What is the Fields Medal?

Now you may have heard of the Nobel Prize, but the Fields Medal might be less familiar to you, but it’s the equivalent of a Nobel Prize for mathematics. This week, the efforts of four mathematicians were recognised with Fields Medals. To explain, Izzie Clarke spoke to Cambridge’s favourite mathematician, University Challenge mastermind, and friend of the show, Bobby Seagull. So what is the background to the Fields Medal?

Bobby - The Fields Medal is awarded for outstanding mathematical achievement for existing work. However, unlike the Nobel Prizes, it’s only awarded every four years at the International Congress of Mathematicians for between two to four mathematicians that, I guess, represent a diversity of mathematical fields.

However there is a catch, there is a condition and medalists cannot be aged 40 at the start of the awarding years, you’ve got to be 39 or under at the start of 1st Jan. And this is because the medal also recognises the promise of future achievement. the prize fund isn’t relevant, there is a 15,000 Canadian dollar award - that’s about 9,000 British pounds.

As you know I’m a quiz fan, so for those of you who are film buffs I have a maths related University Challenge style starter for ten quiz question. Are you ready?

Izzie - Oh, of course.

Bobby - Fingers on buzzers. My best Paxman voice: which 1997 Oscar nominated best picture film has the actor Robin Williams introducing an unrecognised maths genius to his Field Medalist’s university friend?

Izzie - Oh ding, ding, ding. That is Good Will Hunting - Matt Damon.

Bobby - Ten points.

Izzie - Yes!

Bobby - If anyone’s ever seen the film they actually mention the Fields Medal there. So definitely go and watch that film.

Izzie - Okay. So who has won it this year?

Bobby - This year they’ve awarded four. They don’t always have to award four. I think in 2002 they awarded two winners but since then they’ve sort of been encouraged to give four. The youngest person this year is Peter Scholze; he’s only 30 from Bonn University. Then second we have the Italian Alessio Figalli; he’s 34 and works at ETH Zurich. And then third we have a 36 year old Australian who researches in America called Akshay Venkatesh. And then the fourth winner is someone a bit closer to home…

Izzie - Now I’ve heard that this person is at Cambridge University, so what do they work on?

Bobby - Correct. Professor of Maths in the Department of Pure Maths and Mathematical Statistics is Professor Caucher Birkar. He was raised in the Kurdish region of Western Iran and moved to the UK in about the year 2000, and I think he did his PhD at Nottingham before moving to Cambridge. And his work showed that it was possible to bring order and find connections between apparently unrelated algebraic equations.

So just as a reminder, I’m putting on my Mr Seagull teacher hat back on so mathematical equations can be depicted as shapes so, for example, Y = 2X + 3 is a straight line. Another one, X squared + Y squared is 9 - that’s a circle. You still with me?

Izzie - I’m still following yeah.

Bobby - Algebraic geometry is about studying the shapes that can be described by these equations. And as mathematicians, we try and put an order to this variety of equations. As a normal person, if you looked at an animal and you tried to classify them you might think oh, some of these have wings like a seagull, or some of these are carnivorous like a cat eating mice.

But Birkar, his work was trying to understand one of three generic categories and these are called Fano varieties. But, unfortunately for Birkar his original medal, the Field Medal, was stolen after he left the medal in a briefcase with his mobile and wallet on top of the table, and they found the briefcase but not the medal. So this sounds like a sad story but it has a fairytale ending, he was given another medal, and he says he’s become more famous because of this. And actually the Fields Medal has had a bit more attraction in the media because of this. In a way, the theft has worked out well for mathematics.

Izzie - Not that we’re encouraging theft at all!

Bobby - Not at all, not at all.

Izzie - What did the other three work on?

鲍比-彼得Scholtz,他是一个年轻30岁,his research is something called P-adic Geometry. P I think basically stands for prime numbers. And Scholtz’s main innovation was on something called Perfectoid spaces - and that is a real word. This is a class of fractal structures and he’s essentially built a new bridge between arithmetic and geometry, so he’s is the first one.

Then the Italian, Alessio Figalli. His results actually provided a refined mathematical understanding of things that, for example, the shape of crystals or the weather patterns, and even the way that a block of ice might melt. Many of his results actually rely on the use of a technique called Optimal Transport. And curiously enough, this originated in the 18th century when a mathematician was working for Napoleon - and it is the Napoleon in the Abba song. This mathematician was trying to find out an efficient way to build network fortifications, and Figalli essentially - the optimal transport problem is about finding the cheapest way of transporting a distribution of maths from one place to another.

Then we come to the final person, Akshay Venkatesh. And he wasn’t awarded for one specific thing but more profound contributions to a broad range of subjects such as building connections from number theory to something quite distant such as algebraic topology or dynamical systems.