Fiddler And The Proof

There is a link between a great Jewish musical and the search for a historic mathematical proof

Mark Ronan

Hidden away somewhere between Winchester and Basingstoke in blissfully rural surroundings lies a semi-ruined mansion with a delightful theatre attached. This is Grange Park, where each summer they stage operas as the composer intended, rather than using some outlandish directorial concept, as most of the big-name opera houses do these days.

This year the season started with Fiddler on the Roof, admittedly a musical rather than an opera, but performed with great panache and boasting Britain’s great operatic bass-baritone Bryn Terfel in the principal role of Tevye. Suddenly we were transported to the lost world of the shtetl in Tsarist Russia, where a Hasidic Jewish community lives a hardscrabble existence balanced precariously between joy and disaster, like “a fiddler on the roof”. And there is Tevye, a big-hearted paterfamilias with five daughters he needs to marry off, ready to argue and debate with suitors, and with himself. “On the one hand, on the other hand” — the two sides to any argument — are central to Tevye’s life, and to Judaism, which even in its more fundamentalist interpretations legitimises debate. How different from the type of religious fundamentalism where discussion is discouraged and independent thought can lead to rejection or even death.

As I sat submerged in this quintessentially Yiddish story with its feistiness and rational debate, I recalled the intellectual ferment and excitement of my early academic career in America, where the problem of finding and classifying all the basic building blocks of symmetry was under concerted attack. Half the people propelling this forward were Jewish, and I remember witty Yiddish expressions being bandied around during the perpetual search for gaps, the nailing down of loose boards and the huge effort to build a roof over everything so that we knew the essential details were complete.

It took years more to cover a potential gap in the roof, and to rearrange the interior so that future generations would be able to find their way around this massive Rabelaisian world of proofs and refutations, to understand the highly condensed work of the many minds whose efforts frequently overlapped one another. Some 10,000 technical pages in mathematics journals had to be reassessed, pruned and reconstituted. Yet it was already clear to most experts that the building was essentially complete and without leaks.

What got the whole enterprise spinning was a theorem by Richard Brauer, who left Germany in 1933 after Hitler’s anti-Semitic laws had forced him out of his academic position in Königsberg. After spending time at Toronto and various universities in North America, Brauer ended up at Harvard where he provided a tool to help classify all the basic building blocks of symmetry, showing that only a limited number could share the same cross-section. So if you could find all possible cross-sections, you could hope to find all symmetry “atoms”. Yet there was a small problem. Did all atoms of symmetry, except the simplest ones, have such cross-sections?

The people who answered that question — in the affirmative — were the American mathematicians Walter Feit and John Thompson. In 1963 they published a 250-page paper of closely reasoned argument, and the race was on. A mid-1950s paper from France, building on previous work in the 19th century, had provided what might almost be a complete list of symmetry atoms after it was complemented by several variations discovered by mathematicians working in America.

Among these was a list of potential cross-sections that Thompson dealt with, or thought he had dealt with, before a strange letter arrived in Chicago from a Yugoslav mathematician named Janko working in Australia. Armed with an early version of Thompson’s paper, he had noticed a glitch in the smallest case and discovered an awkward new symmetry atom that didn’t fit the known pattern. He called it “J”.

In fact he discovered, and then un-discovered it, several times. Discovery alternated with contradiction until its existence was finally confirmed, and when Thompson asked him what had happened to the last contradiction, Janko didn’t know but it didn’t matter. He had discovered the strangest symmetry atom anyone had ever seen, and new curiosities emerged thick and fast. Whole books have been written about these discoveries, so I shall say no more except to draw together the threads of an extraordinary story.

Two projects were proceeding simultaneously. One was to find gaps in the known pattern and discover new odd-ball symmetry atoms — a sort of Easter egg hunt where those who had looked in strange places and found nothing had neither the time nor inclination to write anything up. That was the task of the other project, opening out all nooks and crannies to show there was nothing new to be discovered. That the entire enterprise succeeded without coming to grief on the reefs of mathematical confusion was due to Danny Gorenstein at Rutgers and, as one of his ex-students told me, visiting his office was like being at Central Command. You went in to discuss your thesis problem and the phone rang. You eventually got back to your own concerns only for the phone to ring again, and so on.

There was endless debate, discussion and questions, but under Gorenstein’s baton, with Michael Aschbacher at Caltech as leader of the orchestra, the whole symphony was moving to its final phase. Ever larger exceptions appeared, including the largest ever found (dubbed “The Monster”), and a final contribution from Janko, but it was all over bar the shouting. Like Tevye in Fiddler, Gorenstein held disparate forces together, but unlike Fiddler there was no pogrom, though forced exile had its impact, in ways not always recognised.

In Paris for a conference, Walter Feit (of the Feit-Thompson theorem) invited me to dine with him. An affable American with a penchant for plaid jackets, he seemed unlikely to have acquired any foreign languages as a young man or even perhaps to have been abroad, and as he had no French I offered to ring the restaurant. Yet years later during a conference at Oxford in his honour he thanked everyone, saying he had once been at school in the city. Parental travel broadening his horizons? No — Walter Feit was on the last Kindertransport from Vienna on September 1, 1939. War broke out two days later, and he never saw his parents again.

At the end of Fiddler, the Jews leave, mainly for America, where a centuries-old tradition of argument and analysis of the Torah feeds an intellectual fervour that continues hugely to benefit the academic world.

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