The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions--not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the caf~ where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathe- matical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the European Editor, Ian Stewart.

Lost Legends of Lvov 2: Banach's Grave Krzysztof Ciesielski

In the previous issue of the Mathematical Intelligencer I described how I found the Scottish Caf6 in Lvov. I wanted to find something else: Stefan Banach's grave. Banach was undoubtedly the best Polish mathemati- cian of his day. The notion of "Banach space" is now known to almost every mathematician in the world. Hugo Steinhaus, author of many important papers and the excellent book Mathematical Snapshots, said that his best mathemat ica l discovery was Stefan Banach. Once, during his evening walk, Steinhaus heard two men using advanced mathematical expres- sions. Banach, then very young, was one of them. Steinhaus told him of a problem that he was currently working on, and a few days later Banach turned up with a correct solution.

Banach was born and went to school in Krak6w, but his mathematical activity is associated with Lvov. On leaving school, he wanted to work in a subject other than mathematics; he regarded mathematics as inter- esting, but felt there would be almost nothing new to be discovered. Fortunately, he changed his mind.

In mathemat ics , Banach was self-taught. After quickly solving some difficult problems, he was of- fered a Chair and lectured in Lvov. The story of how he got his Ph.D. is curious. He was being forced to write his Ph.D. paper and take the examinations, but he kept saying that he was not ready and he still had plenty of time. At last the university authorities be- came nervous. Somebody wrote down Banach's re- marks on some problems, and this was accepted as his Ph.D. dissertation. But an exam was also required. One day Banach was accosted in the corridor by a col- league and asked to go to a certain room, because "someone has come and he wants to know some mathematical details, and you will certainly be able to answer his questions." Banach willingly answered the questions, not realising that he was being examined by a commission that had come to Lvov for this pur- pose. Such a procedure would probably be impossible today.

People who knew Banach said that he wrote down only a small part of his results. He was introducing

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new ideas, solving problems, speaking about mathe- matics all the time. It has been suggested that two mathematicians should have followed him and written down everything he said. Then most of what he in- vented would have been written down. Even without two such scribes, his known achievements are im- mense. Many important theorems are attached to his n a m e - - f o r example, the Hahn-Banach Theorem, the Banach-Ste inhaus Theorem, the Banach-Alaoglu Theorem, the Banach Fixed-Point Theorem, and the Banach-Tarski paradoxical decomposition of a ball.

Banach spent the Second World War in Lvov, living under difficult conditions. After the war he planned to go to Krak6w, where he would have taken a Chair of Mathematics at the Jagiellonian University, but he died in 1945, age 53.

I knew that Banach's tomb must be in Lvov, but I saw little chance of finding it. I didn't expect to find anybody who knew where the grave was. There was no guidebook about cemeteries, and the general Lvov guidebook did not mention Banach.

I went to Lychakov Cemetery (Cmentarz Lycza- kowski), the largest in Lvov. I started walking and looking, cont inuously encounter ing the graves of famous Poles--writers, painters. To my surprise, after a quarter of an hour I found Banach's tomb! If you visit Lvov and want to see it, you should take the first av- enue to the left on entering the cemetery. After ap- proximately a two-minute walk you will find the tomb to the left of the path.

In 1985 the Polish Mathematical Society started pro- cedures for the transfer of Banach's body to Poland. The idea was suggested by the Krak6w division of the

society. This raises a difficult question: Where should Banach's tomb be? In Lvov, the city of the Lvov school of ma thema t i c s , w h e r e he w o r k e d and became famous? Or in a city in present-day Poland, where many people, particularly his family, would have the opportunity to put flowers on his grave? At any rate, so far the attempts of the Polish Mathematical Society have proved unsuccessful.

Mathematics Institute Jagiellonian University Reymonta 4 Krak6w, Poland

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