That’s Maths: The Lvov School and the golden age of Polish mathematics

For 150 years the city of Lvov was part of the Austro-Hungarian empire. After Polish independence following the first World War, research blossomed; between 1920 and 1940 a sparkling constellation of mathematicians flourished in what is today known as Lviv.

Zygmunt Janeszewski, who had been awarded a doctorate by the Sorbonne in 1911, had a vision of Polish mathematical greatness and devised a programme for its achievement. He advocated that Polish mathematicians should specialise in a few clearly defined fields rather than trying to cover too many areas. This would ensure common interests and foster a culture of collaboration.

A plan like this can only succeed if there are talented people to carry it out. Fortunately, while there was no strong tradition of excellence, several brilliant Polish mathematicians emerged around that time. The leading lights were Hugo Steinhaus, who had a doctorate from Göttingen (then the Mecca of mathematics) and Stefan Banach, who would become the greatest ever Polish mathematician.

Fundamental aspects

Diverse contributions to mathematics were made by the Lvov School, earning it worldwide admiration. Names such as Banach, Sierpinski, Kac and Ulam occur frequently in modern textbooks. They were concerned with fundamental aspects of mathematics: axiomatic foundations of set theory, functions of a real variable, the nature of general function spaces and the concept of measure.

The year 1932 saw the publication of Banach’s monograph on normed linear spaces. It contained many powerful results. His genius was to combine different areas of mathematics. He treated functions as points in an abstract space that was linear, with a concept of distance and an absence of “gaps”: a complete, normed linear space, a fusion of algebra, analysis and topology. It proved eminently suitable for the development of the field called functional analysis.

Banach’s monograph was very influential and his notation and terminology were widely adopted. His spaces became known as Banach spaces and they have played a central role in functional analysis ever since. They also served as a foundation for quantum mechanics. Banach’s monograph established the international importance of the Lvov School.

Many of the breakthroughs in Lvov resulted from collaborations, and most publications are the work of two or more authors. The mathematicians used to meet in cafes, discussing mathematics late into the night. Their favourite was the Scottish Cafe, the most mathematically productive cafe of all time. The table tops were white marble on which mathematics could be written and erased. Later, Banach’s wife bought a large notebook for the group – the famous “Scottish Book” – in which problems and solutions were recorded. This was kept in the cafe and was available to any mathematicians who visited. Ultimately, it contained about 200 problems, many of which remain open to this day.

Goose problem

One problem caused a media sensation when it was finally solved. Stanislaw Mazur had offered a live goose for a solution, and, in 1973, the Swedish mathematician Per Enflo travelled to Warsaw to collect his prize from Mazur for solving “the Goose problem”.

The Scottish Cafe exemplified the synergy and camaraderie that pervaded Polish mathematics in the interwar years. The second World War changed everything. Polish culture was systematically eradicated. Steinhaus managed to escape execution by assuming a false identity. Banach survived to witness the defeat of Naziism but died shortly afterwards. Today, Lviv is a major centre of culture in western Ukraine. The golden age of the Lvov School has passed into history.

[ thatsmaths.com ]