Mathematics: Two Public Lectures (Oct 2005)

Subjects: Ramanujan; Applications of Mathematics.

On Thursday (27 Oct 2005) I attended a lecture at Sydney University about The Life and Notebooks of Ramanujan. Ramanujan was an Indian mathematician who worked in England with the famous G. H. Hardy during World War I, and he ranks among the most intuitively brilliant mathematicians ever to have lived; probably the greatest. Merely to hear or read of his achievements is inspiring. In the lecture we learned:

• He grew up in the southern area of India, showed incredible interest in and aptitude for mathematics by the age of 12 (by devouring an advanced university-level text and answering all its problems).
• He loved calculus and hated geometry, and ignored non-mathematical subjects, causing him to get a poor mathematics mark and fail most other subjects at university.
• He worked in isolation on a piece of slate, economising on paper, which was expensive, by recording only his results (i.e. without proofs). He was confident that he could reproduce any of the proofs from memory or intuition. I think I’ve read that he didn’t place great value on proofs – which Western mathematicians and probably most others consider essential – but perhaps this isn’t true.
• He wrote to a few English mathematicians with about sixty of his results. Most foolishly ignored him; Hardy recognised potential genius and wrote back asking for proofs. Ramanujan wrote again without proofs but with another hatful of results!
• He travelled to England to work with Hardy, against the precepts of his Brahmin doctrine, but with the eventual blessing of his mother. Despite this supportive act, she was in some ways a shrew, blocking contact between him and his wife while he was away.
• His time in England was mathematically incredibly productive, but deleterious to his physical and mental health. The privations of war made it impossible to satisfy his diet, and his loneliness in a foreign land accelerated the illness that killed him in 1920 aged 32. If you can imagine Einstein being killed in his prime, you can imagine the impact of Ramanujan’s illness and death on mathematics.
• While working “publicly” by publishing papers with Hardy, he worked “privately” by recording results (again, no proofs) in a series of notebooks. His life’s work apparently consists of over 3,000 proved theorems, so that sounds separate from his notebooks. The notebooks probably began before his work in England, because they are two-thirds original. I.e. by working in isolation, he “discovered” much mathematics that was already known by others. Anyway, his notebooks are an inspiration for current mathematical research nearly a century after their writing.
• Ramanujan’s wife (whom he married when she was 9) died in the 1980s aged 94. The lecturer visited her once or twice and determined some threads from her life. She worked all her life sewing clothes, if I remember correctly, and despite being very poor gave away half her income to charity.

The man delivering the lecture (insert name here) has spent 20 years, with the help of PhD students, integrating into mathematics the contents of … wait for it … one chapter of one notebook. By “integrating”, I mean finding proofs for the results that Ramanujan wrote down.

The lecture included many photos but, sadly, very little mathematics: this was one of several lectures he delivers on Ramanujan, so the mathematics is obviously presented elsewhere. The lecturer strongly recommended the book The Man Who Knew Infinity for more information. In fact, reading the reviews of that book on amazon.com is likely to be more educational than reading this blog post. I'll certainly read the book if it contains some insight into his mathematics.

The second lecture was on Friday 28 Oct 2005 at Macquarie University. Bob Anderson of CSIRO gave the sixth Moyen lecture, in honour of Joe Moyen, who wrote a very influential paper “Quantum Physics as a Statistical Theory”, or some such, in the 1940s. The subject of the lecture was Applications of Mathematics. It was entertaining and informative, with several sub-messages such as “In a crisis, you don’t rise to the occasion; you sink to the level of your training.” (I.e. it’s worth acquiring a lot of knowledge.)

Bob was delivering the lecture because he was awarded the Moyen medal for distinguished contribution to mathematics, physics, or statistics. He has clearly been involved in a lot of mathematical modelling in industry, and dwelled on the following examples:

• determining which grains of wheat are soft (good for pasta) and hard (good for biscuits);
• restricting the strings in the Stuart and Sons piano to vertical movement, thus providing a richer, purer sound than other pianos; and
• analysing the genetic geometry of cotton in an attempt to increase its yield.

All examples made use of sophisticated mathematical modelling to gain an understanding of the physical aspects of the objects under investigation. He convincingly demonstrated the importance of mathematics to these advances in technology, and aimed his talk at a reasonably general audience rather than highly-qualified mathematician. The only suggestion I’d make to Bob is to wrap up each example better, summarising the problem, the mathematics, and the solution.

An ulterior message was clear: significant technological advances can’t be made without people applying difficult mathematics to problems. That obviously can’t happen without lots of people being trained in mathematics, which makes the current decline in high-level mathematics study in Australia, both at high school and university level, rather alarming. It’s a problem that feeds on itself: less qualified mathematicians today means less (good) mathematics educators tomorrow, which means even less qualified mathematicians the day after. Once the impact of this problem becomes clear, it won’t be possible to ramp up production of mathematicians.

Perhaps that’s not a problem. Perhaps we have all the technology we need and can cope with declining research and development for a few decades, but I doubt it. The grain and cotton problems above are good indications of the problems we may face without such research: a reduced inability to improve farming and food production techniques in order to feed and clothe the world.