In 1942, the United States, in collaboration with Canada and the United Kingdom, created the Manhattan Project, with the objective of making the atomic bomb before the Axis powers. In the theoretical design of the bomb, which was carried out in the secret laboratory of Los Alamos (New Mexico), many European scientists who had emigrated to the United States fleeing from the Nazis participated. Among them was Stanislaw Ulam (born April 13, 1909 and died in 1984), a brilliant Polish mathematician who would contribute decisively to the design of the hydrogen bomb.

Ulam grew up in a wealthy family in the city of Lviv (now in Ukraine), where he became part of a vibrant mathematical community. However, the offer of places in Polish universities was scarce, which, together with his Jewish status, led him to emigrate to America in 1935. Four years later, Germany invaded Poland. All of his family would die in the Holocaust, except for his brother, who had accompanied him to the United States.

Ulam tried to enlist in the American aviation, but was fortunately rejected due to his vision problems and continued working at the university until 1943, when he was invited by the German physicist Hans Bethe to go to Los Alamos to work on the design of the atomic bomb.

Atomic energy can be obtained in two ways: with the fission process, which consists of splitting large atoms, such as uranium or plutonium, or with the fusion process, that is, joining small atoms, such as hydrogen. In both cases the process begins with an “ignition” that causes the division (or union) of a few atoms, followed by a “chain reaction”, in which the process spreads to the rest of the atoms.

The melting process is more complicated and much more energized than melting. But igniting the fission process can be carried out with a traditional explosive, while the fusion process requires an enormous amount of energy, which can only be achieved using a fission bomb. Therefore, to build a fusion bomb it is necessary to have obtained the fusion bomb first. When these processes are carried out all at once, an enormous amount of energy is released and a bomb is obtained. The bomb that uses the fusion process is usually called a “hydrogen bomb”, reserving the term “atomic bomb” for the bomb that uses the fission process.

In Los Alamos, Ulam joins Edward Teller’s team, which was investigating the design of the hydrogen bomb. In July 1945 the Manhattan Project successfully tested the first atomic bomb and in August Hiroshima and Nagasaki were devastated by the first nuclear weapons in history. The war was over, and most of the scientists at Los Alamos returned to their universities.

However, four years later, Russia obtained its first atomic bomb and US President Harry Truman then gave priority to the construction of the hydrogen bomb. Teller way brought his team back together and resumed the project, which he directed in a very personal way. The relationship between Ulam and Teller was tense from the beginning, which was not helped by the fact that the Pole, along with his collaborator Cornelius Everett, spent the first six months of their stay making detailed calculations about the viability of Teller’s project.

### Casinos and simulations

To do this, he used a method, devised by himself, which was called the Monte Carlo method in honor of an uncle of his, who frequented the casino. It consists of solving a problem from a large number of simulations. For example, to find the area of a complicated geometric figure (for which we cannot apply the formulas we learned at school) the traditional solution is to approximate the area with larger and larger simple figures contained in the geometric figure. The Monte Carlo method, on the other hand, proposes to first take a square that contains the figure, and then calculate the probability that a random point of the square is in the figure obtained over a large number of simulations. If, for example, the square measures six square meters and we estimate that 33% of the points of the square are in the figure, then we can deduce that the area of the figure will be approximately two square meters.

The Monte Carlo method is usually much faster solving problems than the traditional method. Although he was not the first to devise such a method — it was already used in Buffon’s needle experiment in the 18th century — Ulam was the first to realize its enormous potential, thanks to early computers being developed by his friend, the mathematician The Hungarian John von Neumann. Today the method is still used: it basic in science and engineering and is used in environments as disparate as 3D animation or evolutionary biology.

These results, along with others they obtained with the Italian physicist Enrico Fermi, were withering: Teller’s method did not allow the chain reaction to start or be maintained. Soon after, the calculations would be repeated and confirmed with von Neumann’s MANIAC computer. However, in 1951 Ulam himself discovered that if compressed hydrogen was sufficiently used in an atomic bomb, then the chain reaction would work. After incorporating this change into Teller’s design, known as the Teller-Ulam process, the hydrogen bomb project continued until the first explosion was achieved on Enewetak Atoll in 1952. The power of this bomb was 400 times greater. than the atomic bombs that fell on Japan in 1945. The Russians would not achieve the first functional explosion of a hydrogen bomb until 1955, with Sakharov’s design.

Ulam’s life is recognized in an interesting autobiography *adventures of a mathematician*which was made into a movie in 2020. There he explained his position regarding the investigation of atomic weapons: “Unlike those who violently opposed the bomb […], I never had doubts about purely theoretical works. It did not seem immoral to me to try to calculate physical phenomena […]. What I think is that one should not start projects that lead to catastrophe. But once we know that such possibilities exist, isn’t it better to examine whether they are real or not? An even bigger delusion is to believe that if you don’t do it, it won’t be possible to do it. […]”.

**Federico Cantero Moran*** IT Professor at the Autonomous University of Madrid and member of the ICMAT*

**Coffee and Theorems**** ***It is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”*

*Edition and coordination: ***Ágata A. Timón G Longoria*** ***(ICMAT).**

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