Saturday, 17 January 2009

Ep 97: My Top 5 Mathematics Stories from 2008

Having worked in maths communication throughout 2008, I thought it about time to reflect on the 2008 year in maths, so here are the maths stories from 2008 that caught my eye. I'm sure I have left out many mathematical stories from the year, so please let me know of anything you think I should have included.

1) Solving Tough Modern Problems

Global Financial Crisis: Toxic loans? Greedy bankers? Or mathematics? What is to blame? Whilst the GFC is a very complex system that I won't get into too deeply, the backbone of the crisis lies in a few key points:
  1. In the US (and in Australia), there is a very strong pressure to buy your own home. It's fuelled by the idea that housing prices never fall, low interest rates, ideas regarding how you must get into the housing market, and it's probably helped along by a current generation that doesn't mind a bit of debt, not to mention the ease at which you could get credit in the early 2000s. The problem with all this is that people who could barely afford loans tried to get them, and....
  2. .... the banks happily gave them loans. Once housing prices started to fall, people had mortgages bigger than the value of their homes. These sub-prime loans failed to take into account that plenty of people could not afford to pay them back (they were sub-prime mortgage candidates). In the meantime....
  3. .... banks and other financial organisations were getting clever with their financial instruments and this is where some blame may lie with maths. Securities, backed by mortgage repayments, were repackaged as bonds and sold on to other banks, insurance companies, superannuation funds, governments, everyone. The credit derivatives market was huge! Once housing prices started to fall (housing bubble), people couldn't pay back their debts, the banks didn't get their repayments and this affected not only everyone who owned mortgage-backed securities, but everyone who traded with people who owned mortgage-backed securities. Given credit was so easy to get, this was nearly everyone. Companies such as Lehmann Brothers started to fail and governments had to jump in to bail them out.
Generally I think that if you play the financial game, you need to be cognisant of the risks and if you lose money, it's your bad-luck. But the problem here is that not only have rich bankers failed, everyday people who have been encouraged to put their money into superannuation have lost money. The system did not take into account the very risky business of lending huge amounts of money to people who may not be able to pay it back. Should the mathematical models have predicted this? Some people say that the models were fine, it's just that they were misused by greedy bankers blinded by their million dollar bonuses and not at all focused on the risks. On the other hand, perhaps the models should have taken into account the fact that although the risk of one person defaulting on their loan may be small and therefore lots of people defaulting all at once may be incredibly small, there may be underlying causes that cause lots of people to default at once (as was the case here) and so therefore the risk of a crisis like we have now was not actually negligible. This misunderstanding of risk meant that the mortgage-backed derivatives were mispriced.

I was astonished to hear a finance graduate friend of mine reflect on her time studying finance at University:

"The lecturer put up an equation, the theatre moaned, but the lecturer said 'Don't worry, this is the only equation in the course and you don't even need to know it'."

I seriously hope that's not typical! There is a very interesting podcast on this topic produced by the BBC program Discovery. One of the interesting things they talk about is that it is very hard to mathematically model the human elements of the system - we aren't atomic particles that can be modelled. Read more here on Plus and here on More or Less, and see Flowing Data for a visualisation of the crisis.

Climate Change and Terrorists: 2008 saw the first meeting of the Cambridge Complex Systems Consortium. "This multi-disciplinary team of Cambridge-based academic and industry leaders aims to study complexity across multi-dimensional domains," says Michelle Tuveson of Lockheed Martin UK, who is an associate of Clare Hall College and organised the meeting. "We aim to develop a science around complex systems by determining good frameworks, optimisation strategies, and prioritisation schemes." Issues involving complexity include climate change, terrorists and modelling of the Internet. Read more.

2) The end of a mathematical era

Edward Lorenz, American mathematician and meteorologist, died in his Cambridge Massachusetts home in April 2008 aged 90. Lorenz was known as the father of chaos theory and discovered the Lorenz attractor that often occurs in chaotic systems. Lorenz is perhaps best known for coining the term butterfly effect. Whilst a professor at the Massachusetts Institute of Technology, Lorenz developed the concept that tiny effects in one part of a system can lead to big changes in another part. The term butterfly effect first appeared in his 1972 paper Predictability: Does the flap of a butterfly's wings in Brazil set off a tornado in Texas? Read more.

Walter Warwick Sawyer
was a mathematician and author who made a major contribution to mathematical education. He recently passed away in Canada, aged 97. Sawyer was born in 1911 and was interested in the applications of mathematics to industry. He developed a scheme in which apprentices learnt mathematics by handling physical objects and in 1943 published his first and most successful book Mathematician's Delight, which has been in continuous publication since and has sold about 500,000 copies. Sawyer was a loud voice criticising many aspects of mathematics education and considered that students taught mathematics without an appreciation of its application would have no more understanding of what they were learning than a machine. His love of mathematics is seen in the title of his first book, Mathematician's Delight, whose aim was to "dispel the fear of mathematics":

"Many people regard mathematicians as a race apart, possessed of almost supernatural powers. While this is very flattering for successful mathematicians, it is very bad for those who, for one reason or another, are attempting to learn the subject."

He considered that teachers should work with students' minds and talents, rather than force foreign concepts on them.

"Education consists in co-operating with what is already inside a child's mind... The best way to learn geometry is to follow the road which the human race originally followed: Do things, make things, notice things, arrange things, and only then reason about things." Read more.

3) Solving Tough Old Problems

Road Colouring Problem: Sixty-three year-old Avraham Trakhtman has solved one of the current generation's toughest mathematical problems — the 38 year-old road colouring problem. The road colouring problem, which was first raised by Israeli mathematician Binyamin Weiss in 1970, is a problem concerning synchronised instructions and is expressed simply:

A man reaches a town he has never visited and drives around trying to find his friend's house. Despite the fact that there are no street names, his friend says not to worry and that he will provide instructions (turn left, then right, then left ...) on how to get there. The road colouring problem postulates that no matter where the man starts his journey (that is, no matter which part of the town he encounters first — he could have approached from perhaps the North or the South) there is a set of instructions that will lead him to his friend's house.

Another version of the problem considers a lost e-mail somewhere on the Internet. The sender wants to make sure it is delivered to the right place, no matter where it already is. Read more.

L-Functions: A new mathematical object, an elusive cousin of the Riemann zeta-function, was revealed to great acclaim recently at the American Institute of Mathematics. Ce Bian and Andrew Booker from the University of Bristol showed the first example of a third degree transcendental L-function.

L-functions underpin much of twentieth century number theory. They feature in the proof of Fermat's last theorem, as well as playing a part in the recent classification of congruent numbers, a problem first posed one thousand years ago. Read more.

4) Prizes

One of the most important international prizes for mathematics was awarded jointly in 2008 to two outstanding mathematicians — even though one of them was originally unable to find a publisher for his groundbreaking work. Professor John Griggs Thompson, of Cambridge and Florida Universities, and Professor Jacques Tits, of the Coll├Ęge de France, have been awarded the 2008 Abel Prize, worth £580,000, by the Norwegian Academy of Science and Letters "for their profound achievements in algebra and in particular for shaping modern group theory". Read more.

5) Silly Maths
  • If we are to believe the latest signs from outer space, the local aliens are keen mathematicians and work in decimal. A new crop circle appeared on the 1st of June 2008 in a barley field near Barbury Castle in Wiltshire, England, measuring 150 feet in diameter and correctly representing the first 10 digits of the irrational constant pi. Read more.
  • No matter how many friends you have on Facebook and MySpace, you won't have more real-life friends than the average person. Using mathematics to model online social networks is an evolving field, with techniques that have been used to model human interaction, such as network modelling, moving into the online world. Users of online social networks tend to build up long lists of "friends" with whom they only occasionally interact, if at all. The existing techniques have now been used to model these weaker relationships. Read more.
Well, that's it. Please let me know if I have left anything off you think I should have included. Have a happy mathematical 2009!

Listen to his podcast here:

And remember to tell us your science highlights from 2008 to go into the running for some sciencey prizes. Answers will also contribute to our year-in-review podcast early in 2009. The comp will be open till 26 January. Let us know here.

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