The Lwów School of Mathematics became part of the history of mathematics and a legend. It was born in Krakow’s Planty in the summer of 1916, and flourished at the marble table of the Scottish Café in Lwów (nowadays Lviv). What is left of it is an ordinary notebook with problems written down, later called the Scottish Book, including works that are still the foundation of several areas of mathematics, and stories about extraordinary people who made mathematics an art.

The founders of the school were Stefan Banach, a mathematical genius, son of an illiterate woman living in the mountains and a soldier on leave, and Hugo Steinhaus, who could teach Polish as well as mathematics. They were sometimes treated like a master and an apprentice who surpassed the master, although Steinhaus was only five years older.

Among their most outstanding pupils were Stanisław Ulam, a member of the Manhattan project, co-creator of the atomic and hydrogen bombs, and Stanisław Mazur, the secretary general of the Polish Academy of Sciences after the war. And besides them, many others, such as Juliusz Schauder, also considered a mathematical genius, Herman Auerbach - great chess player, Władysław Orlicz - founder of the Poznań School of Mathematics as well as Mark Kac and Zygmunt Birnbaum, who, like Ulam, worked at American universities after the war.

It all started one summer evening in 1916 on a bench in Planty Park in Krakow, where Otto Nikodym, a graduate of mathematics at the University of Lwów, and Banach, a railway employee after two years of the Lwów Polytechnic, were talking about the Lebesgue integral. Nowadays, there is a bench with both figures commemorating that event.

The conversation was heard by Hugo Steinhaus, a mathematician with a doctorate from the University of Göttingen, who worked in Kraków at the National Reconstruction Centre. Lebesgue's theorem was understood only by the initiated, so intrigued as to who was talking about mathematics in Planty Park in the middle of summer, he approached. This is how he met Banach. Had it not been for this encounter, the history of the Lwów School of Mathematics might never have begun.

In 1917, Hugo Steinhaus became an associate professor at the University of Lwów (the school became the Jan Kazimierz University after the war). He had only a few listeners, but he hoped that when the war ends, the young people would resume studies. He had to wait longer, in November 1918 one war ended, but soon after another one began - against the Bolsheviks. In 1919, he invited Banach to Lwów and arranged for him to work as an assistant at the Technical University, although the protégé did not finish his studies. Besides, he never finished them.

To keep the job, Banach had to prepare a doctoral dissertation. After the publication of several scientific papers, he became known as a mathematical genius, but also a playboy who was bored with writing down the proofs and theorems spoken from memory. “He preferred spending his time at Pogoń Lwów matches or over a beer, of course discussing mathematics. He thought and gushed out mathematical theorems faster than he could jot them down”, those who knew him remembered. “He had the clarity of thinking that Kazimierz Bartel once called unpleasant” – Steinhaus recalled. Dean of the Faculty of Mathematics of the Jan Kazimierz University, Professor Stanisław Ruziewicz, instructed one of his assistants to follow the doctoral student even to a café and write down his mathematical ideas there. Banach only accepted the notes. This is how the dissertation *O operacjach na zbiorach abstrakcyjnych i ich zastosowaniach do równań całkowych *(On the operations on abstract sets and their applications to integral equations) was written. And it was only by trickery that he was made to defend it.

- There are few gentlemen from Warsaw who would like to discuss a mathematical problem with you - the dean told him. The discussion turned out to be a public defence of his Ph.D. thesis.

Without Steinhaus and Banach, the Lwów School of Mathematics would not have existed, but its history and achievements were made up of the works of more than twenty scholars who worked at the Jan Kazimierz University in 1922-1941. Thanks to them, for a dozen or so years Lwów had become one of the capitals of world mathematics. The most important contribution of the School to the history of world science was functional analysis, a new mathematical language that allowed to organize the achievements of mathematicians working in the most important mathematical centres around the world. Suddenly, it turned out that the theorems, proofs, analyses from different places were starting to fit together, although no one had noticed it before.

The creation of functional analysis was the final stage of a long historical process. The list of mathematicians whose research contributed to its birth includes such famous names as Vito Volterra, David Hilbert, Jacques Hadamard, Maurice Frechet and Frigyes Riesz. But it was 1922, when Banach published in French the content of his doctoral dissertation (Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales) in the journal “Fundamenta Mathematicae” which turned out to be a breakthrough year in the history of 20th century mathematics. The several-dozen-page dissertation was of paramount importance not only for the further development of mathematics, but also of natural sciences and physics.

The fame of Lwów mathematicians quickly spread beyond the borders of Poland. In 1929, Steinhaus and Banach founded the journal “Studia Mathematica”. Once a year, works were published in it without translations, in French (that is in the language of the international mathematics), German, English and Italian, and after 1939 also in Russian. Out of 500 copies of the circulation of "Studia", 200 were sold abroad, another 200 were exchanged for mathematical periodicals published at the most important universities in Europe and America, 100 were assigned for sale in the country. Eight issues of the magazine were published before the outbreak of the war, the ninth was published in 1940. "Studia" strengthened the growing fame of mathematicians from Lwów. "This publication can be considered an organ of the so-called Lwów School”, Steinhaus stated.

Lwów scholars met not only during university seminars and discussions. They continued their mathematical sessions first in the Roma café, and later in Scottish café at Akademicki Square, corner of Aleksandra Fredry Street. It was not the most elegant place in Lwów, but it had an atmosphere. The place was frequented by newspaper and radio journalists, students and professors, including mathematicians. Daily meetings were held there, inspired by Banach, who felt best in a café atmosphere. After Scottish was closed in late evening, he would often go to the station's buffet, open 24 hours a day, to continue talking about mathematics over a beer.

The scholars from Lwów and the Warsaw school mathematicians who were guest lecturers at the Jan Kazimierz University: Kazimierz Kuratowski, Bronisław Knaster, Alfred Tarski, and Wacław Sierpiński, visited Scottish Café. Mathematicians from all over the world were also invited to Lwów, including the famous Lebesque, with whose integrals it all began. Years later, Stanisław Ulam wrote that the intensity of thinking and the ability to concentrate during meetings in the Scottish can only be compared only with what was happening in Los Alamos in 1943 and 1944, when scientists working on the Manhattan Project were racing against the Germans as to who would be the first to construct an atomic bomb. “A short but lively discussion, a few lines written on the table, sometimes one of the participants laughed, then a long silence as we drank our coffee and stared at each other without saying a word. The other customers in the café must have been a bit puzzled by this peculiar behaviour. Such perseverance and the ability to concentrate are the most important attributes of genuinely creative mathematical activity”, recalled Ulam. To those watching from the side, they looked like a bunch of insane people, he added.

Tables in the Scottish had marble tops that you could write on with a pencil and, more importantly, you could easily wipe off your notes. “There was a session that lasted 17 hours, the result was a proof of certain theorem from Banach space - but no one wrote it down and no one will be able to recreate it today. Probably the tabletop covered with traces of a chemical pencil after this session was, as usual, washed off by the café cleaner. This, unfortunately, was the fate of many theorems proved by Banach and his students”, wrote Steinhaus

Pushing the table back to the corner and waiting for the students sent by the dean, Zbigniew Łomnicki, to write down the results of the session turned out to be only a half-measure. The system of work in the Scottish changed only thanks to the notebook which entered the history of mathematics under the name of the Scottish Book. It was an ordinary notebook with a marble-like cover, for which Stefan Banach's wife, Łucja, paid 2.50 zlotys in a stationery store and left in the café on 17 July 1935. From that day on, each of the mathematicians could ask for a notebook to write down a problem for colleagues or share one he was currently working on. The interest was twofold. Mathematicians stopped scribbling on marble tabletops, and complex proofs were not lost under the cleaners' cloths.

The first entry in the notebook is dated 17 July 1935. The author was the founder's husband: "When can a metric space (or possibly of type B) be metered so that it becomes a compact complete, and so that are the sequences converging according to the old distance be convergent according to the new one?"

On the same day, problems for colleagues were written in the notebook by three more mathematicians: Ulam, Mazur and Orlicz. They were placed on one side of consecutive pages of the notebook, leaving room for a solution on the other. Most of the problems in the Scottish Book were entered by Stanisław Ulam (62, 40 as the author and 22 as co-author) and Stanisław Mazur (47). Banach wrote 23 problems, Władysław Orlicz 14, Hugo Steinhaus and Józef Schreier 10 each.

Finally, 193 problems were entered into it (in fact there were a bit more, but not all were marked with a separate number), some have not been resolved to this day. One of the simpler ones, entered into the Book by Steinhaus (he himself called them problemaths), read: “A man was using two boxes of matches, taking out the matches at random. After some time it turned out that one box was empty. What is the probability that there were k matches in the second box, if initially there were n matches in each box?".

Problems were labelled with the name and information about the reward that the author declared in return for the solution. Most often it was alcohol. A bottle of wine (bonus for solving Banach, Mazur, Ulam and Sobolew’s problems), champagne (founder Łazar Lusternik), or whiskey ("a measure greater than zero" - promised by a guest from the USA John von Neumann)**. **The value of the prize depended on how difficult the problem was: it could have been a small coffee or beer (sometimes five beers), 10 grams of caviar, a kilogram of bacon, dinner in the restaurant of the best hotel in Lwów - George, and even *fondue à la crème* in Geneva, declared by the Swiss mathematician Rolin Wavre. In 1937, for solving one of the problems Stanisław Mazur promised a live goose. The problem was solved as many as 36 years later. In 1973, the Swedish mathematician Per Enflo received the promised goose from Mazur, but due to customs regulations he could not take it away, so it was eaten in Poland.

After the war, Łucja Banach brought the Scottish Book to Wrocław, Steinhaus sent a copy from the original to Ulam who translated the content of the notebook, made 300 copies and sent them to mathematicians in the United States and around the world. The book was presented in 1958 at the International Mathematical Congress in Edinburgh, causing disappointment among Scottish scientists that its name had nothing to do with their country.

The Lwów School of Mathematics, apart from the scientists who initiated it, was made up of a group of their collaborators and students. Not all of them managed find work at the University and the General Faculty of the Lwów University of Technology, where theoretical mathematics was also thought. “There were not many professors, both at the University and at the Polytechnic, and the salaries were low. People like Juliusz Schauder had to work as teachers in high schools in order to earn a living, and to supplement the meagre income of a lecturer or assistant” - Ulam recalled.

From the end of the 1920s, it was particularly difficult to find a job for scientists of Jewish origin, and more than a half of Lwów mathematicians were of this ancestry. National groups dominated at universities, including Lwów, and Jews were viewed with reluctance. Of course, few professors admitted to anti-Semitism, but when it came to voting on the nomination of a candidate for a vacant chair, it turned out that the "personal qualifications" required by the law made it impossible to choose a Jew. “It is amazing that there was always some professor X who found out something supposedly unflattering in the candidate's past, and even more amazing was the speed with which all doubts disappeared if the candidate was to convert to the Catholic faith", wrote Mark Kac, Steinhaus' student and assistant.

The intensification of anti-Semitic attitudes reached its zenith in the second half of the 1930s. After the introduction of the bench ghettos at the Jan Kazimierz University in Lwów, triumphant demonstrations by students from national organisations turned into brutal fights on the streets of Lwów. Rector Stanisław Kulczyński was forced to suspend classes at the university for several months. Supporters of anti-Jewish decisions pinned green bows to their clothes. "This made it possible to distinguish a handful of decent colleagues who refused to wear these bows", wrote Kac.

The Lwów School of Mathematics and, above all, the name of Banach became famous all over the world. In the 1930s, Professor John von Neumann, co-founder of the first computer visited Lwów several times. He offered Banach a job in the USA in the team of Norbert Wiener, called the father of cybernetics. A few years earlier, Wiener was racing against Banach. The theory, which went down in the history of science as the Banach space, was initially called the Banach-Wiener space.

The last time Wiener sent von Neumann to Lviv was in July 1937. Again, there was an offer to go to the USA, and for a lot of money. "How much does Professor Wiener give?" Banach asked. Neumann handed him a signed check with a number one written on it and said: "Professor Wiener asked to add as many zeros as you find appropriate."

- This is too small a sum to leave Poland - Banach was supposed to answer.

After the outbreak of World War II, several Lwów mathematicians were conscripted, one died during the September campaign, another was sent to the NKVD camp in Starobelsk and was most likely murdered there. Most of them remained in the city occupied by the Red Army.

Lwów became Lviv under the Soviets who introduced the reign of terror. The Polish authorities and officers were imprisoned. In February 1940 mass arrests among the civilian population began, and then deportations to Siberia and the mines of Kazakhstan. The intelligentsia circles were kept under surveillance by the new authorities which demanded absolute loyalty. Ukrainian chancellors and political commissioners were imposed on the universities, At the Ivan Franko University (this was the new name of the Jan Kazimierz University), the Faculty of Theology was dissolved as well as some departments, mainly humanities, many employees were thrown out - they were replaced by Ukrainians and Russians. Mathematicians were treated more leniently. Banach became the dean of the Faculty of Mathematics and Physics, Steinhaus, Mazur, Schauder and Eustachy Żyliński - heads of departments. Professors were enrolled in the Ukrainian Academy of Sciences and given higher salaries. “Together with the university salary, it made our life tolerable”, recalled Steinhaus. The meetings in Scottish café went on, the last entry in the Book waws made in May 1941. Some meetings were attended by Soviet mathematicians who came to Lviv, drawn by the fame of the creators of functional analysis.

Things started to get much worse after the outbreak of the German-Soviet war. Lviv was captured by the Germans in July 1941. This marked the end of the School of Mathematics. Zbigniew Łomnicki, Władysław Stożek (with his two sons) and Stanisław Ruziewicz were murdered in the Wuleckie Hills on the night of 3 to 4 July (a few weeks later also Kazimierz Bartel), which went down in history as the murder of Lviv professors. Stanisław Saks, Meier Eidelheit, Marian Jacob, Józef Schreier and Juliusza Schaudera were murdered by Germans in the following years. Herman Auerbach an Ludwik Sternbach committed suicide to avoid deportation to the extermination camp. Those who, like Ulam, Kac or Birnbaum, left Poland, or like Steinhaus managed to hide in a safe place, or were needed by the Germans, like Marceli Stark, survived.

Steinhaus spent the rest of the war in the vicinity of Gorlice and Jasło, in hiding. Banach, along with many other scholars (including Bronisław Knaster), became a louse-feeder at the Rudolf Weigl Research Institute for Typhus and Viruses, producing a vaccine against typhus. Work at the institute provided very good papers. The information that contact with the holder of the document could be fatal was a life insurance policy during the meeting with German gendarmes. This was the case until the Soviets re-entered Lviv in July 1944.

After 1945, despite attempts to revive it, the Lwów School of Mathematics was not reborn. Most of the Jewish scholars died in ghettos and camps. Lwów, now Lviv, by the decision of the leaders of the great powers concluded in Yalta, found itself outside Poland. The mathematicians who survived settled in various places in Poland and the world. Steinhaus ended up in Wrocław, where a Wrocław Mathematical School was established, co-founded by Knaster and Stark. Banach was also planning to move to Poland. The chair of mathematics at the Jagiellonian University awaited him. He did not make it, he had lung and bronchial cancer and died in Lviv on 31 August 1945. Stanisław Mazur worked at the University of Łódź, and later at the University of Warsaw. In 1968, he resigned from work at the University of Warsaw, protesting against the anti-Semitic purges at the university. Władysław Orlicz living in Poznań established the Poznań School of Mathematics. Stanisław Ulam, Mark Kac and Zygmunt Birnbaum stayed in the United States

According to Professor Roman Duda, a historian of science and mathematics, the Lwów School is the most important contribution of Polish science to general science. The names of mathematicians from Lwów are associated with theorems and theories known to mathematicians on all continents: Banach space, Banach-Steinhaus theorem, Banach-Tarski paradox, Steinhaus-Moser notation, Ulam spiral and Ulam matrix, Borsuk-Ulam theorem, Mazur-Banach game, Schauder theorem, Auerbach’s lemma and many more. Banach, after Euclid, is the world's most often cited mathematician.

The Lwów School of Mathematics became part of history, it found a place in encyclopaedias. The Scottish Book has not yet been published in print in Poland.

*Mariusz Urbanek*

#### Bibliography:

- Duda R.,
*Lwowska Szkoła Matematyczna*, Wrocław 2014 - Kuratowski K.,
*Pół wieku matematyki polskiej**1920-70*, , Warsaw 1973 - Urbanek M.,
*Genialni. Lwowska Szkoła Matematyczna*, Warsaw 2014