The city of Lviv, in present-day Ukraine, belonged to the Austro-Hungarian Empire from the 18th century until its collapse with the First World War, becoming part of Poland between 1919 and 1939. In the interwar period, the community mathematics of Lviv included a good number of the best mathematicians in the world, headed by the distinguished Polish analyst Stefan Banach.
One of Banach's favorite places to meet and work was a tavern near the university called Café Scots. After long hours of discussion, the marble tables of this establishment ended up covered by formulas written in pencil, which were eventually erased by the owner to the dismay of Banach and other outstanding mathematicians such as Stanisław Mazur Y Stanisław Ulam. Therefore, the wife of Banach decided to buy a thick notebook, in which they collected the problems that they could not solve after long hours of work, and that remained in the Scottish coffee.
This notebook, known as the Scottish Book, It contained a legendary collection of open problems, mainly on functional analysis and topology. Whoever solved one of these frequently obtained a prize offered by one of these mathematicians, such as a beer mug, a bottle of good brandy or even a live goose. Obviously, these rewards are more substantial in their historical context: they were the years of the Great Depression before the Second World War. The oca was received by the Swedish Per Enflo in 1972, many years later, at the hands of Mazur, when he met the famous problem 153 when building a Banach space that does not support a Schauder base. The ceremony was broadcast live on Polish television.
Our personal contact with the Book Scottish is through his biggest problem contributor: Stanislaw Ulam, who years later would be described sarcastically as "a pure mathematician who had fallen so low, that his last article had numbers with decimals". This refers to his important contributions to Manhattan Project in the quarantine and fifties, when the United States was working to create the atomic bomb before the Nazi government did. To get an idea of Ulam's scientific legacy, just take a look at the list of contributions They bear his name, which includes the theorems of Borsuk-Ulam, Mazur-Ulam and Kuratowski-Ulam in mathematics, the design of Teller-Ulam and the Fermi-Ulam model in nuclear physics, or the Fermi-Pasta-Ulam system with that the call starts experimental mathematics. Ulam also developed, together with Nicholas Metropolis and John Von Neumann, the famous numerical algorithm based on random numbers that they called Monte Carlo method, in honor of the fondness of an uncle of Ulam for the casino of the same name.
In problem 16 of the Scottish Book, around 1935, Ulam posed a question about the geometry of the magnetic fields created by a cable, through which an electric current flows. When the cable has a circular shape, the behavior of the magnetic field is perfectly understood and is, in fact, a fundamental model in physics and engineering. For more complicated forms of the cable, in particular when it is knotted, such as the lacing of a shoe, the generated magnetic field can not be calculated explicitly. Ulam wondered if the knotting of the wire reflects the geometry of the magnetic field. This geometry is described by the so-called magnetic lines, which are visualized experimentally by the orientation of metal shavings. It is curious that, although the study of knotted cables emerged as a mental experiment in a purely mathematical context, it has recently appeared in the work of a group of Princeton physicists, who have designed a new plasma confinement device called Nudotron.
Ulam thought and did numerical experiments on the problem recurrently throughout his life, and included it in his collections of open problems later published. And although he died without seeing the problem solved, today we can give a quite satisfactory answer: the geometry of the magnetic field does not reflect the knotting of the cable, but almost. We have shown that there are cables knotted in a complex way with magnetic lines of topology as simple as in the case of the circular loop, but that it is enough to carefully create a small frizz in the cable so that magnetic lines appear that faithfully reflect its knotting. Honoring the mathematical school that emerged in Lviv, the demonstration of this fact combines topological ideas with methods of functional analysis.
If the reader wants to taste a jug of beer or a good brandy at the scene, he can go to the Scots Café, which is still open today. If you also want to be invited to the drink, you can request a copy of the Scottish Book that is stored in the premises available to the client, and try to solve some of the problems with awards that remain open. Although we have to warn you that you will have to think hard to solve them, and that the person who would invite you could have died a long time ago.
Alberto Enciso Y Daniel Peralta they are researchers from ICMAT
Coffee and Theorems is a section dedicated to mathematics and the environment in which they are 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 points of contact between the mathematics and other social and cultural expressions and remind those who marked their 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".
Editing and coordination:Agate Rudder (ICMAT).