Today, the University of Auckland put on a morning meeting entitled Excellence in Mathematics: A Celebration of Diversity.

As the program (which is here) makes clear, it is actually a celebration of female mathematicians, and in particular the recent Fields medal for Maryam Mirzakhani.

As you know, I am not generally in favour of singling out any group of mathematicians, be they women, Jews, French citizens, combinatorialists, or whatever (all these four groups have been singled out, some with more serious consequences than others – and you can certainly think of many more examples), nor of Fields medals (most of which reward great contributions but which always have a hint of fashion or politics about them). However, I am very much in favour of celebrating mathematics by talking about our successful practitioners.

The speakers in this celebration were all women with one exception. No woman could be found here to talk about Mirzakhani’s work, so Marston Conder stepped up to the plate, and did a very fine job.

Spaces and hyperspaces

One thing that drives mathematicians is the urge to classify, to understand the members of a large diverse collection. I have seen grown mathematicians quail at the notion of moduli spaces, but the basic idea is simple. We are trying to understand a collection of spaces; we regard our spaces as points in a “hyperspace”, and give structure to the hyperspace which reflects properties of the constituent spaces.

If I want to appreciate the diversity of the New Zealand landscape, the best way is to travel around it observing. Similarly, one basic way to organise and explore our hyperspace is to wander around it, which implies some geometric notion of paths or at least of nearness. Indeed, once I was invited to speak at a conference for János Bolyai’s 200th anniversary; I had the idea of regarding Steiner triple systems on more than 9 points as a particular kind of discrete hyperbolic space, and taking a random walk through it (using a variant of the Jacobson–Matthews random walk for Latin squares).

Another unifying principle is that of an equivalence relation. If we don’t need to distinguish among equivalent spaces, we can regard the points of our hyperspace as equivalence classes of spaces. For a simple example, suppose we want to consider graphs up to isomorphism. The corresponding hyperspace supports various structures, such as a probability measure or a complete metric. Paradoxically, we find that, using either of these structures, there is a single point of the space (the random graph) which makes up almost all the space (its complement is a null and meagre set). Moreover, small moves from the random graph don’t get us away from this point.

It may be that our spaces have various numerical invariants or “moduli”, which can be regarded as “coordinates in hyperspace”. Hence the name “moduli space”.

Here is a very simple example. Consider the space of all normed real vector spaces of dimension 2. What does the corresponding hyperspace look like? Such a vector space is defined by a positive definite quadratic form ax²+bxy+cy² in two variables. So each point of hyperspace has three coordinates (a,b,c), where b² < ac and a > 0. So the corresponding hyperspace is the region of 3-dimensional space defined by these two inequalities.

Things are more complicated if we take our spaces over the rational numbers or the integers rather than the real numbers. Then we find ourselves doing number theory, following in the footsteps of Gauss. Indeed, another of this year’s Fields medallists, Manjul Bhargava, works on this …

Moduli spaces

What follows will not be very precise, and certainly I (rather than Marston) am to blame for any inaccuracies.

A Riemann surface is a closed orientable surface with a complex analytic structure imposed on it. The geometry allows one to talk about geodesics on the surface. It is known that the number of closed geodesics of length at most L grows exponentially, about e^L/L to be precise.

One of Mirzakhani’s achievements was to show that the number of non-intersecting closed geodesics grows only polynomially, like c.L^6g−6, where g is the genus (the number of holes) of the surface.

For this she used the moduli space for Riemann surfaces of genus g. Since there is only one (topological) closed orientable surface of genus g, as in the vector space example the hyperspace for such surfaces is the set of all complex structures on the fixed topological surface. This hyperspace can be parametrised by 6g−6 parameters, called moduli; so the hyperspace is “moduli space”.

What Mirzakhani did, very much simplified, was to show a remarkable connection between volume calculations in moduli space and counting closed geodesics on a Riemann surface corresponding to a point in the space.

Her work has led to a much more detailed understanding of how moduli spaces look. In particular, closed geodesics on moduli space (the natural next step) have remarkable regularity properties, resembling that of dynamics on homogeneous spaces, even though the moduli spaces themselves are far from homogeneous.

Marston also told us a bit about Maryam Mirzakhani herself. For example, she likes to doodle when she is thinking about something; the doodling keeps her engaged. I find the same thing.

Other talks

I enjoyed Hinke Osinga’s talk. Probably anyone who walks in the mountains has thought about watersheds, the phenomenon where raindrops falling on either side of an invisible line in the mountains will end in the sea possibly thousands of kilometres apart. (I went to school a stone’s throw from just such a watershed.) Now there is an object called the Lorenz surface, which plays a similar role for trajectories of solutions to the chaotic Lorenz equation. The dynamics on the surface itself is simple; there is one attracting fixed point, at the origin. But just off the surface, trajectories have very different behaviour depending which side they are on; and the surface is dense in space, explaining the enormously complicated behaviour of the system. Hinke first devised crochet instructions for producing a model of the surface, and then worked with an artist who produced a hammered steel model. (Think of the surface growing outward, parametrised by the time to reach the origin. The sculpture consists of a band between successive “circles”, and has a remarkable shape, smooth in parts, intricately convoluted in others.)

The other talks were mostly applied. (Perhaps making art out of the Lorenz surface is applied maths?) Gill Dobbie talked about big data, which is currently in the trough of disillusion after the wave of hype in the Gartner hype cycle for emerging technologies. Rosemary talked about her work with ecologists, and how after converting them to her viewpoint (even getting Hasse diagrams and a picture of the Fano plane published in biology journals) found that she had to question some of her own assumptions about which design is best. Tava Olsen talked about operations management, and Cather Simpson on how to use femtosecond lasers in real industrial processes. One thing I got from this talk is that, for things like artists’ pigments, the shorter the relaxation time of the molecule after excitation by a photon, the greater the long-term stability of the pigment. Having all that energy hanging about in the molecule is very destructive, as she said like a child who has been binging on chocolate let loose in a china shop: get it out as soon as possible!

Summing up

Maybe you want to learn about the beautiful landscape of New Zealand. There is no real substitute for going there, travelling about, and experiencing it first-hand. But maybe that is too expensive, or the travel is dangerous, or you are too busy (or you can’t get a visa) – then what do you do?

You could invite people from different parts of New Zealand to come and tell you about their area. With skill, they could convey something of its essence.

That was the strategy here, and worked successfully, making an entertaining and instructive morning.