Two conferences, two mysteries

Two conferences in recent days. Time permitting, I would say more about each. I just want to point to two mysteries and some synergy.

Last week I was at the British Mathematical Colloquium in Manchester. The last plenary speaker at the first was Nina Snaith, who talked about one of the most striking recent mysteries, the connection between zeros of the Riemann zeta function and eigenvalues of large random Hermitian matrices. Her talk had the title “Every moment brings a treasure”, a quote from Gilbert and Sullivan but also reference to her work on moments of these distributions.

She described the encounter between Hugh Montgomery, a mathematician interested in the Riemann zeta function, and Freeman Dyson, a physicist who had been looking at random matrices, at Princeton in 1972. If the Riemann hypothesis is true, all zeros of the zeta function lie on the line in the complex plane with real part 1/2, while of course the eigenvalues of a Hermitian matrix are real. But, up to a rather trivial translation and rotation, the distributions have very similar properties.

This started a lot of activity. Nina described how, in the late 1990s, she and her supervisor Jon Keating had made a conjecture about the 8th moment of the zeta function using random matrix theory, following a challenge from Peter Sarnak. They met Brian Conrey in the corridor at a meeting on the subject; he was about to present a conjecture about this based on number theory. They went into a room and compared formulae; their conjectured formulae looked different but, after some nervous moments, calculation showed that they were equal. (The actual value is still unknown.)

After a lively question and answer session, and audience member remarked that face-to-face interactions of these types would be impossible at an on-line meeting, and for this reason we must preserve actual get-togethers.

The following Monday I was at a strange meeting, the 7th International Conference on Mathematics and Statistics. I spoke first, and described the ADE affair and my encounters with it. What I hadn’t realised is that this conference was joint with the 6th Applied Mathematics, Modelling and Simulation meeting. By lunchtime, I was beginning to think I was the only mathematician there; but things looked up later in the afternoon.

My talk was followed by polite applause and no questions. But despite my fears, it turned out to be, not the tree falling in the forest, but the frog jumping into the ancient Japanese pond. One person had understood the message. This was Jérôme Dollinger from Louvain. He spoke to me afterwards, and asked me a question which shed new light on the difference between the ADE mystery and the RZF–RMT mystery.

Jerome asked whether the ADE classification could be used in other classification problems, say in hypergraph theory. On thinking about it, I had to admit that almost all the applications we know had arisen because the ADE connection had mysteriously appeared in the solution to the problem; it doesn’t seem to be a tool which easily applies to unrelated problems. This is in contrast to what I had seen from Nina Snaith’s talk. After the discovery by Montgomery and Dyson, many people started thinking that perhaps the Riemann hypothesis could be proved by finding an infinite-dimensional Hermitian operator whose eigenvalues (after simple transformation) are the Riemann zeros, and which is approximable by finite random Hermitian matrices. Several people tried, but no solutions survived. But what happened instead was that the connection has give the random matrix community plenty of problems to work on, inspired by known properties of the RZF.