When describing John Forbes Nash, Jr (13 June 1928 – 23 May 2015), it’s hard to be more succinct than Richard Duffin, a professor at the Carnegie Institute of Technology, who wrote, in his letter of recommendation to Princeton, that ‘this man is a genius’. It was 1948: Nash, having abandoned a degree in Chemical Engineering for one in Mathematics, was only just embarking on a journey that would ultimately make him one of the most famous mathematicians of the 20th Century. Despite the interest of Harvard University, Nash eventually decided to pursue his graduate studies at Princeton and it was there that he published the 317 word paper, *Equilibrium points in N-person games*, that introduced the Nash Equilibrium and won him the Nobel Prize for Economics (jointly with Reinhard Selten and John Harsanyi) in 1994. As a result of this work in game theory, Nash was appointed to the RAND Corporation, which applied this relatively young field to the pressing policy issues of the time: nuclear weapons, the space race, the Cold War.

# Tag Archives: issue-02-article

# Fermat Point by Suman Vaze

Suman Vaze sits on her small balcony in crowded, bustling Hong Kong, with a view, just about, of a beautiful Chinese Banyan tree tenaciously growing on a steep stony slope, and paints mathematics. Inspired by the abstract expressionism of Rothko, the radical and influential work of Picasso, and the experimental models of Calder, she fully embodies Hardy’s belief that mathematicians are ‘maker[s] of patterns’. Our front cover is one of her pieces: the bold colours proclaim the eponymous *Fermat Point* – the point that minimises the total distance to each vertex of a triangle – along with its geometrical construction. Add an equilateral triangle to each side of the original triangle then draw a line connecting the new vertex of the equilateral triangle to the opposite vertex of the original: the intersection of these lines gives the Fermat point. Not only do these lines all have the same length, but the circumscribed circles of the three equilateral triangles will also intersect at the Fermat point.

# In conversation with Artur Avila

Sitting in a pub in Leicester Square, talking to one of the most brilliant mathematicians of our generation, is not the way one would normally expect to spend a sultry evening in early June. But it turns out that Artur Avila, winner of the 2014 Fields Medal, takes very spontaneous holidays, and is a big fan of the pub.

In the UK, Artur certainly does not conform to the stereotype of a mathematician. He is good-looking, stylishly dressed in a white T-shirt and designer jeans and asking about the best London nightclubs. However, Avila was born and bred in Rio de Janeiro, famous for its spectacular parties and beautiful beaches. He still spends half his time there, based at the National Institute of Pure and Applied Mathematics (IMPA), and spends the other half at the French National Centre for Scientific Research (CNRS) in Paris, where the nightlife is terrible, according to Avila.

It’s perhaps unusual for an exceptional mathematician not to spend all their time in the USA or Europe, where there are higher numbers of world class research institutions, but Avila believes ‘it’s significant that I studied at IMPA because it shows that Brazil has institutions that can prepare someone to do maths at a high level, and it’s not necessarily true that you always have to go to the United States or to Europe to advance.’

And, of course, Rio has many appealing features: ‘I have several times brought collaborators to the beach with me and we would just sit and share ideas with each other, with the sound of the sea in the background.’

# The joys of the Jacobian

I am an applied mathematician. Like, *very* applied. I use mathematics, sure, but I’m far more interested in what maths can do to solve real problems. Specifically, how can mathematics be used to tackle issues in infectious diseases?

The best thing about maths, from my point of view, is that it has one incredible superpower: it can predict the future. That’s amazing. And very useful, of course.

When teaching my undergraduate students the required details to make these predictions, I stumbled upon a very profound realisation. Namely, that the Jacobian matrix – a technical thing from linear algebra – is a) just about the most massively useful thing ever and b) a glorious way to reconcile two apparently disparate strands of mathematics.

Don’t believe me? Read on and hopefully you too will become a convert to the church of the Jacobian…

# The Mpemba paradox

If you take two identical cups, fill one with warm water and one with cold and put them in the freezer, you’d expect the cooler one to freeze first, but it doesn’t always. In fact, in many circumstances, it is the warm water that freezes first.

This is the Mpemba paradox, named after Erasto Mpemba, who observed it as a schoolboy in Tanzania in the 1960s. When physicist Dr Denis G. Osbourne visited his school, Mpemba took the opportunity to ask about his strange observation. Although initially skeptical, Osbourne later reproduced the observations and several years later, in 1969, they jointly published the result. Since then, it has been reproduced in many experimental studies and its origins have been debated extensively.

# Breaking out of the prisoner’s dilemma

The prisoner’s dilemma often features in television programmes (such as ITV’s Golden Balls) where two contestants have to decide whether they want to share or steal a pot of money. They make their choices in secret from one another and then their decisions are simultaneously revealed.

Let the tuple $(a, b)$ mean that you get $a$ and your opponent gets $b$: so $(3, -1)$ represents you winning £3 and your opponent losing £1. We introduce the pay-off matrix for a non-iterated (played only once) prisoner’s dilemma:

You \ Opponent |
Cooperate |
Defect |
---|---|---|

Cooperate |
(1, 1) | (-1, 3) |

Defect |
(3, -1) | (0, 0) |

Each player is given the opportunity either to defect or to cooperate. In the original set-up with prisoners the option of defecting meant betraying the other and testifying whilst cooperation represented remaining silent.

# The perils of p-values

*This article has code for you to run! Skip to the downloads.*

Imagine that we have a group of 20 volunteers. We give all 20 people identical pills, and measure a response in each of the people. The responses would not all be the same—there is always some variability. If we divide the 20 responses randomly into two groups of 10, the means of the two groups will therefore not be identical.

If we had instead given each group of 10 people different pills (say drug A and drug B) then we would also find that the means of the two groups differed. If drug A was better than B then the mean response of the 10 people given A would be bigger than the mean of the 10 responses to B. But of course the response of group A might well have been bigger, even if drugs A and B were actually identical pills.

It is one of the jobs of applied statisticians to tell us how to distinguish between random variability and real effects. They can tell us how big the difference between the means for A and B must be before we believe that A is really better than B and not just the result of random variability.

It is the aim of this article to persuade you that the ways of doing this that are commonly taught give rise to far more wrong decisions than most people realise. This is not trivial. It gives rise to the publication of discoveries that are untrue. For example, it may result in the approval of medical treatments that don’t work.

# Roots: All things being equal

Imagine the scene: The year is 1557. Henry VIII’s eldest daughter, Mary, is on the English throne. It’ll be another year before her younger sister, Elizabeth, becomes queen. You’ve published a fair few mathematical texts, and you’re halfway through writing your latest book ‘The Whetstone of Witte‘, the second in a pair of books on Arithmetic (the title was a pun about sharpening your mathematical wits).

You’re determined to only use English language in the book but you are getting really frustrated with having to write ‘is equal to’ every time you note down an equation. Then it dawns on you! Why not use a symbol to represent ‘is equal to’? It’ll save time. It’ll save ink. After all, isn’t mathematics all about efficiency? But what symbol to use?

# Fractograms

Bored one day in a staff meeting, I took to playing around with numbers – a nice way to pass the time. I wondered if chaotic numbers might exist; that is, numbers whose digits at first might look quite random, but hidden within this apparent disorder would be the signature of order that lies at the heart of chaos. I had big notions that maybe the digits making up famous irrational numbers like $\pi$ or $\sqrt{2}$ might be such numbers, but I decided to start with more simple numbers, those of the recurring decimals. I took the fraction 1/7 as my starting point.