Anne Street

Anne Penfold Street died this week.

Anne arrived at the University of Queensland as a lecturer while I was a student there. She taught me measure theory; a feature of her well-organised lectures was that all results were numbered in sequence, so that we reached Theorem 100 before the end of the course.

But her main field was combinatorics, where she wrote several books: an introduction, a book on combinatorics of experimental design with her daughter Debbie, and research monographs.

Anne was very active in the combinatorics community, as president of the Combinatorial Mathematics Society of Australia (a sister body of the BCC) and recipient of their medal, and as president of the ICA.

She was the third female professor of mathematics appointed in Australia, and the Australian Mathematical Society has an award named after her.

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Prospects in Mathematics, 2

I didn’t say anything about my own talk at the Prospects in Mathematics meeting. But soon afterwards I was re-reading Lee Smolin’s 2006 book The Trouble with Physics, and it resonated with some of the things I said.

My topic was the Classification of Finite Simple Groups, and its impact on research in many parts of mathematics and computer science.

I began with one of my favourite examples. In 1872, Jordan showed that a transitive permutation group on a finite set of size n > 1 contains a derangement (an element with no fixed points). The proof is a simple counting argument which I have discussed here. In 1981, Fein, Kantor and Schacher added the innocent-looking postscript that the derangement can be chosen to have prime power order; but the proof of this requires the Classification of Finite Simple Groups.

In outline, easy reductions show that we may assume that the group is simple and the stabiliser of a point is a maximal subgroup; so we have to show that, if G is a simple group and H a maximal subgroup of G, then there is a conjugacy class of elements of prime power order disjoint from H. This has to be done by taking the simple groups one family at a time (or one group at a time, for the sporadic groups).

The subtext of my talk was: the proof of CFSG is very long, and is the work of many hands; it is certain that the published proof contains mistakes. (The proportion of mathematics papers which are error-free is surprisingly small, especially when possible errors in cited works are taken into account.) So can we trust it, and if not, should we be using it? Mathematicians have always had as a guiding principle that we should take nobody’s word for anything, but check it ourselves; CFSG makes that principle almost impossible to follow.

I told the students that, when they arrived at university from school, they were probably told that school mathematics is not “real” mathematics, and that now they would see the real thing, with an emphasis on proofs and building on secure foundations. Now that they are about to embark on a PhD, they have to be told something similar. In a university mathematics course, they are given the statement of a famous theorem, say Cauchy’s, with an elegant proof polished by generations of mathematicians. Now they are entering territory where proofs don’t exist; they will have to build proofs themselves, and sometimes they might make mistakes.

Smolin, in his book, has some hard words to say about the sociology of string theory (a subject in which he himself has worked). He was asked to write a survey paper about quantum gravity, and wanted to include the result that string theory is a “finite theory”. If this seems a little odd, string theory (like quantum electrodynamics) is a perturbative theory, where the answer to a calculation has to be found by summing infinitely many terms. (I remember my feeling of shock when I learned this from Mike Green quite a long time ago.) In the case of QED, it is well established that the sum converges, and according to Smolin, string theorists accept that the same is true for string theory. But when he went looking for a proof of this, he found that everyone referred to a paper of Stanley Mandelstam which showed only that the first approximation was finite. It seems that the assertion that finiteness had been proved was never checked by the people who quoted it; according to Smolin, the ethos of the field was to believe that such a statement must be true.

I do believe that finite group theory avoided this horror. First, it was never “the only game in town”, monopolising grants and postdoc appointments the way that string theory did. Second, everyone knew that the proof was not complete; many people hoped it could be completed (as it eventually was – the delay was partly caused by the fact that it was a big job, and experts were reluctant to commit themselves to it), but anyone who used it noted that it was being used. There were many such papers based on the assumption of CFSG, and had it proved to be false, revising them all would have been a huge job; but at least we had a good idea where to look.

However, it is true that the proof of CFSG is so long that it is unreasonable to expect a mathematician who uses it to have read and checked the proof. This is especially true for the many uses of the theorem outside group theory (in the theory of other algebraic structures such as semigroups and loops, in number theory, in computational complexity, and so on). This is probably the biggest change in a subject which has always taken the statement “Take nobody’s word for it” as a guiding principle. (In the paper of Fein, Kantor and Schacher, the assertion about derangements of prime power order is really a lemma in the proof of the theorem that the relative Brauer group of a finite extension of a global field is infinite – whatever that means!)

What about the mistakes, which undoubtedly occur in the published proof?

We have to hope that mathematicians will continue to recognise the importance of CFSG, and will continue to apply it, and even (in the case of an honorable few) to revise and improve it. How do errors in published mathematics come to light? By several methods. People read the papers and notice a problem; or they apply the result and are led to a contradiction; or they discover something which conflicts with the statement of the theorem. We are more likely to do this if the theorem in question remains an active part of our mathematical practice than if we put it on a dusty shelf somewhere and ignore this.

I believe that CFSG is too important to be put on a dusty shelf. So I am hopeful that it will stand up to the test.

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Prospects in Mathematics

York

I’ve just spent two days in York, at the annual Prospects in Mathematics meeting.

This LMS-supported meeting is aimed at undergraduates or masters students who are thinking about doing a PhD, and tries to give them information about what research in mathematics is like, what different branches of mathematics entail, where you might go to work on a topic you are interested in, and so forth.

Apart from the mathematics talks, there were presentations about Centres for Doctoral Training, about what life as a PhD student is like (by three local PhD students), and a question and answer session; a reception, and two very good lunches and a dinner (York’s attempt to persuade people to apply there for their PhD?)

As always at these events, I got a very strong impression about the great enthusiasm of the students, as well as their clear-eyed view about what they are letting themselves in for. When I am talking to prospective PhD students, I often find it necessary to warn them that a PhD is not necessarily a passport to a well-paid job. I didn’t mention this once, since the students seemed aware of that; they were there because they loved mathematics and wanted to do more.

A little about some of the more traditional talks.

Tim Spiller talked about quantum information theory. Three important aspects of quantum mechanics are fairly directly connected to potential applications: superposition is what would allow a quantum computer (were one ever built) to solve certain kinds of problem exponentially faster than a classical computer; uncertainty allows detection of unauthorised interception of communications and allows quantum cryptography (in the form of key distribution); and entanglement allows the use of quantum techniques in sensing and imaging. The UK is currently putting substantial resources into this area, and they are hoping to have practical devices available quite soon (maybe not quantum computers, however).

Vicky Henderson talked about the meeting point of economics and psychology. Psychologists know that we are not rational agents, although economists still mostly assume that we are. She talked about prospect theory, proposed by Tversky and Kahneman some time ago, which modifies the utility and probability functions of classical economics which incorporates the fact that we tend to be risk averse on gambles unless the chance of success is very small, in which case we overestimate this chance. The modifications seemed a bit unmotivated to me, and I saw no account of the fact that for some people gambling seems to have positive utility.

Sarah Waters talked about fluid mechanics applied to tissue engineering: a nice talk, but not my thing, I’m afraid.

Katrin Leschke started with soap films, which form minimal surfaces (minimal area for given boundary). She explained that these are harmonic, and so are real parts of holomorphic null curves, with an integral representation in terms of Weierstrass data. She also brought good news about the (threatened) mathematics department at Leicester.

Martin Hairer told us that probabilists are good at deriving results for given probability distributions, but that choosing these distributions is more problematic. The guiding principles are symmetry (e.g. the six outcomes from a well-made cubical die should be equally likely) and universality (the distribution shouldn’t depend on details of the random events causing it). The classical examples of universality are the central limit theorem and Brownian motion; we learned some interesting history of the latter (for example, it was discovered by Ingenhousz half a century before Brown; and Bose, the manufacturer of noise-cancelling headphones, was founded by a student of Norbert Wiener, who gave the mathematical description of Brownian motion as a random function from the Wiener measure). His main interest was a recently discovered universality class described by the KPZ equation, where the most general universality result has not been proved; but he showed us some beautiful simulations, e.g. of dropping Tetris blocks randomly.

Ruth Gregory studies higher-dimensional black holes. While the event horizon of a black hole in ordinary space-time is typically spherical, adding a dimension allows a variety of shapes: spheres, cylinders, tori, and so on. So questions about stability arise. The principles that the entropy of a black hole is proportional to its area, and that entropy cannot decrease, show that ordinary black holes cannot split up into smaller ones; but in five dimensions they can. (So “cosmic censorship” fails in five dimensions.) The cylindrical black holes are unstable, and tend to wobble; it is thought, though not proved yet, that they can break up into spheres with a fractal pattern along the axis of the cylinder. This is the same phenomenon as the flow of water from a tap breaking up into drops as the flow rate changes. I learned from Ruth’s talk that black holes have something in common with Black–Scholes: there are theorems, but these are extrapolated beyond the region where their assumptions hold, by non-mathematicians (physicists or bankers).

Julie Wilson talked about a career which has taken her from a PhD in number theory through crystallography, pattern recognition, and machine learning to metabolomics, food fraud, and archaeology. I learned a new word; “undeamidated”.

Victor Beresnevich talked about Diophantine approximation and metric number theory. Diophantine approximation quantifies and analyses the fact that the rational numbers are dense in the real numebrs, and extends the results to higher dimensions and to manifolds. Metric number theory is mis-named since it is concerned with measures rather than metrics: how big is the set of badly approximable numbers? (Hausdorff dimension 1, but Lebesgue measure 0.) He gave us Khintchine’s theorem, some extensions, and some related open questions.

Finally, my colleague Mark Chaplain told us about the different techniques required in modelling cancer on different scales (within a cell, between cells, or at tissue scale), and recent attempts to construct models which span several scales, aiming eventually at a “virtual tumour”.

The meeting inevitably made me wonder what I would do were I starting a PhD today. I hope I would be brave enough not to go to a CDT; I am not a herd animal. I think what I actually did was what was best for me.

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Continuous functions

What does it mean for a function to be continuous?

Received wisdom is that this was sorted out in the nineteenth century by Cauchy and Weierstrass; even if the epsilon-delta definition is a bit of a mouthful for students, once they have mastered it there is no further problem.

But an interesting article by J. F. Harper in the most recent BSHM Bulletin (doi 10.1080/17498430.2015.1116053) looks at definitions used by Bolzano, Cauchy, Duhamel, Weierstrass, Heine, Jordan, Whittaker, Goursat, Pierpont, Hobson, Hardy, Hausdorff, Courant, Siddons, Rudin, Bartle and Sherbert, Carlson, Gowers, and Kudryatsev, and finds numerous inconsistencies, some of them between different works by the same author.

The procedure is to compare these authors’ definitions on four particular real functions:

  • the function which is 0 at the origin and undefined elsewhere;
  • the function which is 0 on the positive reals and undefined elsewhere;
  • the function which is 0 on the rationals and undefined elsewhere;
  • the function which is 0 at the origin and sin(1/x) at the non-zero real number x.

But hang on; the first three are not functions on the real numbers!

So there is a more basic confusion, about what is meant by a function.

I recommend to you Littlewood’s article “From Fermat’s Last Theorem to the abolition of capital punishment”, which appears in his Miscellany (a book you should certainly read!). Littlewood quotes, as an “intellectual treat”, the muddled ramblings of Forsyth, in his Theory of Functions of a Complex Variable, stretching over two pages. Littlewood goes on to say,

Nowadays, of course, a function y = y(x) means there is a class of “arguments” x, and to each x there is one and only one “value” y. After some trivial explanations (or none?) we can be balder still, and say that a function is a class C of pairs (x,y) (order within the bracket counting), C being subject (only) to the condition that the x‘s of different points are different.

Actually, even these two definitions are not equivalent; the second allows all four functions above, while the first forbids the first three.

We normally define a function f : X → Y. The correct logical definition is as a set of ordered pairs, but we should replace Littlewood’s second condition by the condition that each element of X occurs once and only once as the first component of an ordered pair in the set. In other words, the domain is part of the definition of a function.

I tend to explain this to students by saying that a function is a black box: you put in an element x, and out comes an element y. (Unlike earlier mathematicians, we neither know nor care how the black box actually operates; it is defined solely by the inputs and corresponding outputs). But, like all good black boxes, it comes which a guarantee, which states that if an element of X is put in, then out will come an element of Y. If you put in an element which does not belong to X, the behaviour is not guaranteed; anything might happen (but to agree with the logical definition, it is perhaps better to say that nothing comes out).

This definition has the disadvantage that the codomain is also part of the definition of the function. So the squaring function from the natural numbers to the real numbers is a different function from the squaring function from the natural numbers to the natural numbers.

I am not too troubled by this. After all, in the standard constructions of the number systems, the natural number 1, the integer 1, the rational number 1, the real number 1, and the complex number 1 are completely different objects; yet we use them as if they are all the same thing, without getting into trouble.

If the domain is part of the definition of a function, then Harper’s first three functions are not functions from the real numbers to the real numbers. I would be much more interested to have genuine examples of real functions which discriminated among proposed definitions of continuity.

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All kinds of mathematics …

Please reserve the dates 24-27 July 2017 in your diary!

Next year, I will turn 70. Some good friends (notably João Araújo) are arranging a conference in Lisbon to mark the occasion, and many other good friends have agreed to come to speak.

The list of speakers, not yet complete, already includes R. A. Bailey, W. Bentz, R. Calderbank, P. J. Cameron, G. Cherlin, P. Diaconis, C. Freer, M. Giudici, R. Gray, A. Hulpke, M. Kinyon, D. Leemans, H. Leitão, D. Macpherson, A. Malheiro, F. Matucci, J. Meakin, J. D. Mitchell, P. M. Neumann, J. Nešetřil, P. P. Pálfy, C. E. Praeger, C. M. Roney-Dougal, B. Steinberg, P. Silva, L. H. Soicher, P. Spiga, E. Szemerédi, and M. Volkov. All are my colleagues, co-authors, teachers, students, fellow-students, or people I have talked about on this blog. What a line-up!

It looks to me that this will be the event of the year 2017. Add to this the fact that Lisbon is one of the world’s most beautiful cities, and you really can’t afford to miss it!

The webpage is here. Bookmark it, and be sure to come. I look forward to seeing you.

I don’t know how to begin to thank the organisers and the speakers at this event, so I won’t even try now. But they can be assured that I feel deeply touched and grateful.

The full title of the conference is “All kinds of mathematics remind me of you”, subtitled “Gathering excellence where Cameron excels: Combinatorics, Groups, Model Theory, Number Theory, Semigroups, Statistics, and more …”.

If you are close to my age, the title of the conference might remind you of something. In 1970, at around the time I was born (in the sense that Paul Erdős used the word), a lass from Derry called Dana Rosemary Scallon won the Eurovision Song Contest with a song called “All kinds of everything remind me of you”; she was the first of a considerable number of Irish winners of this competition.

As you can see from the list of speakers, the variety of mathematics in Lisbon in July will not be inferior to the variety of things that Dana found evocative of the subject of her song.

You have put the dates in your diary, haven’t you? and bookmarked the web page?

For friends on other continents, there is a good selection of conferences in Europe in summer 2017 that might tempt you to spend more than a week here (and, despite what you may have heard, Britain is still part of Europe):

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Two excursions

Last Friday, my friend Jan Kratochvil spoke to the Edinburgh Mathematical Society at the University of Stirling. I decided to go and hear him.

The situation he talked about was typefied by this example. Suppose that you have a large network which you know can be drawn in the plane. (Planarity testing for graphs can be done in polynomial time.) But, for historical reasons, part of the network is already built in the plane; you are not allowed to knock it down and start again. This makes the task harder; is it possible to extend this embedding to the whole network? A related problem: if you have several graphs, sharing a common part, can you embed all the graphs so that the embeddings coincide on the common part?

As well as planar graphs, this question can be asked for various geometrically defined graphs such as interval graphs. Honza and his collaborators have uncovered a wealth of detail about these situations.

One “classical” result I didn’t know is the following. The question involves representing a graph as the intersection graph of a set of continuous functions on a closed real interval. (Think of the graphs – in the other sense – of the functions; now “intersect” has the expected meaning.) An old result of Golumbic characterises such graphs. If two functions do not intersect, then by the Intermediate Value Theorem, one lies entirely above or entirely below the other. Thus if a graph can be represented in this “fun” way, its complement is the comparability graph of a partial order. Golumbic showed that the converse is also true.

But the general questions of extendability and embedding several graphs can be asked for the “fun” class, and some very entertaining things arise.

I decided to go from St Andrews to Stirling by bus; although the bus service is not very frequent and takes two hours, it is much more convenient, since it stops very close to the University of Stirling. The outward journey was in daylight, and an interesting trip. After leaving Fife at the hamlet of Burnside (where road signs said “Burnside: Please Drive Carefully”), and passing through the curiously-named Drum, Crook of Devon, Rumbling Bridge (it did rumble, but only because the road surface was so rough), and Yetts o’Muckhart, the rest of the journey followed a nearly straight road under the shadow of the Ochil Hills (or would have been if the sun had been on the other side). These hills are a lava extrusion, but show a very dramatic scarp on the south side as a result of a fault line. Here is the last of the Ochil Hills, Dumyat, above the University of Stirling.

Dumyat

On Saturday, we decided to take a trip on Scotland’s newest railway, the Edinburgh to Tweedbank line. Quite the wrong time to make such a journey: the hours of daylight are short, and the trains into Edinburgh in the morning and out in the evening are crammed with shoppers (so much so that announcements at the stations warn of overcrowding on the trains).

The trip involved a slow and dreary journey through the Edinburgh sprawl, until the line turned south and entered the Moorfoot Hills, where it followed the Gala Water (lovely name, pretty stream) down to Galashiels, where it enters the Tweed. The disused railway line east of Galashiels was a footpath carrying the Southern Upland Way; walkers have been provided for, but have been sidelined by the newly restored tracks.

Nevertheless, we had time to walk along the banks of the Tweed (the station is not on the river, despite its name), climb to the pass in the Eildon Hills, and have a pub lunch, before returning. I was unable to get a decent photo of the three peaks which led the Romans to call the place Trimontium: they are best viewed from the other side. But here is a spectacular sunset that entertained us as we waited for the train home.

Tweedbank station

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h-index

To return to an old theme:

Today my h-index has reached 50, at least in the opinion of Google Scholar. According to this site, I have 10259 citations.

In case you think this is unreasonably high, you might turn to a more reliable source like MathSciNet. This lists 2766 citations and a h-index of 24.

No prizes for guessing which I would use if I were applying for promotion! How fortunate that all that is (hopefully) behind me.

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