My favorite equation

Aside: To comment on my recent Gresham College lectures please go to this blog entry. My most recent Gresham College lecture was on Monday 15 April.  I talked about proof, by human and by computer.  The lectures can be viewed on the Gresham College website.

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I have been reading William Poundstone's excellent book about the interview questions asked by Google, Microsoft and co, Are You Smart Enough to Work at Google?  Amongst many others, he discusses how a candidate should answer the question "What is the most beautiful equation you have ever seen?"

For me, and I suspect a great many mathematicians, the natural answer is "Euler's formula e^i.pi+1=0" (which is much more beautiful when properly set out!)  (This is the special case of Euler's more general formula for e^ix.)  Why do we like this equation so much?  Well, first because it is astonishing.  I don't have any natural intuitive understanding of what it means to raise a number to an imaginary power, but this equation shows that doing so is amazingly powerful.  This equation demonstrates the relevance of "imaginary" numbers to the real world.  Who would have thought in the sixteenth century that imaginary roots of negative numbers could lead to the incredible developments in our understanding of the universe exemplified by Schrodinger's Equation, for example?  (Even if I'm not sure "understanding" is the right word when one is talking about quantum theory!)

Euler's equation is remarkable because it involves five very special numbers, zero, one, i, pi and e, and it includes  the fundamental mathematical operations of addition, multiplication, exponentiation as well as the notion of equality.

So is Euler's formula my favourite?  Well, some of my favourites change over time.  Is my favourite composer Bach, Schubert, or Monteverdi?  At different times I would have answered any of these, and it depends on my mood (as well as who's performing).  On the other hand, I can't imagine ever changing my favourite football team, however depressing their results (and I fear that the Pars' sensational 6-1 win yesterday in the relegation / promotion play-off semifinal has just set us up for greater disappointment in the final).  I feel that Euler's as favourite equation is probably more permanent than Schubert but may be less lifelong than my love for the Pars.

But apparently Euler's is not the right answer for Google - it's not original enough.  (Understandably.)  So i wanted to be provocative I'd have to give another favourite equation.  And I think I might put forward John McKay's equation.  Here is the equation, written by McKay himself in my visitor's book.  

So what is this about?  The equation 196884 = 196883+1 is certainly plausible, but why is it any more interesting than any other trivial arithmetic sum?

Well, McKay works on sporadic simple groups, the existence of the largest of which, the Monster, was conjectured in 1972 and confirmed in 1980.  The smallest non-trivial irreducible representation of the Monster has degree 196883.  McKay's wife works in the area of modular forms (a quite different area of mathematics) and McKay happened to see that she had written down an equation which included the coefficient 196884.

McKay thought the similarity of the two numbers could not be coincidental.  It turns out that the other coefficients in the elliptic modular function also relate to the representations of the Monster.  McKay's observation has led to the discovery of very deep (and very obscure) connections between apparently totally different branches of mathematics.  This whole area has been given the evocative name "Monstrous Moonshine".  It's much too difficult for me, but I believe considerable progress is being made, although at the time he wrote his formula in my visitor's book, McKay told me that he thought it likely the matter will never be fully understood - there being little likelihood of anyone ever having deep enough knowledge of both of these two subjects to be able to investigate the connection.

So McKay's formula may not be as immediately beautiful as Euler's, but it has something of the same spirit (and perhaps even importance).  It demonstrates a very deep connection between group theory and modular forms; it's mysterious and hard to understand, and it's inspiring important mathematics.  And it says a lot about the serendipity which lies behind insights even in a subject as apparently logical and rigorous as mathematics.   If I can't use Euler's equation then when Google ask me this question I'll go for McKay.

A book which changed my view of linear algebra

Aside: To comment on my recent Gresham College lectures please go to this blog entry. My most recent Gresham College lecture was on Monday 15 April at 6pm at Barnards Inn Hall, near Chancery Lane tube station, central London.  I talked about proof, by human and by computer: all readers of this blog are very welcome.  The lectures can be viewed on the Gresham College website.

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(Just in case any of my students are reading this: just because I didn't use to much like linear algebra doesn't mean you won't!)

When I was an undergraduate I wasn't very excited about linear algebra.  It was worthy stuff, but it wasn't as attractive as group theory and combinatorics were.  Even when I was employed to write mathematical modelling software and relied heavily on matrix methods, I felt the applications of linear algebra were useful but not really very interesting.  Even when the founders of Google have made millions by exploiting the Power Method, I still found it hard to be wildly enthusiastic about this (very important) area of mathematics.

But I have just read a wonderful book which has changed my mind completely.  It's Jiri Matousek's Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra.

The book consists of (you've guessed it!) thirty-three chapters, generally of four to six pages, each describing an entertaining problem which can be solved by linear algebra.  The applications are staggering - this isn't boring applied mathematical modelling or billionaire-making search engines, but REAL mathematics - combinatorics, geometry, coding, probabilistic algorithms.

For example: we begin by finding the formula for Fibonacci numbers.  We show that there are no four points in the plane such that the difference between any pair is an odd integer.  We learn about turning ladders around in a finite field.  We have the wonderful matrix-tree theorem which counts the spanning trees of a graph. There is the wonderful account of the information that can be transmitted by a secret agent whose only means of communication is to choose the colour of umbrella he uses each day, to be photographed by a satellite which can't tell the colours apart.   And my favourite - how do you tell whether a given binary operation on n objects is associative?  It appears you have to test n cubed cases.  Even if you only want a probabilistic answer, sampling doesn't appear to help us much: if the operation has one non-associative triple, you have to sample half the triples to have an even chance of detecting the offending triple.  But no - there is an ingenious algorithm which does very much better.

The amazing thing for me is that we are using linear algebra in all these diverse areas of pure mathematics where its relevance seems quite unexpected.  This is a sensational book!

Advancing Women in Mathematics

Aside: To comment on my recent Gresham College lectures please go to this blog entry. My next Gresham College lecture is on Monday 15 April at 6pm at Barnards Inn Hall, near Chancery Lane tube station, central London.  I will be talking about proof, by human and by computer: all readers of this blog are very welcome.  The lectures can be viewed on the Gresham College website.

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On Wednesday 27 February the London Mathematical Society launched a report "Advancing Women in Mathematics".  There are too few female mathematicians.  While this may be because women are too sensible to want to do mathematics, I believe that mathematics has enriched my life and given me opportunities and I am dismayed that these benefits are being taken up in the UK largely by representatives of only one half of the population.  Only 6% of professors of mathematics in the UK are women.  (Most other countries do a lot better.)

There are some promising signs.  There are now sufficiently many successful mathematicians to show conclusively that women can do mathematics at the highest level. But they face obstacles.  Chris Good and I recently did a project about "Being a Professional Mathematician", which produced resources including interviews with mathematicians, and the interviews with Gwyneth Stallard and Sue Merchant, in particular, have interesting insights into their lives as female mathematicians.

Women now make up 40% of undergraduate mathematicians (and were a slightly smaller proportion of the speakers at the recent undergraduate conference Tomorrow's Mathematicians Today).  But the proportion of women in mathematics drops as one lists PhD students, postdocs, academics and professors.  Even 40% of undergraduates could be seen as a rather disappointing proportion when more women than men go to university.

The LMS report contains useful examples of initiatives and actions which may help.  The LMS and its excellent Women in Mathematics Committee is to be congratulated for so publicly raising this issue and for its commitment to improving the situation.

Tomorrow's Mathematicians Today 2013

(To comment on my recent Gresham College lecture please go to this post.)

Yesterday the University of Greenwich hosted the second UK conference for mathematics undergraduates. Tomorrow's Mathematicians Today 2013.  (The first TMT was also held at Greenwich, in 2010.)  About 150 students attended, and there were 30 student speakers from 15 different universities, as well as an excellent keynote address by Robin Wilson.  The conference was sponsored by the Institute of Mathematics and its Applications, GCHQ (a big employer of mathematicians) and the publishers Taylor and Francis, and the OR Society was also present.

I thought it was a wonderful day.  (The programme and abstracts can be found on the conference website.)  The atmosphere was remarkable - friendly and supportive of the speakers.  The contents of the talks varied enormously.  Some students were simply telling us about mathematics that excited them: others were reporting on their own original individual or collaborative work.  Some talks were relatively elementary, others sophisticated.  But there wasn't a single dud in the sessions I attended.  (Sadly, with parallel sessions, one couldn't attend all the talks.)  

In the first morning session I learned about applications of category theory in computer science, "funny functions" which show up misconceptions and misintuitions in analysis, and the applied group theory of "speedcubing" - how to find the most ergonomically efficient ways to solve Rubik's cube quickly.  Later in the morning I attended talks on Shor's algorithm and the threat quantum computers pose to conventional cryptography, and Conway's Angel and Devil game on an infinite chessboard - a simple game which took thirty years to solve.

In the afternoon I heard three fascinating talks about applications of mathematics in biology and medicine - the strengths and weaknesses of models in epidemiology and in the study of how bacteria move - and about financial mathematics and students' perception of GCSE mathematics.  The afternoon ended with Robin Wilson's keynote talk about Euler's life, labours and legacy.  The audience was enthusiastic throughout.  

The prize for the best paper, sponsored by GCHQ, went to Jason Young of Cardiff University for his talk on "Understanding the Effect of Individual Behaviour in Hierarchical Queues" - about how individuals' selfish or unselfish behavour affects queuing times for everyone.  It was an excellent example of interdisciplinary work with real practical consequences leading to insights which might, for example, reduce waiting times for medical treatment.

It was a wonderfully stimulating day. I am sure I am not the only person there who found myself unusually exhausted early last evening.  Thinking about mathematics, even as enjoyably as yesterday, is hard work!

Are mathematicians and artists opposites?

(To comment on my recent Gresham College lecture please go to this post.)

A newspaper feature today ("My funny Valentine: Do opposites really attract?" in the Independent)
presents as its prime example of the attraction of opposites a couple com[prising an artist and a mathematician.

Are mathematicians and artists really opposites?  I know many mathematicians and artists who get on.  Why wouldn't they?  Both professions require creativity, thinking for oneself, a willingness to challenge (or at least test) received opinion, perseverance in the face of difficulty, preparedness to wait for inspiration, and integrity.  And sometimes, perhaps, occasional use of drugs for stimulation (the drug for choice of mathematicians being coffee, and the use generally rather more than occasional.)

The public perception that mathematics is routine drudgery should be challenged!  Mathematicians and artists have a lot in common.

Gresham College Lectures

On Monday February 4th, I gave the first of a series of three free public lectures on computng and mathematics at Gresham College, Barnards Inn Hall, Holborn, Central London.  Any reader of this blog is welcome to attend these lectures.

The first lecture was entitled  "Arithmetic by Human and Arithmetic by Computer".  Video and audio recordings are available at the Gresham College website.   This lecture featured some cool tricks for doing arithmetic, a performing monkey and brilliant Scottish inventions.

The second lecture was "How Computers get it Wrong: 2+2=5".  Video and audio recordings are now available at the Gresham College website.  This lecture discussed different kinds of computer error.

The third lecture will be given at Gresham College (Barnards Inn Hall, Holborn, central London) on Monday 15th April.

Please post any comments or questions about the lectures as comments on this blog post.

Maths and arrogance

Earlier this week a popular-science Facebook page posted a letter apparently from a schoolteacher to a pupil's parent complaining about the pupil questioning the teacher's assertion that a kilometre is longer than a mile.  The teacher felt that even though he was wrong he should not have been questioned by a student.   This brought back memories of a similar occasion in my youth (when, on my parents' insistence, I privately asked my teacher whether she had meant "Fahrenheit" when she said the average summer temperature in Nebraska is 95 degrees Centigrade, and was roundly abused for my temerity.)   Others have reported similar experiences.  Being told "You mustn't question me - I am a teacher" is something many people seem to remember.

I find this baffling.  Teachers may say all sorts of things under stress, but in some of the cases cited there was no excuse.  I hope times are now more enlightened.  Children have to learn that everybody gets things wrong sometimes, and that when you make a mistake, it's best to admit it and learn from it.  Authority should be questioned when necessary.  The way to respond to a challenge is to show that the challenge is wrong. Someone who forbids questions should not expect to be believed.  Even in primary school, I don't think there is a good argument for saying children should not question a statement that they think is wrong.

I do remember one very distinguished mathematician responding angrily to a question in a postgraduate lecture.  We were all puzzled by a definition, so a friend of mine asked "Why have you defined that in this way?"  The lecturer's response was "Because it bloody well works, that's why!", which didn't help us much nor did it gain him our respect.

Despite this example, I sometimes like to think that one of the ways in which mathematics is good for the soul is that studying mathematics gives excellent protection against arrogance.   It's hard to take oneself too seriously when there are simple problems, like saying whether every even integer is the sum of at most two primes, which one cannot solve.  Mathematics is not a subject in which one can fool oneself into over-estimating one's abilities: there is always a reality check.  I might mistakenly believe that my poem is an unprecedented masterpiece but I know that my proof of the Twin Primes Conjecture doesn't stand up.

I'm probably wrong, but I feel that if mathematicians ruled the world, they wouldn't have the over-confidence to lead us into unnecessary wars.  Self-questioning and self-doubt should be encouraged (unless you're a sportsman!)

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On Monday 4th February I am giving the first of three free public lectures on computing and mathematics at Gresham College, London.  Readers of this blog are very welcome!