Busy times, 9: Beyond the limit in St Petersburg

Anatoly Vershik

Anatoly Vershik is a universal mathematician, with influential work in asymptotic combinatorics, groups and group actions, probability, mathematical physics, and many other areas.

This week, I was in St Petersburg for a conference with the wonderful title “Representations, Dynamics, Combinatorics: In the Limit and Beyond”, celebrating his eightieth birthday. The meeting was at the Euler International Mathematical Institute, commemorating a universal mathematician who spent much of his career in St Petersburg.

Leonhard Euler

In their talks, both Benjamin Weiss and Hillel Furstenberg quoted Psalm 90, which in the King James version reads

The days of our years are threescore years and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow; for it is soon cut off, and we fly away.

Both of them used something like “heroism” in place of “strength” here. It is heroic of Anatoly to be still as active in mathematical bridge-building as he is. He is a hero in another sense too. As someone who has always chosen his own path, he had a lot of trouble from bureaucrats during his life, even condemned to work in an Operations Research department for a time. But these stories are better told by someone else.

He is more than twelve years older than I am, but perhaps I am scattier. I was in the agency that processes Russian visa applications for an hour and three-quarters, at the end of which I know what the stupid pupils in the class feel like. I took the form to the desk three times; the first two times I was told no, this wasn’t right, I should go away and do it again. They had computers in the place which applicants could use to re-do their forms, but although the form asked for the name, address, and telephone of every educational institute you had been to, access to all educational institutes was blocked, so I had to be a bit creative. Eventually the form was done and submitted, and I was told to come back the following afternoon. I spent the intervening thirty hours worrying about all the reasons they might have for turning down my application, but in the end my passport was there with a visa in it.

There was far too much interesting stuff at the conference for me to mention all but a small selection.

  • Jarik Nešetřil gave a nice talk about the connections between Ramsey theory (specifically, the classification programme for Ramsey classes of structures) and homogeneous countable structures, and also his recent work with Patrice Ossona de Mendez on the connections between homomorphism counting and graph structure.
  • Tatiana Smirnova-Nagnibeda and Slava Grigorchuk talked about Vershik’s notion of “totally non-free actions”. These are measure-preserving actions of a group on a probability space in which almost all pairs of points have distinct stabilisers (if I have it right). In trying to find an example to think about to fix my ideas, I realised that the action of the full symmetric group of countable degree on the space of graphs on the countable vertex set has a stronger property, which I have known about for some time, and which I will describe here sometime soon, I hope.
  • Sergey Fomin talked about the fact that forbidding subtraction in arithmetic computations can make the number of operations required exponentially greater, even when the function can be calculated. Unsurprisingly, cluster algebras made a brief appearance,
  • Andrei Okounkov told a beautiful story about some formulae which arise in connection with box-packing in two and three dimensions. The former is equivalent to counting partitions, and according to Hardy and Ramanujan the number is the exponential of a constant times the square root of n; he explained the square root of n as the average boundary of a Young diagram with n boxes, and the constant as twice the square root of ζ(2), where ζ is the Riemann zeta-function; the celebrated ζ(3) occurs in the three-dimensional case. Then he threw out the mystifying remark that anyone meeting this for the first time wants to generalise to higher dimensions, but this is not possible because 2×(2+3) = 10, where the first 2 is the dimension of the complex numbers over the reals, and 10 is the dimension of string theory. I got distracted by this, and observed that 2×(22+32) = 26, which was the dimension of string theory in the early, fermionic versions; then 2×(23+33) = 70, whose significance is reflected in the fact that the sum of the first 24 squares is 70 squared, giving an embedding of the Leech lattice into the 26-dimensional spacetime; then 2×(24+34) = 194, a number whose significance in this game I don’t know – it is not the dimension of E7.5 (which is 190).
  • I mentioned Igor Frenkel’s talk on categorification in the preceding post. A very impressive cocktail of things from apparently different parts of mathematics.
  • Dmitry Shirokov gave a talk scheduled to be given by Victor Maslov on the foundations of classical thermodynamics, where they have no van der Waals forces but explain the phase transitions by clustering instead.
  • Yury Neretin talked about the diagonal action of the product of three copies of the finitary symmetric group, showing phenomena which don’t occur in the finite case: a parametrisation of the double cosets by diagrams, which give a combinatorial description of a topologically defined multiplication. I haven’t really got to grips with this yet.

You can find slides of my talk in the usual place.

St Petersburg

On Monday there was a party, at which various people gave eulogies about the birthday boy, and his daughter cut an amazing birthday cake. On Tuesday, he very generously took the foreign guests to a pie shop, a café where the speciality was pies (meat, cabbage, berry, and various other things), where we spent a congenial couple of hours. On Wednesday a leisurely boat trip in bright sunshine took us to the extraordinary conference banquet: walking home with Greg Cherlin through the white night was quite an experience. On Thursday there was a chamber music concert, featuring music by St Petersburg composers of the early eighteenth century, including Madonis (I didn’t catch the names of the others). Friday was the excursion, to Menshikov’s Grand Palace at Oranienbaum and then Peter the Great’s naval base at Kronstadt.


On Saturday I was able to give my talk and attend all but the last talk of the conference, and still be back home in London well before the Underground closed for the night.


About Peter Cameron

I count all the things that need to be counted.
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6 Responses to Busy times, 9: Beyond the limit in St Petersburg

  1. John Duncan says:

    It seems worth mentioning that 194 is the number of conjugacy classes in the monster! Also, I think you meant bosonic, rather than fermionic, when describing the significance of 26.

    • Thanks – I had completely forgotten that! So the next one is 550 – any takers for that?

      I can never remember the difference between fermions and bosons – does that make me supersymmetric?

      • John Duncan says:

        So far I have nothing on 550. But it’s tempting to reinterpret 26 as the number of conjugacy classes in M24. Is there a nice simple group out there with 1586 classes?

  2. One of the most remarkable things about the Monster is how few conjugacy classes it has, relative to its order. If you take the symmetric group S44 of comparable size, it has 75175 conjugacy classes, whereas for the most prolific type of simple group, PSL(2,q), the number of conjugacy classes is roughly the cube root of the order. So probably this line is going to fizzle out.
    If it is true that there are only finitely many “interesting” numbers in the sequence, then perhaps the index of the last “interesting” number is significant.

    • John Duncan says:

      That is a nice way to view the situation. So, if we understand 194 to be the number of conjugacy classes in the monster, how are we to understand 2(2^4+3^4)?

  3. There is an album of conference photographs here.

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