**One day in the 1930s a group of men met at the Scottish Café in Lviv. Over glasses of cognac, they scribbled on the cafe's marble tabletops, and history was made. They were no ordinary guests. These men were Hugo Steinhaus, Stefan Banach and Stanisław Ulam – mathematicians who dreamt big, wrote poems, constructed the atomic bomb and helped organise the first flights to the moon.**

A recently published book – The Brilliant Ones by Mariusz Urbanek – sheds light on an unusual group of men brought together by their love for maths. It all began one summer day when a mathematician, Hugo Steinhaus, heard someone talking about the Lebesgue integration in Krakow's Planty Park. At that time, this calculus was only known by a handful of mathematicians. That someone turned out to be Stefan Banach, the son of an illiterate servant, a genius without a diploma, one of Poland's biggest scientific talents and the second most frequently invoked name in common mathematics after Euclid.

They would lay the basis for new findings on the marble tabletops of the ambient Scottish café in pre-war Lviv. There professors, associate professors and people with doctorates from the Lviv Technical University and the Jan Kazimierz University would meet over coffee and cognac to discuss maths for hours on end. The results of these gatherings were twofold. They gave rise to many anecdotes and a thick, lined notebook with 193 equations (The Scottish Book), some of which have yet to be resolved. These meetings lay the foundation for the Lviv School of Mathematics – the most important Polish contribution to world science, entangled in the whirlwind of history that was World War II. Unfortunately, little is known of the school except for the great talent of its members. Culture.pl traces their footsteps.

## Hugo Steinhaus: Humour is a code

Educated in Göttingen, Lower Saxony, Steinhaus was a very peculiar mathematician. He came up with aphorisms (picked up by poets such as Julian Tuwim), was a fan of charades and speaking Polish perfectly. He paid attention to manners, grooming and etiquette. He would send letters back to their senders if they were incorrectly addressed and could ruin the career of a PhD student who introduced himself surname first. The strict and pedantic Steinhaus was probably the only teetotaller among the group.

He got the idea for a cypher in Joyce's Ulysses, while the concept of a spatial x-ray locating device came to him during a winter stroll spent observing snowflakes falling on his fur coat. He is one of the early founders of the probability theory which was used by courts to establish paternity before the development of genetic tests.

He lived through the war under the name Grzegorz Krochmalny and hid in an estate near Lviv and at a manor neat Nowy Sącz. He gave private lessons in exchange for firewood, oil and milk. In his spare time he played chess and worked on designing a solar clock. During the war he resorted to solving serious mathematical equations by post but continuing his research from before the war was out of the question.

After the war he settled in Wrocław. There, under a pseudonym, he published cycles of aphorism in weekly newspapers. "A joke cannot be shot," he said, "it has to be scored". The poet Julian Tuwim instructed him in publishing aphorisms in the monthly science review Problems. They were very popular among students who called them "Huguenots" or "steinhausens". Critics compared them to Stanisław Jerzy Lec's Unkempt Thoughts. The mathematician authored the Mathematical Kaleidoscope, later translated into many languages. He's the one who convinced Stefan Banach to move to Lviv before the war. He later said that Banach was "the greatest discovery of his life". In the Scottish Book, next to intricate calculations, Steinhaus jotted down situational jokes with a mathematical edge. This one describes Banach searching for a box of matches in his coat:

A certain man had two boxes of matches and took out matches at random from one and the other. A while later it turned out that one was empty. What is the probability there will be an x amount of matches in the other if initially there were y matches in each box?

## Stefan Banach: revolutionising mathematics over booze

*Stefan Banach/ Forum*

He was a nonchalant mathematician who didn't fit in with the crowd of intellectuals and professors. He never graduated from university. Instead of playing the popular game of tennis in his leisure time, he preferred carambole, football matches and films with cowboys. Unlike the others he didn't wear a tailcoat and a top hat. Everyone in Lviv knew about him and talked about him. On top of that, he could hold his liquor. "Banach's space", "Banach's integral", "Banach limit" and "Banach's algebra", all eminently complex mathematical equations and functions were coined over a glass of cognac, a pint of beer, loud music and in the midst of a cloud of tobacco smoke. His fame grew and in 1932 he published his life's work – Theory of Linear Operations. A couple of years later he was invited to present it at the International Mathematical Congress in Oslo. Theory of Linear Operations has been translated into hundreds of languages.

It's hard to believe that a couple of years prior to this, he was working as an extra at the opera to earn some extra money. For dancing the mazurka in Stanisław Moniuszko's Halka, he received 20 groszy and his science career was slowed through his lack of a masters degree. Steinhaus intervened and asked the minister to bend the university rules in Banach's case. There was only one condition, that he completed a PhD within a year:

I managed to achieve that in half a year, but in a rather particular way. Banach's guardians gave him assistants who followed him and jotted down his thoughts and ideas, his statements and arguments. All Banach had to do was accept the written version. They used a trick to get Banach in front of the examining committee – Banach was informed that a group of gentlemen had arrive from Warsaw to ask him to solve an interesting mathematical equation for him and were waiting in the secretary's office. Those were the circumstances under which Stefan Banach's public PhD defence took place. – says the author of the book The Brilliant Ones.

In 1939, he received the Grand Prize of the Polish Academy of Arts and Sciences. Unfortunately, most of the 20 thousand złoty in prize money disappeared from the bank at the beginning of World War II. On September 1st, 1939, he was at a health resort and against the will of his son Banach returned to Lviv, which had been bombed by the Germans. After the Lviv enclave of science died, the mathematician found shelter and work at the Typhus Research Institute. There, he survived the war as a lice breeder. He and the other scientists found solace in solving the biggest dilemmas in science, listening to lectures by specialists from different fields, which they continued to do. In their talks they also spoke about the war and the future of Lviv.

Following the Russian takeover of the city, Banach returned to work as a lecturer. There were rumours going around Lviv that Stalin himself offered Banach the seat of the President of the Polish Soviet Republic, numerous awards and accolades. He died of lung cancer in August 1945. He lectured till the last moment.

## Stanisław Ulam: behind the scenes of the Manhattan project

His life would make for a good film. He wanted to be a lawyer or an astronomer – he got his first telescope at the age of 10. He wasted no time in learning the names of the different constellations and their distances from Earth. Jules Verne was one of his favourite authors. He attended university lectures on the theory of relativity at the age of 12. Little did he know that decades later he would present his ideas about going into space and flying to the moon to the advisor of President-elect John F. Kennedy.

Let's take a look back at his university days. Ulam joined the Scottish table thanks to Stanisław Mazur, whom he met in his second year of mathematics. "Maths for this group of fascinated people was like some sort of fever", reminisces Zygmunt Birnbaum in The Brilliant Ones. On cold winter evenings, the students and teachers kept warm by resting their backs against hot tiles and for hours at a time they discussed integrals and root numbers. Ulam was one of them. After graduating he went to Vienna, then Zurich, Paris, Cambridge. He even received a letter of invitation to the United States. In 1935 at Princeton he met Albert Einstein and received an internship offer from Harvard. Nevertheless the rising star decided to return to Lviv. But not for long.

News of the Ribbentrop-Molotov pact reached him aboard the transatlantic liner Batory. By the time he reached New York, war had broken out. He listened to the radio day in and out and read all the newspapers in search for information about Lviv. In one he found a picture of his brother with the subtitles "A student from Poland is wondering whether his home had been destroyed". He also wrote that he feels as if on the margins of history. At the time the Los Alamos and Manhattan Project were coming together. Their aim? To construct the deadliest bomb in the history of the world. The most accomplished scientist from around the world and future Nobel prize winners gathered in one place. Following the first successful trials Robert Oppenheimer, the head of the project said, "it crossed my mind that I became Death, the destroyer of worlds". The bomb that reached Hiroshima on August 6th killed over 80 thousand people and marked the dawn of the atomic age. What happened to Ulam?

When I found out about Hiroshima and saw the pictures of destruction, at first I felt puzzlement. I made a mental leap: the numbers written with a white chalk on a black board and then straight away a city swiped from the surface of the earth – Ulam disclosed to Olgierd Budrewiczow.

The memories of Hiroshima and Nagasaki chased him till the end of his life. However in 1950 he was one of three scientists at Los Alamos who worked on the hydrogen superbomb project, with a strength equivalent to 700 of the bombs dropped on Japan. What would Newton and Archimedes have achieved had they spent their time worrying about the consequences of their discoveries, he used to explain to his wife. In the 60s he was a frequent guest in Washington, he worked on a moon flight programme which John F. Kennedy spoke about in an address to the American people.

## Stanisław Mazur: Mathematicians don't always know how to count

In spite of family traditions, he didn't become a pastry chef but spent a lot of time in cafés anyway. He had lightning quick reflexes and a good sense of humour, he was smart and fit right in with the other brilliant minds at the Scottish table. There was nothing about mathematics he didn't know, but he solved equations only in his head. He was notable to many for abstaining from publishing his works and his leftist views A follower of the communist ideology and an antifascist. Urbanek describes him as such:

He was always distracted. He once didn't recognise his younger daughter in the street. Half a year went by before he found out that she dropped out from school to become a ballerina. With his mind constantly on equations, he couldn't deal with the simplest calculations. The newspaper seller from whom he purchased the paper every day asked him to add up the prices, thinking that the famous mathematician shouldn't have any problems with that. – But he broke out in a sweat and complained that he really doesn't know how much 2,40 plus 3,65 and 4,80 is – Mazur's daughter, a famous ballet dancer and choreographer Krystyna Mazur divulges in the book.

During the war, just like other mathematicians he wanted to continued to work and hid from the Gestapo and the Ukrainians. Once the war was over he worked in Łódź and at the University of Warsaw. His students loved him. One of them, Bogdań Miś, recalls how he once debunked the theory of a world-class mathematician evidenced in an extensive essay by presenting a simpler solution which he and Banach had come up with in 1937. It was so simple that the academics didn't bother to publish it.

Later on, he went into politics. In 1947 he became a deputy to the Sejm (lower house of Polish Parliament) from the wing of the Blok Demokratyczny party and in 1952 as part of the Polish United Workers' Party. He was an idealistic communist and he believed that it was the best system in the world. His disillusionment with the system came in 1968. Nevertheless, he never emigrated. Some time later in front of cameras he handed over a live goose in a big decorated wicker basket to an academic from Sweden who solved equation number 153 from the Scottish Book.

## The rest of the Scottish round

The Lviv café marked the most productive period in the academic lives of a few other mathematicians: Professor Kazimierz Kuratowski, Władysław Orlicz, Kazimierz Bartel, as well as Leon Chwistek, who was a logician and an avant-garde painter, theoretician of modern art, literary critic, philosopher and mathematician, and a friend of Witkacy and Marek Kac.

And finally Antoni Łomnicki and Włodzimierz Stożek. Their lives and careers were cut short by Hitler's shots. Associate professor Herman Auerbach killed himself with cyanide when the Gestapo was taking him from the Ghetto hospital. What is left of the Lviv School of Mathematicians? Apart from the academic essays and anecdotes, a monument to the professors murdered in Lviv was erected in Wrocław. New York based sociologist Susan H. Case, a survivor of the attacks on the World Trade Center wrote one volume of poems in which she compares the devastation caused by 9/11 to the end of the Lviv School.