Genialni. Lwowska szkoła matematyczna by Mariusz Urbanek
My rating: 3 of 5 stars

"'Mathematics mediates between spirit and matter'"
The inscription on Hugo Steinhaus' gravestone (My own translation of the Polish sentence 'Między duchem a materią pośredniczy matematyka.')

Mariusz Urbanek's Genialni (2014) (in my lame translation the English equivalent would be The Geniuses) chronicles life stories, achievements, and careers of a group of extraordinary and world-famous Polish mathematicians who were jointly known as the Lwów School of Mathematics and who offered momentous contribution to various branches of mathematics. As far as I know, the book has not been yet translated into English; in fact, four years ago I resolved to attempt the task, but quickly decided that a person who can write in English well would do a better job.

I have a strong personal connection with one of the remarkable mathematicians whose life and work are depicted in the book. Hugo Steinhaus was the PhD advisor of Jan Oderfeld who was my PhD advisor. Thus, in the sense of mathematical genealogy, I am a grandson of Steinhaus. What's more, Steinhaus, in addition to greatly contributing to such branches of mathematics as functional analysis, geometry, and logic, was one of the founders of probability theory which is my favorite branch of mathematics, and which I often teach at my university.

The narration begins on July 17, 1935, as the members of the Lwów School gather for their regular mathematical discussions in the traditional meeting place, "Scottish Cafe" in Lwów. Following the author's lead I will focus here only on three members of this illustrious group: Steinhaus, Stefan Banach, and Stanisław Ulam. Banach, the founder of functional analysis in its modern shape is "generally considered one of the world's most important and influential 20th-century mathematicians" (Wikipedia). John von Neumann, an American mathematical genius, traveled to Lwów several times in the 1930, trying to get Banach to move to the US. Mr. Urbanek quotes a facetious statement that Steinhaus' greatest contribution to mathematics was "discovering" Banach. Seriously though, Steinhaus was Banach's informal PhD advisor.

Of the three mathematicians Stanisław Ulam is probably most famous in the U.S. because of his contributions to the development of nuclear weapons. He began his American career in the famous Institute for Advanced Studies in Princeton. Later came his participation in the Manhattan Project in Los Alamos - development of the atomic bomb - where he was one of the two leading mathematicians. Still later Ulam was the second of command, after Edward Teller, in the team that developed the hydrogen bomb. In fact, many scientists consider his contributions more significant than Teller's, as he solved the ignition problem. Ulam is also considered the father of the so-called Monte Carlo methods in mathematics, which is my most favorite field in math and in which I often dabble.

The author's style is very readable, perhaps even too "gossipy" and unfocused in several places. Numerous anecdotes about the mathematicians enliven the book. However, the reader should be aware that a substantial portion of the book is very serious and tragic. After Germany attacked Poland from the west on September 1, 1939, thus starting World War II, the Soviets invaded the eastern part of Poland, the part that included Lwów. Persecution began, with frequent arrests and deportations. Then the Germans attacked the Soviet Union in June 1941 and the mathematicians suffered the same horrors as all other Polish people: planned or random executions (25 Polish professors were executed in Lwów on July 4th, 1941), deportations to labor or concentration camps, or - if they were lucky - escape from their home and emigration. After the horrors of German occupation came the terror, almost equally severe, imposed by the Soviet-controlled Polish "Communist" government. Banach and Steinhaus had to wait until the mid-to-late 1950s for some semblance of normality return to life in Poland.

An interesting and readable book, but perhaps one that will be of interest mostly for people connected with mathematics in some way.

Three-and-a-quarter stars.

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