- First, there are many different ways in which a group may be presented to the computer: by a set of concrete generators (which may be permutations, matrices, or something else entirely); a presentation by generators and relations; as the symmetry group of something; as the homotopy group of some topological space; and so on. These may require very different techniques. For example, given generating permutations on a finite set, we can learn everything about the group in a finite amount of time; but given a presentation, it is undecidable even whether the group is trivial or not.
- What seems easy to a human may be hard for a computer, and
*vice versa*. I can say, “Let*p*be a prime divisor of the order of*G*, and let*P*be a Sylow subgroup of*G*.” But, even if I know the group order and can factorise it(!), standard proofs of the existence of Sylow subgroups are worse than exponential. It was a huge and somewhat shocking breakthrough when Bill Kantor showed how to find them in polynomial time. - Randomness plays a big part in computational group theory. Many important algorithms are either
*Las Vegas*(if they run, they give the right answer, but they may fail with small probability), or, even worse,*Monte Carlo*(which might give the wrong answer with small probability). This is disturbing to mathematicians, even those who entrust their personal and financial details to a system whose security depends on numbers which are “probably prime”. - There is also a distinction between theorists and practitioners, which was graphically highlighted by Laci Babai at a conference where he referred to them as the “reds” and the “greens” (I cannot now remember which was which). A theorist devises an algorithm which runs in time “soft O of
*n*squared”, with constants depending on something or other, and “soft O” allows an arbitrary power of log*n*; the practitioners have a program which runs on huge groups in a few seconds or hours when programmed in GAP or Magma and run on such and such a machine.

(In connection with the last point, Dick Lipton and Ken Regan have discussed the concept of galactic algorithms, which are theoretically efficient but will never be run, either because the constants are too large, or because the power of *n* is too large. They hope that some of these can be brought down to earth in the future.)

The basic things that a computer can do with a group are to multiply group elements, invert an element, and return the identity. This led to the notion of a *black box group*, where the group elements are represented (not necessarily uniquely) by bit strings of fixed length, and the black box performs these operations in a specified time which may be taken as the unit for measuring the complexity. I remember a talk at the ICM in Kyoto in 1990 in which Babai illustrated this by a picture of a Japanese-style street vending machine, with three slots: you put group element *g* in the first slot, *h* in the third, and ¥100 in the third, and the product *gh* is delivered to you. A black box algorithm will thus run on any finite group in which the basic group operations are computable, and multiplying its complexity by the time taken for a multiplication will give the complexity of the algorithm in the practical situation.

So, for example, if your groups consist of permutations on *n* symbols, you can encode elements as bit strings of length *n* log *n* and apply black box algorithms to learn about your group. But you probably wouldn’t do that. Given a permutation, it is easy to compute its cycle lengths; their least common multiple is the order of the element. Given a set of elements, the Screier–Sims algorithm, and various refinements, efficiently give a canonical form for elements, the group order, and a membership test.

A compromise is provided by various sorts of “grey boxes” which provide a bit more information, for example the orders of group elements.

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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.

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 *ax*^{2}+*bxy*+*cy*^{2} in two variables. So each point of hyperspace has three coordinates (*a,b,c*), where *b*^{2} < *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 …

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 6*g*−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.

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!

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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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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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.

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.

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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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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Suppose that we have a string C-group *G* of rank *d*, generated by involutions ρ_{i}, for *i* in {0,…,*d*−1}. As earlier, let *G _{S}* be the subgroup generated by the involutions ρ

So we can take the faces to be the cosets of the subgroups *H _{i}*, for

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

- ρ
_{0}is the reflection swapping 1 and 2, so as permutation it is (1,2)(3,4)(5,6)(7,8). - ρ
_{1}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); - ρ
_{2}is the reflection in the plane through 12 and the opposite edge 78; so ρ_{2}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 *i*th colour if (*x,y*) is a cycle of the permutation corresponding to ρ_{i}. From the edge-coloured multigraph, we can recover the permutations ρ_{i} (their non-trivial cycles are the edges of the *i*th 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 *S _{n}* is

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 *G _{S}* generated by subsets of the given generators form a lattice isomorphic to the Boolean lattice of rank

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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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.)

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:

- 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. - 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. - 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 *S*_{4}×*C*_{2} 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},…*f _{d}*). Now, for any

For example, in the cube, ρ_{0} reflects the cube in the plane of symmetry bisecting the edge *e*; ρ_{1} reflects the cube in the plane of symmetry through *v* bisecting the face *f*; and ρ_{2} 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 ρ_{i} 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 ρ_{i}and ρ_{j}commute, so their product has order 2. (For example, in the cube, ρ_{0}and ρ_{2}are reflections in perpendicular planes.) - For any subset
*S*of {0,…*d*−1}, let*G*denote the subgroup of_{S}*G*generated by the elements ρ_{i}with*i*∈*S*. Then the intersection of*G*and_{S}*G*is equal to_{T}*G*_{S∪T}. 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 ρ_{0},…ρ_{d−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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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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At roughly the same time, I worked with Cheryl Praeger on designs with flag-transitive but point-imprimitive automorphism groups. These are fairly rare, and always have a beautiful structure involving number-theoretic or group-theoretic coincidences. Symmetric designs are even rarer. But we were led to suspect the existence of a symmetric 2-(1408,336,80) design. (Why one with one-tenth the number of points of the Rudvalis design? I have no idea!)

This never got published. The reason was that we (well, mainly Cheryl) developed a very general construction method, extending an earlier idea by Sharad Sane. The ingredients are three designs (one symmetric, one resolvable, and one group-divisible) with parameters related in a certain way, together with some bijections with appropriate properties. Our construction of our new design (and indeed, some known designs with subgroups of their full automorphism groups which are flag-transitive but block-imprimitive) were purely group-theoretic, and to a casual glance bore no resemblance to our general methods. Indeed, it can be quite hard to say exactly what designs and bijections should be put into the general method in order to produce these designs.

I will describe here the construction of the symmetric 2-(1408,336,80) design, because I have a small apology to make.

The group 3.M_{22} has a 6-dimensional representation over the field GF(4), giving rise to a semi-direct product *G* = 2^{12}:(3.M_{22}). (Matrices generating 3.M_{22} can be obtained from the on-line Atlas of Finite Group Representations, and downloaded into a GAP program.) Restricting to the subgroup 3.M_{21}, the 6-dimensional module has a 3-dimensional submodule, and so we obtain a subgroup *H* = 2^{6}:(3.M_{21}) of *G*. So we can represent *G* as a permutation group of degree 22×64=1408 on the cosets of *H*. (Computationally, constructing this permutation representation is by far the most time-consuming part of the exercise.)

Now *G* is imprimitive, with 22 blocks of size 64; the group permuting the blocks is the 3-transitive M_{22}. The stabiliser of a point has an orbit of length 336, which meets every block except the one containing the stabilised point in 16 points. The 1408 images of this point under *G* are the blocks of the required design.

My apology is for claiming, in various places, that the automorphism group of this design is the group used in the construction. In fact it is twice as large (though still flag-transitive and point-imprimitive). The outer automorphism of M_{22} acts as a field automorphism over GF(4), so is not visible in the linear action on the 6-dimensional module; but it does preserve the design. So the full group has structure 2^{12}:((3.M_{22}):2).

It is no coincidence that I am thinking about this while Cheryl and I are in the same town, as you will not be surprised to learn if you have ever worked with Cheryl!

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Because she arrived so early, her hotel room was not yet ready, and so Rosemary and I suggested that she come to our room to leave her belongings. Cheryl was keen to come when we suggested doing the Coast-to-Coast Walk – indeed I can think of few better ways to cope with a change of time zones.

There are few countries the size of New Zealand that can be walked across in a few hours; indeed, in most of New Zealand this would be out of the question. But Auckland is on a very narrow neck of land, and a 16km walk takes you from Viaduct Harbour on Weitamata Harbour, on the Pacific Ocean, to Onehunga Lagoon on Manukau Harbour, on the Tasman Sea. When I was in Auckland on the Forder lecture tour, I walked across and back before lunch.

The path takes in the summits of two of Auckland’s largest extinct volcanoes, Maungawhau (Mt Eden) and Maungakiekie (One-Tree Hill). It was a day of breathtaking clarity, warm in the sun though the air was cold; we went much slower than on my previous trip, stopping to look at things and potter round interesting sites.

At the end, the new electric train service took us back from Onehunga station to Britomart Travel Centre in under half an hour.

The most remarkable incident of the walk occurred in Cornwall Park, below Maungakiekie. We had had an excellent lunch in the Aspire Café in Manukau Road, but decided to defer coffee to the restaurant in Cornwall Park. But the restaurant seemed to be closed. So we went in to the tourist information office next door, to ask whether there was anywhere else we could get coffee.

There was one person there, Philippa Price, whose job title is Cornwall Park Information Centre Manager, but the only person she was managing was herself. On such a beautiful day, the park was crowded with tourists, and she had to deal with all who came to the information centre: one to register a dog with the Cornwall Park Dogs scheme, others just asking for directions, and so on. So she would have been perfectly entitled to say “No, the restaurant is closed for refurbishment, I’m afraid”.

What she actually said was “I’ll make you some”. Between other jobs, she brewed up a pot, sat us down at a table in one of the many rooms in the Information Centre (illustrated with stunning photographs of the park), and stopped to chat when other business allowed while we drank it. And at the end, she wouldn’t charge us anything for it!

I had on my Prague MCW T-shirt, and the word “Combinatorcs” seemed to ring a bell; she was sure she had seen me before. We established that it was probably when I talked about infinity on the BBC Horizon programme. Talking about other media appearances, I mentioned that I had been on the Kim Hill show when I was here on the Forder tour. She is a great fan of Kim Hill, and indeed of the radio in general, her window on the world, and before we left she had found the podcast of the interview.

It proved, if nothing else, that I don’t handle fame well. My two companions are probably more famous than I am, and I felt a little embarrassed about being in the limelight.

A final note on geology. There are about fifty extinct volcanoes in the Auckland volcanic field, a World Heritage Site for its combination of natural and cultural features. (Different authorities quote slightly different numbers.) It is near-certain that there will be another eruption one day; the Auckland City Council website estimates that an Auckland resident has about an 8% chance of experiencing one in his or her lifetime. Almost the only other things that experts agree on is that the next eruption will not be one of the existing volcanoes; it is completely unpredictable where and when it will be, and it is likely to cause very severe disruption and loss of life.

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