Tuesday, September 03, 2013

Lev Pontryagin: 105th anniversary

Lev Semenovič Pontryagin was born in Moscow, Russian Empire, on September 3th, 1908, i.e. 105 years ago, and died in May 80 years later, about 25+ years ago.

Interestingly and sadly enough, a primus stove explosion made him legally blind at the age of 14. That didn't prevent him from becoming a top mathematician.

On the other hand, it didn't stop him from being a jerk of a sort, either. In 1936, he warned the Soviet officials that the mathematics community was full of counter-revolutionaries in the so-called Luzin affair. People were losing jobs. He was not only an aggressive commie, he was a sort of fascist, too. During mathematical conferences, he would scream that a pro-Israel Jewish scientist named Nathan Jacobson was a mediocre mathematician and racist because he was a Zionist. Another, even better Jewish mathematician, Grigory Margulis, won the Fields medal but couldn't get the permission to leave the USSR after Pontryagin painted him as a dirty Jew, too.

Much like other anti-Semites, he would claim that he wasn't one – he was just an anti-Zionist, everyone was told. Good try but my suspicion isn't quite gone (although I made a different conclusion 5 years ago).

Later in his career, he would work on optimization; Pontryagin's minimum principle is behind the bang-bang control. But string theorists primarily know him because of his earlier work on algebraic and differential topology.

Needless to say, the most famous concept named after him is a characteristic class now known as the Pontryagin class. (Although the Pontryagin duality for the Fourier transform on locally compact groups is also deep.) If you search through Google Scholar for papers mentioning both "string theory" and "Pontryagin class", you get 276 hits dominated by papers written by Witten, Vafa, Harvey, Moore, Sethi, Mukhi, and a few pals.

The Pontryagin class is a complexified even Chern class within a cohomology whose degree is a multiple of four. Needless to say, I don't really understand these matters well. The people for whom it's their cup of tea must think about many things in terms of vector bundles. It's probably great for them and it allows them to see and calculate many interesting things but despite a course by GM, I just couldn't learn to use those things. I need to translate bundles to some fields with some properties or physical conditions, otherwise I don't really understand them. In some sense, I feel that mathematics and not physics must be the "mother tongue" for those folks even though many of them are stellar theoretical physicists, too.

A longer CV was written 5 years ago.


  1. ... and speaking of supercentenarian anniversaries, Nobel-winning economist Ronald Coase almost but not quite made it to celebrating his 105th birthday. He died yesterday aged 102.

    These passages, from two obituaries republished on Benny Peiser's GWPF website, should warm the cockles of TRF reader Luke's heart:
    The Royal Swedish Academy of Science awarded Coase the 1991 Nobel in economics “for his discovery and clarification of the significance of transaction costs and property rights for the institutional structure and functioning of the economy.”
    Unusual for an economist, Coase had concluded early in life that mathematics “was not to my taste.” So he built his career offering insights on the legal precepts and institutions, such as the corporation, on which the field is built. (...)
    Despite the claim that his theory is “mathematical,” Coase’s work lacks
    even a single equation. Coase’s ideas are about reality, not theoretical
    math — a reason why he rejects what he calls ”blackboard economics” because “it does not study the real world.”

  2. This kind of algebraic topology is actually my field but I don't think I would like to to try to discuss it here in any detail ;-)

    Pontryagin's contributions to topology are of course huge. From my point of view an even more important thing than the Pontryagin classes is the Pontryagin-Thom construction. It was first introduced by Pontryagin in 1950 in connection with the earliest work on stable homotopy groups of spheres - a central subject in homotopy theory. A more general version was later rediscovered by Thom and now Thom's name is more often used in this context (e.g the so called Thom space of a vector bundle) but the original discoverer was Pontryagin.

    Pontryagin was indeed an anti-semite and so were several other famous Soviet mathematicians, particularly the number theorist Vinogradov and the great algebraic geometer Igor Shafarevich (still alive) although the latter case is more complicated for Shafarevich is above all a Russian nationalist (he was always an anti-communist and a close friend of Solzhenitsyn). His views of this matter are rather nuanced and he certainly does not consider himself an antisemite (he has had Jewish students such as Moishesohn to whom he was close even after the latter emigrated to Israel). But so was Pontryagin.

    Vladimir Arnold, whose mother was Jewish, has no reason to be fond of Pontryagin since it is well known that it was Pontryagin who prevented him form getting a Fields medal (in those days Western mathematicians carefully consulted Soviet authorities, which meant Pontryagin, before awarding any prizes to Soviet mathematicians or even inviting them to the West. Not doing so meant that nobody would be allowed to travel).

    But in his memoirs Arnold tells not only about Pontryagin's well known anti-semitism but also a story of how Pontryagin took great risks to save his Jewish student Vladmir Rokhlin from a labour camp. The following is my rough translation from Arnold’s memoir:

    "When I was a student at Moscow State University Pontryagin was already very weak. He was probably the most brilliant of our lecturers. He had only just left topology and became occupied with control theory and his personality also underwent significant changes. Later he explained the reasons which lead him to take up applied mathematics and his antisemitic views in his autobiography published in “Uspiehi Matematicheskikh Nauk”. When he submitted his work for publication, a representative of the KGB told him that should not be published in this form as everything was stated too openly. I wish the original text would be published, the one that appeared is far too polished.

    Some believed that Pontryagin’s antisemitism was the result of his fear that he himself had Jewish blood and that this fact could be made public.

    However, Pontryagin was not always like that! During the war his best student V.A. Rokhlin was wounded and taken prisoner by the Germans. (He survived by pretending to be a Muslim). Later Rokhlin was liberated by the allies, returned to the USSR and continued to serve in the army, while the war still went on. When he was taking a captured German offcier to his commander he met a drunk NKVD officer who wanted to immediately shoot the prisoner. Rokhlin resited and fortunately at this moment Rokhlin’s commander saved both by taking them immediately elsewhere. But in the end, like all those who were liberated by the allies, Rokhlin was sent to a labour camp (in the north of Russia).

    After a few months a fellow prisoner of Rokhlin’s arrived in Moscow and told Pontryagin that Rokhlin is alive but is dying of hunger in a camp. Pontryagin, with the support of Kolmogorov, Aleksandrov and others, wrote to Beria, asking for the immediate release of the talented mathematician. Beria signed the order for the release of Rokhlin, who was given a machine gun and continued to serve as a guard in the same camp. Pontryagin together with the others wrote a second letter to Beria and finally Roklhin was released and allowed to return to Moscow."

  3. I am happy to see experts in topology around here too! I just hope you'll start criticizing whenever something someone says about topology is not right... someone like me, I mean... ;)