Toral Communities

In dynamical systems, a wild torus is an invariant torus whose cross-section is a nowhere differentiable closed curve. The picture above, taken earlier this summer, is rather different: it is a torus in the wild. I noticed some last year, but wasn't able to get a good photo. They returned this year and the light was better, but even so I have had to enhance the contrast to make the torus clearer.

Tori in nature are usually either somewhat contrived (humans, earthworms, ...) or the result of a complicated extrusion process (smoke rings). This, I believe, is a community of simple cellular organisms that has formed a natural torus. For a while I thought it must be an egg-containing jelly (extrusions again) but a bit of research on the web makes me think it is much more likely to be a jelly-like community that can create columns, and the ends of the cylinder have joined to form a torus.

Why would this be beneficial? Perhaps it is a mechanism to limit further growth or recruitment. The active area of a cylinder is likely to be the ends, so closing in on itself like this could be a way of preventing further organisms from joining. A gated toral community! If the column becomes too large it might be more easily torn or snagged on other objects, so some limitation on size might be necessary.

Fairy rings (holes in the growth of a clump of plants) may also be toral in nature, extending below the surface. But here I think the mechanism is different -- either nutrients in the central region have already been used up, or dying plants leave behind a chemical residue that prevents new growth in that region. Of course the reason the latter mechanism is beneficial might be due to the former suggestion.
 
On the subject of natural tori, I seem to remember Bilbo and Gandalf engaging in a smoke ring competition in Lord of the Rings (The Fellowship of the Ring). Ah yes, here's a CGI version!

Tribute to a quiet wonderwoman

Anne Bennett died on 6 September 2012. She was not old and not young (she left two teenage children -- she was my sort of age), and her name probably doesn't mean much to many mathematicians. It should, and it does to those who worked with her.

Anne was part of the secretariat of the London Mathematical Society; she was also seconded to the Council of Mathematical Sciences, which is how I knew her. During the past year the profile of mathematics within Parliament has increased enormously. We have had several articles on mathematics in the magazine of the Parliamentary and Science Committee (PSC), the first seminar on mathematics in the Houses of Parliament that anyone can remember, and mathematicians were invited to the 60-year celebration of science in the UK (jubilee related) seminar and dinner of the PSC.

This unprecedented coverage of mathematics was orchestrated, quietly and effectively, by Anne. It was Anne who knew who to talk to, Anne who knew when to press home an advantage (and when not!) and Anne who knew what was possible.

After her death, two emails (alarmingly alive) came to mind. The first arrived just after the first parliamentary seminar on maths, when I had tried to follow up the success of the occasion by encouraging my co-chair,  Andrew Miller (MP and Chair of the PSC), to  move forward on some of the ideas we had discussed about  the engagement of the mathematics community with Parliament. Anne wrote (and I should say that she also sent me a word of congratulations on my performance, but later!)

Sorry Paul but this is what I did not want you to do right now.
The inside information I have received is NOT to do this.  I need to talk further with my internal contacts first.
Anne

The second, on the day before she died, was a simple word of thanks:

Thank you to Paul and Ken for your considerable efforts in producing the copy for the Science in Parliament magazine and to everyone for their comments.
Best wishes, Anne


These two emails sum up her contribution perfectly: she understood what we were trying to do and was not afraid to tell us what was needed if she felt that we could do with a bit of direction; but she also recognised when people had put effort into projects she valued, and had the grace and confidence to tell people when she thought they they had done something well.

Isn't that what leadership is about?

You can find a more complete obituary on the LMS web pages.

EBacc to the future

Michael Gove, the Secretary of State for Education, has just announced plans for a radical change in school exams for those aged 16. On the face of it this tougher, more rigorous system looks good for mathematics. I am concerned that in fact it will reduce mathematical activity in the UK.

I do think that the examinations have become easier over the years; this is an inevitable consequence of the incentives on student success for both teachers and the examination boards.

I do think that there is a consequent danger that the brightest students will become bored; though there is nothing to stop teachers from introducing more demanding and varied material (why is non-examinable a dirty word in education?).

However, this does not mean that I support the proposals outlines by Gove and Clegg a couple of days ago. Some of my reasons are fairly standard:

1.   Weaker students  (and students with less academically supportive family backgrounds) are likely to struggle  to achieve anything – particularly in all or nothing subjects like mathematics
2.  Girls are likely to fare worse in the more traditional exam-only culture.
3. Clegg’s intervention to ensure that there is only one examination may backfire: many people will leave school with only a transcript of attainment showing very little achievement.

An inevitable consequence of (3) is that a second exam system will eventually be brought in (sound familiar?). The system is to be brought in in two stages – English, Maths and Science starting in 2015 (examined in 2017) and the rest the following year. This creates a strange hybrid generation in the middle!

But there are other knock on effects of the current plans.

     4.  Wales. My understanding is that Wales is not forced to follow England’s lead here. Will it go it  alone, keeping GCSEs going, or will it join Scotland’s system? Or move with England? 
     5.  A levels: a harder exam system at aged 16 is likely to discourage students from continuing to A level, leading to a less well-educated work force, precisely what this initiative is trying to address!
     6. In particular, the numbers taking maths, which is seen as hard, at A level is likely to fall and hence… 
     7. …. the numbers taking maths at University will also decrease.

Teachers and Universities have been working hard to overcome the sudden drop in numbers taking mathematics at A level following the 2000 A level exam fiasco. Those in favour of this policy argue that those coming to university will be better prepared. I won’t hold my breath, and anyway, we need good mathematicians getting good degrees to be the next teachers, industrial innovators and financial marketeers! Maths education is not just about the elite next generation of academics.

So my concern is that whilst the best young mathematicians will thrive in the new environment, most of them would  have succeeded anyway, and there is a danger that this policy will actually reduce the supply of good mathematicians!    

Be careful what you wish for.

Wither Hefce (Hefce withers?)?

The International Review of Mathematics (see earlier blogs) praised the breadth and depth of UK research, commenting on how research excellence is distributed across the country. This excellence is possible because of the current dual funding system: researchers can apply for support for research assistants on specific projects through the Research Councils (RCUK). There is not enough money to support all mathematics research through this mechanism, so the majority of research is supported through Hefce's research contribution to the universities, which makes it possible for people to go to conferences and keep in touch with the international community.

Hefce, in case you didn't know, is the Higher Educational Funding Council for England (other UK countries have their own versions). As the name suggests they are a funding body, and have been awarding money to universities for teaching (the Hefce T) based on agreed numbers of students, and to support basic research activity (the Hefce R contribution) based on the RAE and now REF assessment exercises. Here's a quote from their webpage:

The Higher Education Funding Council for England promotes and funds high quality, cost-effective teaching and research, meeting the diverse needs of students, the economy and society.

However, the change in the student fee system means that effectively all the Hefce T support will be withdrawn over the next few years. This is, of course, the majority of the Hefce funds, and Hefce is by definition a funding body. So will Hefce continue to act as a funding body, disbursing the Hefce R? If it does we will be in the curious state of having two research funding mechanisms, and this is bound to create tensions as to the different roles, and attract attention as a target for efficiency.

My crystal ball suggests that Hefce's days are numbered. Universities will be expected to use student fees to support the basic (non-RCUK funded) research of their staff.

But will they? Can they? And if they do, will it be with strings (preference to those who already have RCUK grants or matching funds from other sources). If the system evolves in this direction we may well see the UK maths research output drastically cut.

Two codas.
1. It could be argued that the RCUK supports the best UK research and this is all we need. Wrong on two counts! Fields medallist and blogger extraordinaire Tim Gowers has never held an RCUK grant. Also, the EPSRC and other research councils are becoming every more involved in choosing research directions, so different sub-disciplines are not treated equally. It is the dual funding mechanism that ensures that as fashions change we have the expertise to adapt and lead.
2. Research informs our teaching in many ways. Losing that research base will impact on the quality of teaching in universities. 

The Oak's Shilling

My new book, Maths in Minutes, is formally published on 1 March. It's a great format due to Quercus: 200 short articles on areas of mathematics, intended to introduce the non-mathematician to the subject. Even with 200 sets of 200 words it is impossible to cover the whole of mathematics, so I have concentrated on core mathematics rather than applications.

It looks good, and some of the entries are (I hope) very good, though there are inevitably some weaker areas! I am particularly proud of the introduction of the ideas of proof, and how Holmes' deductive method fits into this (the role of the contrapositive). On the other hand, the short topic-led format makes it harder to bring out the deep inter-connectedness of mathematics. You win some, you lose some.

You can buy it at all good bookstores, but I get no royalties (boo hoo).

Versatile Virtuosity

The special issue of Dynamical Systems, and International Journal in memory of Jaroslav Stark is published next month. It is always sad to think that Jaroslav has gone, but writing the editorial reminded me that Jaroslav was an amazingly versatile mathematician. In his early career his interests spanned from pure mathematics to engineering via rigorous computer proofs and he went on to  become an important voice in Systems Biology. He also published on statistics and data interpretation.

This versatility reminds me of another of my heroes. The other day I came across Boris Vian on youtube. Vian was an engineer (he is credited with the design of the rubber wheels in the Paris Metro, which ensure a smooth and relatively quiet journey) but he is better known as a writer and one of the people responsible for bringing jazz to France (he died in 1959). His books were required reading for arty  French adolescents when I was young (I'm less keen on the best known J'irai cracher sur vos tombes, but L'herbe rouge, L'Ecume des jours and L'Arrache-coeur are amazing). I wonder whether he is still read.


I had not realized that Vian wrote music too. His anti-war song, Le Deserteur is wonderful; a very simple but powerful song written as a letter to the President on having received his military papers to leave for the war `by Wednesday evening'.  Another example of versatile vituosity.

I suppose both Stark and Vian show that extraordinary people are often extraordinary in a variety of ways. The rest of us just get on with what we do.

Welcome Back

Such a long absence requires a short explanation (not sure of the blog etiquette here). A lump in my leg grew over the summer and after extensive screening I had a small operation in October to remove it.

Fortunately it was not what they thought it was (I was diagnosed with a neuroma, a tumour of the nerve sheath but a nodular fascia, which is non-malignant and boring) so normal service will be resumed shortly....