At the age of ten, Ulam entered the gymnasium in Lvov and, about this time, he became interested first in astronomy and then in physics. An uncle gave Ulam a telescope when he was about 12 years old and later Ulam tried to understand Einstein's special theory of relativity. However this required an understanding of mathematics and so, at age 14, he began to study mathematics from books, going well beyond the school level mathematics he was learning. Ulam said ( or ):-
... I was sixteen when I really learned calculus all by myself from a book by Kowalevski, a German not to be confused with Sonia Kovalevskaya .... Then I read also about set theory in a book by Sierpiński, and I think I understood that. We had a good professor in high school, Zawirski, who was a lecturer in logic at the university. I talked to him about it then and when I entered the Polytechnic Institute.Now with interests in astronomy, physics and mathematics, Ulam entered the Polytechnic Institute in Lvov. In 1927, his first year at the university, he was taught by Kuratowski who had just been appointed to Lvov. Ulam said ( or ):-
[Kuratowski] gave an elementary course on set theory, and I asked some questions, then I talked to him after classes, and he became interested in a young student who evidently was interested in mathematics and had some ideas. I was lucky to solve an unsolved problem that he proposed.Ulam obtained his Ph.D. from the Polytechnic Institute in Lvov in 1933 where he studied under Banach. He investigated a problem which originated with Lebesgue in 1902 to find a measure on [0,1] with certain properties. Banach in 1929 had solved a related measure problem, but assuming the Generalised Continuum Hypothesis. Ulam, in 1930, strengthened Banach's result by proving it without using the Generalised Continuum Hypothesis.
In 1935 Ulam received an invitation from von Neumann to visit the Institute for Advanced Study in Princeton for a few months. Planning to spend three months there he sailed from France to New York. At the Institute for Advanced Study he met G D Birkhoff who invited him to Harvard University. Ulam said ( or ):-
... I went back to Poland, but the next fall I returned to Cambridge as a member of the so-called Society of Fellows, a new Harvard institution. ... I started teaching right away: first, elementary courses and then quite advanced courses. I became a lecturer at Harvard in 1940, but every year during that time I commuted between Poland and the United States. In the summers I visited my family and friends and mathematicians. In Poland mathematical life was very intense, the mathematicians saw each other often in cafes such as the Scottish Cafe and the Roma Cafe. We sat there for hours and did mathematics. During the summers I did this again. And then in '39, I actually left Poland about a month before World War II started.You can see a picture of the Scottish Café at THIS LINK.
In 1940 Ulam was appointed as an assistant professor at the University of Wisconsin. In 1943 Ulam became an American citizen. In that year von Neumann asked him to undertake some very important war work. They agreed to meet ( or ):-
... in Chicago in some railroad station to learn a little bit more about it. I went there, and he could not tell me where he was going. There were two guys, sort of guards, looking like gorillas, with him. He discussed with me some mathematics, some interesting physics, and the importance of this work. And that was Los Alamos at the very start. A few months later I came with my wife ... arriving for the first time in a very strange place.He worked on the hydrogen bomb at the Los Alamos National Laboratory in New Mexico. This work is described in :-
Working with physicist Edward Teller, Ulam solved one major problem encountered in work on the fusion bomb by suggesting that compression was essential to explosion and that shock waves from a fission bomb could produce the compression needed. He further suggested that careful design could focus mechanical shock waves in such a way that they would promote rapid burning of the fusion fuel. Teller suggested that radiation implosion, rather than mechanical shock, be used to compress the thermonuclear fuel. This two-stage radiation implosion design, which became known as the Teller-Ulam configuration, led to the creation of modern thermonuclear weapons.Ulam, with J C Everett, also proposed the 'Orion' plan for nuclear propulsion of space vehicles.
Rota  describes how Ulam's personality changed in 1946:-
One morning in 1946, in Los Angeles, Stanisław Ulam, a newly appointed professor at the University of Southern California, awoke to find himself unable to speak. A few hours later, he underwent a dangerous surgical operation after the diagnosis of encephalitis. ... In time, however, some changes in his personality became obvious to those who knew him. ... his ideas, which he spouted out at odd intervals, were fascinating beyond anything I have witenssed before or since. However, he seemed to studiously avoid going into details. ... he came to lean on his unimpaired imagination for his ideas, and on the [hard work] of others for technical support. ...While Ulam was at Los Alamos, he developed the 'Monte-Carlo method' which searched for solutions to mathematical problems using a statistical sampling method with random numbers. It is now widely used in computer implementations of mathematical software.
A crippling technical weakness coupled with an extraordinarily creative imagination is the drama of Stan Ulam. Soon after I met him, I was made to understand that, as far as our conversations went, his drama would be one of the Forbidden Topics. ... But he knew I knew, and I knew he knew I knew.
He remained at Los Alamos until 1965 when he was appointed to the chair of mathematics at the University of Colorado. At the time of his death he was professor of biomathematics at the University of Colorado.
When asked to sum up his work, he said (, ):-
Originally I worked in set theory and some of these problems are still being worked on intensively. It is too technical to describe: measurable cardinals, measure in set theory, abstract measure. Then in topology I had a few results. ... Then I worked a little in ergodic theory. Oxtoby and I solved an old problem and some other problems were solved in other fields later. In general I would say luck plays a part, at least in my case. Also I had luck with tremendously good collaborators in set theory, in group theory, in topology, in mathematical physics, and in other method, which is not a tremendously intellectual achievement but is very useful, a few things like that.Ulam's writing include A collection of mathematical problems (1960), Sets numbers and universes (1974) and Adventures of a Mathematician (1976).
He was described by Rota in the following way:-
Ulam's mind is a repository of thousands of stories, tales, jokes, epigrams, remarks, puzzles, tounge-twisters, footnotes, conclusions, slogans, formulas, diagrams, quotations, limericks, summaries, quips, epitaphs, and headlines. In the course of a normal conversation he simply pulls out of his mind the fifty-odd relevant items, and presents them in linear succession. A second-order memory prevents him from repeating himself too often before the same public.His wife, Françoise Ulam, writing in  described Ulam's working methods:-
Ulam ... is almost exculsively a talking man, a verbal person. When not thinking ... what he enjoys most is to talk, to discuss, to argue, to converse, with friends and colleagues. Relying on his phenomenal memory, he carries everything in his head. ...
The physical act of taking pen to paper has always been painful for him. His mind and his eyes are the obstacles. His mind, because it works much faster than his fingers...; his eyes because one is very myopic the other very presbyotic. ... From childhood fears, then from youthful vanity he spurned wearing glasses, until very recently. Thus Ulam has always had a very hard time bringing himself to write anything for publication, either in long hand or with a typewriter. Machines and other mechanical objects have always turned him off. ... How then does he ever produce a written text? Mainly by talking ...
Article by: J J O'Connor and E F Robertson