 he was back at Los Alamos, where he remained, with interruptions, until 1965.

The hydrogen bomb

After the first Soviet atomic test in 1949, President Truman ordered a crash programme to build the Super. Teller’s design, the “classical Super”, relied on a fission explosion to ignite deuterium directly. In 1950 Ulam and Cornelius Everett, computing by hand and with slide rules, and then Ulam with Fermi, showed that it would not sustain burning: the fuel would cool faster than it reacted. Electronic calculations later confirmed this.

Early in 1951 Ulam proposed a different arrangement. The fusion fuel should be placed apart from the fission primary and strongly compressed by the energy of the primary before ignition. Teller saw that radiation from the primary, rather than mechanical shock, could do the compressing. Their joint report of 9 March 1951 set out what is now called the Teller–Ulam configuration. It was tested at full scale in the Ivy Mike explosion of 1 November 1952 at Enewetak Atoll. The relative weight of the two contributions was disputed for decades. Teller minimized Ulam’s role in later accounts, while many physicists and historians credit Ulam with the essential idea of staged compression. Ulam himself wrote little about it.

Computers and propulsion

Ulam was an early and enthusiastic user of the MANIAC computers built at Los Alamos under Metropolis. Beyond Monte Carlo and the FPUT study, he used them for experiments in iterated maps, in growth patterns of cellular figures, and in number theory. He had suggested to von Neumann the discrete grid model that became von Neumann’s self-reproducing cellular automaton. In 1955 Ulam and Everett proposed propelling a spacecraft by a sequence of small nuclear explosions behind a pusher plate. The idea became Project Orion, studied from 1958 until 1965.

Boulder, death and legacy, 1965–1984

In 1965 Ulam left Los Alamos for the University of Colorado in Boulder, where he became professor and, from 1967, chairman of the mathematics department. He kept a consulting connection with Los Alamos. From 1968 he also held a professorship in the medical school, reflecting a growing interest in mathematical biology: he wrote on the comparison of DNA sequences, proposing metrics on strings that anticipated sequence-alignment methods. He retired from Colorado in 1975 and afterwards spent winters as a graduate research professor at the University of Florida. His doctoral students included Paul Kelly at Wisconsin (1942), George Estabrook (1969), Leonard Gallagher (1972) and Celestino Mendez (1974) at Colorado, and Edward Howorka at Florida (1979).

He wrote for wide audiences. Adventures of a Mathematician (1976) is one of the best autobiographies by a mathematician, rich in portraits of Banach, von Neumann, Fermi and the Lwów café. With his friend Mark Kac he wrote Mathematics and Logic (1968). His selected papers appeared as Sets, Numbers, and Universes (1974). He was elected to the National Academy of Sciences, the American Academy of Arts and Sciences and the American Philosophical Society, and in 1976 received the Commander’s Cross with Star of the Order of Polonia Restituta.

Ulam died of a heart attack in Santa Fe on 13 May 1984, aged 75. Essays and lectures collected after his death appeared as Science, Computers, and People (1986) and Analogies between Analogies (1990).

Legacy

• Set theory and topology: measurable cardinals opened the theory of large cardinals; the Borsuk–Ulam theorem is a basic tool of topology and combinatorics.
• Probability and computation: the Monte Carlo method and branching processes are standard in physics, statistics, finance and machine learning; the probabilistic programming language Stan is named after him.
• Nonlinear science: the FPUT experiment made the computer an instrument of discovery and led to solitons and the study of chaos.
• Problems: the Ulam spiral, Ulam numbers, the liar game, Ulam’s problem on increasing subsequences, the reconstruction conjecture of Kelly and Ulam, and stability of functional equations all remain active.
• Memory: the Stanislaw M. Ulam Distinguished Scholar programme at Los Alamos has run since 1985.

Connections

People linked to Stanislaw Ulam across this collection. A name without a link has no page here yet. See Stanislaw Ulam in the network of all 337 lives →

Teachers and advisors

• Kazimierz Kuratowski 1927-1933
• Włodzimierz Stożek 1933

Students

• Celestino Mendez 1974
• Edward Howorka 1979
• George Estabrook 1969
• Leonard Gallagher 1972
• Paul Kelly 1942

Influenced

• Donald Hyers 1941

Colleagues and collaborators

• Alfred Tarski
• Enrico Fermi 1950-1954
• Hans Bethe
• John von Neumann 1935-1957
• Stefan Banach 1930s
• Cornelius Everett 1944-1955
• David Hawkins 1944
• George David Birkhoff 1936-1940
• Hugo Steinhaus
• John Oxtoby 1938-1941
• John Pasta 1953-1955
• Józef Schreier 1933-1937
• Karol Borsuk 1933
• Mary Tsingou 1953-1955
• Nicholas Metropolis 1947-1960s
• Paul Stein 1960s
• Stanisław Mazur 1932

Rivals

• Edward Teller 1944-1984

Friends

• Paul Erdős 1935-1984
• Mark Kac

Family

• Adam Ulam

Also mentioned in this life

• Ludwig Boltzmann
• Augustin-Louis Cauchy
• Leonhard Euler
• Joseph Fourier
• Georg Frobenius
• G. H. Hardy
• Andrey Kolmogorov
• Henri Lebesgue
• Isaac Newton

Fields

• set theory 12 lives
• measure theory 5 lives
• topology 9 lives
• ergodic theory 7 lives
• probability theory 23 lives
• mathematical physics 45 lives

On CoursesHub

• Topology in Physics · Differential Geometry

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Stanislaw Ulam, MacTutor History of Mathematics.
2. Stanislaw Ulam, Wikipedia.
3. Stanisław Marcin Ulam, Mathematics Genealogy Project.
4. S. M. Ulam, Adventures of a Mathematician, Scribner, 1976; University of California Press, 1991.
5. N. G. Cooper (ed.), From Cardinals to Chaos: Reflections on the Life and Legacy of Stanislaw Ulam, Cambridge University Press, 1989 (originally Los Alamos Science 15, 1987).
6. R. D. Mauldin (ed.), The Scottish Book, 2nd ed., Birkhäuser, 2015.

Mathematics and physics

1. S. Ulam, Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930), 140–150.
2. S. Mazur and S. Ulam, Sur les transformations isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932), 946–948.
3. K. Borsuk, Drei Sätze über die \(n\)-dimensionale euklidische Sphäre, Fund. Math. 20 (1933), 177–190.
4. J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. 42 (1941), 874–920.
5. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224.
6. N. Metropolis and S. Ulam, The Monte Carlo method, J. Amer. Statist. Assoc. 44 (1949), 335–341.
7. E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems, Los Alamos report LA-1940 (1955).
8. T. Dauxois, Fermi, Pasta, Ulam and a mysterious lady, Physics Today 61 (2008); arXiv:0801.1590.
9. M. L. Stein, S. M. Ulam and M. B. Wells, A visual display of some properties of the distribution of primes, Amer. Math. Monthly 71 (1964), 516–520.
10. S. M. Ulam, A Collection of Mathematical Problems, Interscience, 1960.
11. J. Spencer and P. Winkler, A simple solution to Ulam’s liar game with one lie, arXiv:0705.1220.
12. D. Romik, The Surprising Mathematics of Longest Increasing Subsequences, Cambridge University Press, 2015.
13. J. Matoušek, Using the Borsuk–Ulam Theorem, Springer, 2003.
14. T. E. Harris, The Theory of Branching Processes, Springer, 1963.