Another classic

I was clearing out some papers in my office today and came across this classic article on proof techniques called “The Uses of Fallacy” which always gives me a laugh whenever I read it. It is especially relaxing reading after a few hours of grading student assignments which inevitably make heavy use of “Proof by Misdirection”, and should really be compulsory reading for beginning students of logic.

There are innumerable variants of this that have appeared in many places, but this is possibly the original?

(This PDF was sent to me by Marston Conder a few years ago.)

Halmos’s “How to Write Mathematics”

As I’m teaching a course called “Scientific Communication” to a few Honours students this semester, I’ve been reading or re-reading many of the standard articles on “How to Write Mathematics” and doing quite a bit of online searching on the topic.

After reading many of these articles, a bunch of blog posts and several sets of slides, the main conclusion that I’ve reached is that Halmos’s classic article “How to Write Mathematics”, written nearly 40 years ago, is an absolute gem. It’s still totally relevant to today’s mathematicians and moreover, it is beautifully written itself. I actually went down to our library and got out the book in which it appears, but there seem to be plenty of copies on the web.

The basic rule is simple: focus on trying to convey your ideas to the reader as clearly and simply as possible. The aim is not to impress the reader with your vocabulary or erudition or by labelling difficult logical jumps or complex calculations as “obvious”. Instead the object is to use simple language, clear and consistent notation, and plenty of examples and figures to convey the ideas to your intended audience.

Of course, looking at the still-too-low average standard of mathematical papers, this is obviously more easily said than done!

The ARC, the ERA and the EJC

The ERA, or Excellence in Research in Australia, is a government attempt to measure the quality of research being done in Australia, for as-yet-unspecified reasons. It is being run by the Australian Research Council (ARC).

Rather than go to the difficulty and expense of getting people to actually read papers (as they did in Britain’s RAE), they decided to try to rank every journal – not just in mathematics – and then judge each paper by the journal it appears in, rather than the paper itself (though see the footnote below).  It’s a bit like judging a person by the school they went to, rather than who they are, but it’s probably a little better than the previous method of simply counting all papers. Maybe it’s not even that different to how we rate mathematicians in fields distant to our own? In any case, like it or hate it, ignoring the ERA is not an option for someone in my position.

The ranking itself is fairly coarse-grained and was intended to put all the journals in one discipline (such as mathematics) into A* (top 5%), A (next 15%), B (next 30%) and C (the rest). Of course this creates immense problems for multi-disciplinary journals (and subsequent problems for inter-disciplinary research) and even within a discipline there can be endless arguments about the boundary cases.  For mathematics, the job of ranking the journals was partly undertaken by the Australian Mathematical Society and their rankings seemed pretty much as I would have expected, at least for the few journals I know about (that is, few relative to the hundreds ranked).

But then something went screwy with The Electronic Journal of Combinatorics. This is a well-respected online journal in combinatorics that has been going for 16 years and boasts a stellar editorial board including such people as László Lovász, the current President of the International Mathematical Union. When the AustMS did the rankings, it was given an A rating, so when I got my list of papers to check, I was surprised to see that both of my EJC papers were ranked in the bottom C category!

Collaboration Tools: Google Wave

Another potential collaboration tool that we’ve been playing with is the much-hyped Google Wave. Although it is very early days, and those who believe the more extreme hype will inevitably be disappointed, I think that it’s going to be a really useful tool.

Google Wave is a browser-based “collaboration platform” which is basically a combination (“mash up”?) of email, wiki, instant messaging and social networking. But, because of the way I’m using it, I really just think of it as “souped-up email” and I like it because it makes it natural and easy to combine and organize messages and resources at a “project” level.

I work on a number of different projects at the same time, with various differing combinations of collaborators, both here at UWA and remotely across the world. Most of our communication is by email and for any one project, the exchange of ideas, questions, references, papers, computer programs, computer output and so on will be scattered over hundreds of messages, usually with dozens of different Subject: lines. If I am looking for the answer to a question that I recall being answered earlier by, say, Michael then while I can easily locate all of his messages there will be lots of messages about other projects, seminar announcements, coffee times etc. that I have to wade through or filter out. If there are three or more collaborators, then each person has a possibly-different subset of the messages organized or otherwise in their own way. Using a Wave for each project eliminates or at least ameliorates many of these problems.

Collaboration Tools: Instiki

A frequently-asked (but infrequently resolved) question is whether there is anything more effective than plain old e-mail for mathematicians to use for collaborative work – anything from simple systems for keeping ones ideas organized to fully-blown version control systems for papers and books. Browsing the relevant question on mathoverflow.net shows a wide variety of more-or-less effective systems in use. One of the major issues for mathematical collaboration is how to get around the problem of mathematical typesetting on the web.

During Chris Godsil’s recent visit to Perth, we investigated a number of possible systems, and it seems that while there is no single tool that does everything, there are some neat tools that in combination can be very handy. One of these tools is the very neat little Wiki engine called Instiki and I’ll let Chris take it from here in his role as guest blogger.

Why Use Instiki? (Guest Blogger – Chris Godsil)

Gordon and I wanted something online that we could use as a scratchpad/scrapbook for joint projects.  Almost any wiki type software would do for this but, for us, Instiki has two considerable advantages.  First, it is easy to set up.  Second, it allows us to use TeX easily and effectively.

For a real example of what can be done with Instiki, see the nLab site

One disdvantage is that the TeX will only render when viewed by a MathML aware browser.  Basically this means Firefox, not Safari.  There is a plugin for Internet Explorer, I’m told. Here’s a screenshot showing some simple TeX markup as rendered by Firefox.

Scientist of the Year

I’ve just got back from the Western Australian Science awards luncheon, and am delighted to report that Cheryl Praeger won the award, and is now officially Western Australian Scientist of the Year for 2009.

It’s quite rare for mathematics to be acknowledged in this way in direct competition with much “sexier” fields such as nanobiotechnology, and in her speech Cheryl emphasized the integral role that mathematics plays in all the sciences, and pointed out that Australia needs to lift its performance in this area given that the percentage of graduates majoring in Maths iss less than half the OECD average.

Congratulations, Cheryl!

Projective Planes I : PG(2,q)

This is an elementary description of the finite desarguesian projective plane and its automorphism group . I needed to explain this to a non-combinatorialist, so thought I would just add it to the blog. The required background is just elementary linear algebra, elementary group theory and the concepts of a finite field and an incidence structure.

We start with the finite field where is necessarily some power of a prime . An automorphism of a field is a permutation of the field elements such that

and the collection of all automorphisms forms a group. The automorphism group of is the cyclic group of order generated by the automorphism .

Next we construct the three-dimensional vector space with vectors being triples of elements of which we shall view as row-vectors. If is an invertible matrix with entries in , then the map is a permutation of (fixing ). The collection of all such invertible matrices forms a group called the general linear group and denoted . We can build a matrix in by picking an arbitrary non-zero vector for the first row, then choosing any vector that is not a multiple of for the second row and then any vector not in the span of for the third row. Therefore the order of the general linear group is given by

Laptop etiquette?

Back when I first started attending maths conferences, laptops barely existed. In those days, if you got bored or lost during a seminar or conference presentation, the normal thing to do was to drift off, daydream or discreetly work on your own problems on the conference-supplied pad of paper. It seemed important – perhaps out of courtesy to the speaker – not to make it obvious that you had tuned out, and so working on your own overhead transparencies or actually falling asleep was frowned upon, though neither was completely unknown.

Nowadays of course, almost everyone brings their laptop to a conference and so now the temptations of email, web-browsing or finishing up corrections to a paper are ever-present. However it is impossible to use a laptop discreetly and so it is immediately obvious, in particular to the speaker, that you’ve probably stopped paying attention to the talk. In some talks that I’ve been to, half the audience is working on their laptop after the first ten minutes.

What is the appropriate etiquette relating to laptop use during seminars? Is it disrespectful and off-putting to the speaker, or just a sensible use of valuable time? Of course the issue is clouded by the fact that there are perfectly legitimate reasons to use a laptop during a seminar – during Terry Tao’s plenary at the Aust MS conference, the guy in front of me was using his laptop to look up definitions or follow up on details on various things that cropped up during the talk. And if Michael starts “live-blogging” the Phan systems seminar, then he’ll need to be using a laptop for that.

Overall, I’m just not sure what to think.

On the other hand, some people obviously know exactly what they think. At last years ACCMCC meeting in Auckland, Doron Zeilberger gave one of the plenary talks. Doron, who is well known for having many strongly held opinions, started his talk by announcing that laptop use during his seminar was forbidden, and ordered everyone there to close their laptops or leave. My former PhD supervisor Brendan McKay and a certain prominent NZ mathematician who I shall name only as M were well known at the conference for seemingly being permanently immersed in their laptops. Luckily Brendan was there at the start of the talk to hear the warning, but for some reason M was about ten minutes late in arriving. We watched him with bated breath, and sure enough, after a few minutes the urge overcame him and out came his laptop, at which point a fired-up Doron strode up the steps to confront him, and shut it down.  As M plaintively remarked later “I was only doing what he told us to do and looking at his website”.

Just as I finished writing this, I realized that one of Doron’s opinions gives an explicit description of the conduct he expects from his audience, and also mentions this same episode, including a link to the video.

(This talk was also notable for being one of the loudest seminars that I have ever attended. Doron is a very passionate speaker talking about things that he cares about deeply, and he doesn’t skimp on the volume.)

Mathematical heaven

In the July 2009 issue of the Gazette of the Australian Mathematical Society, the president Nalini Joshi described spending time at the Isaac Newton Institute in Cambridge as being “in mathematical heaven” and I must say that I agree.

The Newton Institute is one of the buildings in Cambridge University’s Centre for Mathematical Sciences which is a pretty spectacular modern complex (though I’m not sure what the “towers” are for?) devoted entirely to various flavours of mathematical science.

The Newton Institute itself seems to have been designed simply by asking mathematicians what sort of environment would be most conducive to producing mathematics. An open central area is dotted with chairs and tables, with small shared offices around the perimeter. An automatic coffee machine supplies reasonable quality raw material to be turned into theorems. Even though Cambridge University’s mathematics library is about 50m from the Institute, the Newton Institute has a small private library that contains a good selection of well-known books.

The main sign that the building was designed by mathematicians though is that on almost every wall, there are the highest quality ground glass chalkboards, complete with signs saying “Please Leave” on one side and “Please Erase” on the other, to be left for the cleaners. And when I say “almost every wall”, I really mean it!

I was there for about four months during the first half of 2008, attending the program on Combinatorics and Statistical Mechanics, and it was a wonderful experience. Cycling round Cambridge, thinking, doing and talking mathematics, Wednesday evening soccer and the English pubs – heaven indeed!

What staff shortage?

There was a story in The Australian newspaper last week warning that universities faced a “looming staff shortage” as thousands of academics approach retirement and that widespread dissatisfaction with high workloads and the corporate management culture would mean that there wouldn’t be enough new blood to replace them. (Here’s a link to the article.)

It sounds convincing enough, but – at least in mathematics – it just doesn’t seem to gel with what we actually see on a day to day basis, either at UWA or elsewhere. With a couple of exceptions, it seems that most Australian universities are still reducing staff in mathematics departments, and actively welcome retirements rather than worry about replacing them. Certainly our department has shrunk more or less monotonically for more than 20 years, and there are no signs that it has bottomed out.  And of course there are well known examples like Flinders, which was rather embarrassed when Terry Tao won the Fields Medal and the media descended on his alma mater only to discover that they had essentially shut down all higher-level maths.

Meanwhile, despite the constantly deteriorating conditions, there seems to be no shortage of really good applicants for the occasional positions that come up, especially in Pure Mathematics. I personally know several really first-rate young pure mathematicians currently on fixed-term research-only positions who are searching, with varying degrees of desperation, for any continuing position (where “continuing” means “not fixed term” – the word “tenure” in Australia no longer has any connotations of job security or permanency because anyone can be fired at any time simply on financial grounds) and several more who have simply given up.

Perhaps this is unique to mathematics and other disciplines really are finding a looming staff shortage? (But here again, obviously the University of Melbourne is not too concerned, given that they’re busily trying to get rid of more than 200 staff, though not all are academics.)

So what gives?

[Update (9 October): See Terry Tao’s recent post on the “Maths in Australia” blog about the new AMSI director writing to Victoria University, one of the universities that is proposing further cuts in maths]