Last Friday I went to the London Mathematical Society general meeting, at the BMA building in Tavistock Square. On a beautiful warm day I walked along the Regents Canal to Islington and then down through back streets past the former end of the New River, to the cool and impressive BMA building.
As the President, Terry Lyons, said, the official business of the meeting was pleasant: apart from admitting new members and giving people the opportunity to sign the book, it consisted in the announcement of prizes and honorary memberships. There was a longer list than usual of the latter, to commemorate the sesquicentenary of the society; and an impressive list it was too: Joan Birman, Robert Calderbank, Shafi Goldwasser, Donald Knuth, Robert Langlands, and Maryam Mirzakhani.
I was also delighted that the Hirst Prize was awarded to my colleagues John O’Connor and Edmund Robertson, for their wonderful history of mathematics website.
As usual, the real business was the two lectures; the second was the final performance of this year’s Hardy lecturer, Nalini Joshi, while the warm-up act was a talk by Marta Mazzocco.
Both lecturers spoke about Painlevé equations. These are non-linear second order differential equations, which I don’t know anything about. There are six Painlevé equations, numbered PI to PVI, with varying numbers of parameters from zero to four; I believe that in some sense they cover all cases. But the two talks were concerned to connect the Painlevé equations to things that I do claim to know something about: cluster algebras, and root lattices. Each speaker assumed (I think) that we know a bit about Painlevé equations but less about the other topics, and so spent more time on the things I knew a bit about – not that I minded!
But this means that my account of the lectures is going to be very sketchy.
Marta Mazzocco introduced cluster algebras by way of Pythagoras’ theorem, a2+b2 = c2. Slightly reinterpreted, this says that a rectangle which has one pair of opposite sides a and the other pair b, then the diagonals are c. In this form it generalises to arbitrary cyclic quadrangles: if one pair of opposite sides are a and a‘, and the other pair b and b‘, and the diagonals c and c‘, then aa‘+bb‘ = cc‘ (this is Ptolemy’s Theorem).
Now this equation defines the mutation rule for a cluster algebra. Clusters are triples of numbers; if we use the mutation rule x1x1‘ = x22+x32, we obtain an infinite cluster algebra. If we start with (1,1,1), then all the clusters satisfy the equation x12+x22+x32 = 0. This procedure generates the Markov numbers, the solutions of this equation in natural numbers. (The fact that they are all integral is a consequence of the famous Laurent phenomenon for cluster algebras.)
Now this account disappears into the stratosphere. This example, generating the Markov numbers, is related to the “quantum cohomology” solution to PVI. There is a representation on the punctured Riemann sphere, where the punctures become cusps; the surface can be triangulated by geodesics starting and ending in cusps, and everything can be expressed combinatorially in terms of the orders of the geodesics at each cusp.
Nalini Joshi saw her job as building a bridge between different areas, and illustrated her talk with pictures of the building of an icon of her home town, the Sydney Harbour Bridge.
She started out with the A2 root system, the vertices of a regular hexagon centred at the origin. If we divide it into six triangles, then the reflections in the sides of a fundamental triangle cover the plane with copies of this triangle; we can regard the structure as the root lattice (it is closed under translation) or the corresponding affine root system.
Choose coordinates of a point in the fundamental triangle which are its perpendicular distances from the two sides. These coordinates have constant sum (which can be assumed to be 1). They extend, allowing negative values, to the whole plane; the reflections, translations, and rotations of the fundamental triangle have simple expressions in these coordinates. However, in terms of “Cremona isometries”, something much more complicated occurs. We get a non-linear discrete dynamical system, which is non-autonomous (the time parameter n appears explicitly in the transformation formulae), and is described by a certain discretisation of a Painlevé equation!
All this holds too for other types of root system satisfying the crystallographic condition (not G2 as yet, apparently). There are relations between root systems and the corresponding Painlevé equations, which were worked out by Sakai. The simplest example, in root system terms, is a map from B3 to A2, which is easily described: look at a cube (the B3 system) along a diagonal; what you see is a hexagon (the A2 system).
There is also a duality which I didn’t quite catch, which interchanges large and small, so that the A2 system is replaced by E6. This is related to the McKay correspondence.
There was a lot more too. For example, take functions of four variables which are satisfied by the vertices of a square. Fit these together on the square faces of a cube or hypercube; this leads to an obvious consistency condition, which yields only a few solutions, one related to elliptic curves. They also discovered a case missed by Sakai, involving the root system F4.
That is the limit of what I can say. But I hope that I have communicated some of the pleasure I took from the occasion.