A celebration of diversity

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 eL/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.L6g−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.

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Open access and metrics: the Ball committee report

I mentioned this report in an earlier post; I am grateful to John Ball for directing me to the report on the web (here; the press release is here).

The overall conclusions are clear. The ICSU goals for open access are that the scientific record should be

  • free of financial barriers for any researcher to contribute to;
  • free of financial barriers for any user to access immediately on publication;
  • made available without restriction on reuse for any purpose, subject to proper attribution;
  • quality-assured and published in a timely manner; and
  • archived and made available in perpetuity.

These goals and their implication are discussed in detail in the report, which I urge you to read. Some related complications discussed include availability of data (this is very important in science but less so in mathematics); copyright issues; and legitimate constraints on open access (the report says “openness should be the norm which is deviated from only with good reason”).

The reason why bibliometrics are also in the title is that these are used in research evaluation, often in a rather crude way which will have to change as publication norms change. The panel says,

Metrics used as an aid to the evaluation of research and researchers should help promote open access and open science … If the full potential of open access to science is to be realised, new metrics will be required that incentivise open-access approaches and value research outputs that go beyond traditional journal publications.

Good news to colleagues whose outputs are, for example, widely used computing packages, or web-based information sources.

On another issue of serious concern to mathematicians, the report says,

The goals of open access advocated above can be satisfied … only if robust procedures are in place to ensure that those who do not have the means to pay for publication or access, or who are not affiliated to recognized institutions, are not disadvantaged.

All in all, it is good that some people with some influence are aware of our concerns.

In my view, there is one very important thing missing from the report. Part of an academic’s job has always been external activities: refereeing, both of papers and of grant proposals; editing; work for learned societies and their subcommittees; running information-rich websites; and so on. Since a lot of this relates to publication, and the burden is likely to increase when it is recognised that diamond access is the best way to go, this is closer to the subject of the report than might first appear. It would have been good to have seen a statement that university management should recognise these activities as part of our job, and should reward them (and adjust other loads) appropriately.

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Waitakere ranges

Tree fern

Yesterday we went to the Waitakere ranges.

We took the train (Western line) to Swanson. This is one stop short of the terminus at Waitakere, but trains don’t run to Waitakere on Sundays, and after the imminent electrification of the line, this will become permanent: there is a tunnel between Swanson and Waitakere which is too small to double-track or electrify and too expensive to enlarge.

It was a beautiful day, though the trails were very wet and muddy after the recent rain. The mountains are only a little over 300 metres, but after a gentle start they slope up more and more steeply as you approach from the east, so no roads from Swanson reach the scenic drive along the top, and the last bit of track is quite challenging. The view from the top was quite remarkable.

Auckland from Waitakere ranges

On the way down, we passed many fine kauri trees. These are currently threatened by kauri dieback disease, and trampers are instructed to scrub their boots and spray them with disinfectant before walking these trails. I hope that we didn’t spread the disease; this was our first trip to the country since our arrival, and our boots hadn’t been bought last time we were in this country. Any residual British mud was probably fairly benign.

Kauri tree

The greatest joy of the forest for me was the birdsong, especially tui. This is still new enough for me that I stop and listen when I hear a tui getting into its stride. There are some on the university campus, competing with traffic noise, and also with the local common mynas which seem to have learned to copy tui (among other sounds). In the forest, they compete only with other birdsong, and the effect, among the magnificent trees, is remarkable. But I’m afraid I can’t share it here.

Back in Swanson, we had a very good and leisurely lunch at the Station Café before catching the train back to town.

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Bibliometrics, open access, and all that

On Saturday, the delegates to the ICSU in Auckland were supposed to be taken on various sightseeing trips around the city before getting down to serious work the next day. Unfortunately, the weather put paid to most of that: it was no weather for harbour cruises for example. Some gannet-watching was possible, though even this was apparently a bit fraught.

On Saturday night we were invited to a “barbecue” at Gaven and Dianne Martin’s beautiful house at Albany Heights, along with mathematical delegates to the meeting and a few others. The kitchen was large enough that it was not necessary for Gaven to stand in the rain preparing sausages for the assembled company. With plenty of good NZ wine, it was a very pleasant occasion.

At dinner, I talked to John Ball, who is chairing a committee producing a report on bibliometrics, open access, and all that. I was very heartened by his account of what the report is going to say. I will pass on one particularly nice story.

Detailed statistics are available for the football games in the recent World Cup. In one particular match, reported here, the statistics show the teams to be very evenly matched: ball possession 52 to 48, attempts on goal 18 to 14, free kicks 14 to 14, and so on. You might think it was a close and exciting game, until you look at the one statistic that really matters: goals. These were 1 to 7. (The game was Brazil v Germany.)

Statistics for Wiles’ proof of FLT, or Perelman’s proof of the Poincaré conjecture, anyone?

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Regular polytopes, 2

In the preceding post with this title, I showed how to translate the existence question for regular polytopes into one concerning groups, specifically string C-groups. I will begin by saying a bit more about the reverse construction.

Suppose that we have a string C-group G of rank d, generated by involutions ρi, for i in {0,…,d−1}. As earlier, let GS be the subgroup generated by the involutions ρi for iS; and let Hi be the group GS where S consists of all the indices except i. Then Hi is the stabiliser of the flag fi in our standard flag. By transitivity, the other i-dimensional faces are parametrised by the cosets of Hi in G.

So we can take the faces to be the cosets of the subgroups Hi, for i = 0,…d−1; two faces are incident if the corresponding cosets have non-empty intersection. This recovers the structure of the polytope. By the intersection property, G is the trivial group, and G{i} is the subgroup of order 2 consisting of the identity and ρi.

In our cube example, let us number the vertices from 1 to 8, so that the special vertex v is 1, the special edge e is 12, and the special face f is 1234; let 5,6,7,8 be the vertices on the other face adjacent to 1,2,3,4 respectively. Then

  • ρ0 is the reflection swapping 1 and 2, so as permutation it is (1,2)(3,4)(5,6)(7,8).
  • ρ1 is the reflection in the plane through 15 bisecting the angles between the adjacent faces; so it is (1)(3)(5)(7)(2,4)(6,8);
  • ρ2 is the reflection in the plane through 12 and the opposite edge 78; so ρ2 is (1)(2)(7)(8)(3,6)(4,5).

It is easy to verify that this group has the required properties.

The group can be encoded if we have a faithful permutation representation of it. Suppose that G acts faithfully on the set {1,…n} for some n. Now form the edge-coloured multigraph on this vertex set, in which x and y are joined by an edge of the ith colour if (x,y) is a cycle of the permutation corresponding to ρi. From the edge-coloured multigraph, we can recover the permutations ρi (their non-trivial cycles are the edges of the ith colour) and hence the group G.

This graph is called a CPR graph (for “C-group permutation representation graph”).

If we take the set on which G acts to be the set of maximal chains, we obtain the Cayley graph of G with respect to our distinguished generators. Another natural choice for the set is the set of j-dimensional faces (if this action is faithful); this is what we did above for the cube, with j = 0.

A set S of elements of a group G is said to be independent if no element of S is contained in the subgroup generated by the remaining elements. By the Intersection Property, the distinguished generators of the group of a regular polytope are independent.

A theorem of Julius Whiston (discussed here) shows that the largest size of a set of independent elements in the symmetric group Sn is n−1, and that if equality holds then the independent set generates the symmetric group. Philippe Cara and I found all the independent generating sets of size n−1 in Sn. The only case in which they are all involutions is where they correspond to the edges of a tree, and the only such case in which we have a string C-group is when the tree is a string (the Coxeter–Dynkin diagram of type An (as described here). The corresponding polytope is the (n−1)-simplex (the tetrahedron for d = 3, n = 4).

So we see that a regular polytope having a CPR-graph with n vertices must have rank at most n−1, with equality if and only if it is a simplex.

Dimitri Leemans and his co-authors have been extending this result, as I hope to describe soon.

In the meantime, let me remark a curious connection with another recent post here. Given a regular polytope of dimension d, the subgroups GS generated by subsets of the given generators form a lattice isomorphic to the Boolean lattice of rank d. As we saw, the proof of this requires the Intersection Property. If this property does not hold, then we only have a join-semilattice. I discussed this issue here: the existence of a Boolean meet-semilattice of the subgroup lattice of G is equivalent to the existence of a Boolean join-semilattice of the same rank, but not to that of a Boolean lattice of the same rank. I think there are things here deserving further explanation.

To conclude, Marston Conder told me that he and Deborah Oliveros published a paper in the Journal of Combinatorial Theory last year, in which they showed that in order to verify that a group generated by involutions does satisfy the Intersection Property, it is not necessary to check all possible intersections.


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Regular polytopes, 1

One of the topics I am thinking about with Dimitri Leemans at present concerns regular polytopes. He and his co-authors Maria Elisa Fernandes and Mark Mixer have produced some nice results and a tantalising problem about these objects. I will give a brief introduction here; hopefully later I will have some more to report on.

A polytope is a higher-dimensional generalisation of a polygon in 2 dimensions or a polyhedron in 3 dimensions. Rather than stretch your geometric intuition, I will describe polytopes combinatorially. Keep the cube in mind as an example. Here is the cube; I have selected a particular vertex v, edge e and face f. (Ignore the primed vertices for the moment.)

A cube

The faces (of whatever dimension) are partially ordered by geometric incidence. (In the cube, a vertex lies on an edge, an edge on a face, or a vertex on a face.) We “complete” the cube by a minimal and maximal element, corresponding to the empty set and the entire polytope.

The polytope, as partially ordered set, has the following properties:

  1. Every maximal chain has the same length. (In the case of the cube, a maximal chain has the form (empty set, vertex, edge, face, cube).) So we can talk about the dimension of a face: the empty set has dimension −1, a vertex dimension 0, an edge dimension 1, and so on.
  2. A connectedness condition: we can move from any face to any other by a sequence of steps in which consecutive faces are incident; we can further assume that every face in the sequence except the first and last has dimension i or j, where i and j are two given dimensions.
  3. If f and g are faces of dimensions i and i+2 respectively, then there are exactly two faces of dimension i+1 incident with both f and g. (In our picture of the cube, v and v’ are the faces incident with the empty set and e; e and e’ are incident with v and f; and f and f’ are incident with e and with the whole polytope.)

An automorphism of a polytope is a permutation of the faces preserving the partial order (and hence preserving the dimensions of faces). The collection of all automorphisms is closed under composition and so forms a group, the automorphism group of the polytope. The automorphism group of the cube is, as you would expect, the group S4×C2 of order 48.

It follows from the three conditions above that the identity is the only automorphism fixing a maximal chain. (In the case of the cube, suppose that an automorphism fixes v, e and f. Then it must fix v’, the only other vertex incident with e; similarly it must fix e’ and f’. Using the connectedness, we can work from any maximal chain to any other, and find that everything is fixed.)

So the number of automorphisms does not exceed (and, indeed, is a divisor of) the number of maximal chains. The most symmetric polytopes are thus the ones in which the number of automorphisms is equal to the number of maximal chains, and so the group of automorphisms acts transitively on the maximal chains. These are the regular polytopes.

Suppose that P is a regular polytope. Then there is a unique automorphism of P mapping any maximal chain to any other. We fix a maximal chain C = (f−1,…fd). Now, for any i with 0 ≤ d−1, there is a unique maximal chain Ci which agrees with C in every dimension except i, and so a unique automorphism ρi which maps C to Ci. Then ρi also maps Ci back to C, and so ρi2 = 1, where 1 denotes the identity automorphism.

For example, in the cube, ρ0 reflects the cube in the plane of symmetry bisecting the edge e; ρ1 reflects the cube in the plane of symmetry through v bisecting the face f; and ρ2 reflects the cube in the plane of symmetry through e and bisecting the angle between the faces f and f’.

A connectedness argument shows that, using these reflections in a suitable sequence, we can map C to any maximal chain. So the group G generated by the automorphisms ρi acts transitively on the maximal chains, and so must be equal to the automorphism group of the polytope.

So the automorphism group is generated by d involutions (elements of order 2).

These involutions have two more important properties:

  • If |i−j| ≥ 2, then ρi and ρj commute, so their product has order 2. (For example, in the cube, ρ0 and ρ2 are reflections in perpendicular planes.)
  • For any subset S of {0,…d−1}, let GS denote the subgroup of G generated by the elements ρi with iS. Then the intersection of GS and GT is equal to GST. This is called the intersection property.

A group generated by involutions with this property is called a string C-group. (“String” because we can imagine the involutions ρ0,…ρd−1 arranged along a string, so that non-adjacent involutions commute; the convention for Coxeter graphs is that involutions are joined by an edge if and only if they do not commute. The “C” stands for “Coxeter”.)

Conversely, any string C-group can be shown to be the automorphism group of a regular polytope.

This material is discussed in a paper by Daniel Pellicer, “CPR graphs and regular polytopes”, in the European Journal of Combinatorics 29 (2008), 59–71.

In the next part, I hope to discuss how we represent and recognise string C-groups, and how this contributes to the theory of regular polytopes.


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Hood fellowships

On Monday, we went to a celebration of 10 years of the Hood Foundation, in University House, a lovely building which had been a synagogue and then a bank and was now offices for part of the university administration (with a central area for functions like this). A young man played Bach cello suites from the balcony (where women used to sit when it was a synagogue), but unfortunately was completely inaudible over the rising level of conversation.

The most interesting part of the evening was an inspirational talk from John Hood. He said, in essence, that support for curiosity-driven research is vital for all our futures, and that New Zealand is very poor at supporting it compared to countries of similar size and wealth. What could we do? Only two things. First, try to persuade politicians of the importance of research. Second, encourage philanthropy. A large number of people are friends of the university (in some sense), and they can be encouraged to put their hands into their pockets.

I feel a little uneasy about all this, and I am not quite sure why. American universities have seen their alumni as a resource for many years now, but this has been slower coming to Britain (and, I suppose, New Zealand too). I don’t like relying on charity for support, though that is what I am doing at the moment, and having a very productive time of it. (Today, a paper submitted, a paper accepted and sent to the journal production department, progress on two further projects, and a very nice colloquium talk connecting C*-algebras, graphs, and dynamical systems.) Will the donors feel that I am using their money well? Should I be even thinking about this while I am so busy with the research?

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