Fields Medal

 

One of the biggest stories in mathematics when I was in secondary school was hearing that there was this guy in Princeton called Andrew Wiles who had been working very long and hard on Fermat's Last Theorem.

Back then, I was one of the more talented people at mathematics in my school, although I had been regularly accused of underachieving by my teachers. Mathematics was, for me, an easy A. I did hang out with the mathletes, took part in school competitions. But I never made it to the regular maths team. I wonder what it would have been like to been on track to be a maths quiz star. You had to be patient enough to study mathematics.

I didn't need patience to do the maths that appeared on regular homework. They were concepts that I grasped pretty quickly. I don't remember having to expend much effort on mathematics until I reached JC, and although I had to work for my As, it was relatively straightfoward. And it was easy because everybody around me was kicking and screaming, so I just had to progress at their pace, which I could well do with one eye closed.

So in hindsight, Andrew Wiles was an important parable for me at an important time. He locked himself away from the world and hammered away at a problem for 7 years until it got solved. Mathematics wasn't really about sprinting. It wasn't about Archimedes having a flash of insight and jumping out of his bathtub in joy. It was about taking a very long hike up a very tall mountain, and doing it over and over again.

Given the experiences I had as a teenager, it would be very churlish for me to wish that things were different, but I wish I managed to put together a successful run at being a mathlete, and I wish I had somebody to help me along with that. (I actually did, but I didn't know how to make use of that resource). I wish I had put together a regimen, where some form of intense focus and some ritual / regime helped me go up that mountain. I chose the more flashy route, which was an artistic life whose path was littered by flashes of inspiration. That is nice, but mathematics helps you think about structure, architecture and the intricate patterns behind putting things together.

Mathematics has gotten to the point where a lot of it is abstruse. It's easy to hide in a forest of complexity. The truly interesting results are the ones which lead to applications, or forming unexpected connections between superficially dissimilar instances. I wish I had at least learnt some fundamental results like Galois theory or saw the proof of the fundamental theorem of algebra (that polynomials always have complex solutions). I never learnt anything about diophantine equations. Those were the regrets I had about mathematics.

I heard about how mathematicians lived like recluses, and spent their days and nights chipping away at hard problems. There was June Huh, who was the very epitome of the tortoise brain, who worked slowly, but somehow always saw the deeper connections underneath the concepts.

There was Paul Erdos, and I remember Fan Chung describing him as somebody who always led you on an adventure that was very strenuous. There was Zhang Yitang, who had a story similar to Andrew Wiles: he toiled in obscurity for a long long time before coming up with his twin prime work.

Genius in mathematics wasn't so much the “AHA” flash of insights that come by. People have different styles of trying to find things out. I'm probably too messy a person to be a truly great mathematician, and after all, you need to have done most of your work before 40.

It's great that this time, the class of 2026 has 2 ethnic Chinese Fields medallists. The first two were Yau Shin Tung and Terence Tao, and they are legends. Yau Shin Tung didn't prove the Poincare conjecture: that was Grigori Perelman. But he laid down a big part of the foundations. Wang Hong and Deng Yu have big shoes to fill. Yau Shin Tung wasn't a Chinese national when he was awarded the prize, but he should be considered a mainlander too.

I'm more interested in prizes like the Fields Medal, the Abacus prize, the Turing prize and various other computer science and mathematics related work. I think that the Nobel prizes in Economics and Life Scences are still very vital. I am less sure about Chemistry and Physics. They gave the Chemistry Nobel to AlphaFold, and that to me was a bit like cheating, because why would you give a Chemistry prize to machine learning?

World Cup Cinderellas

There are some teams which capture the imagination of the world at a major footballing event. In 1990, it was Cameroon. In 1994, we had a few, because traditional powers like England and France didn't qualify, and Argentina and Germany were knocked out early. We had Romania, Bulgaria and Sweden as Cinderellas. In 1998, the Cinderella was Croatia. In 2002, Senegal, Turkey and South Korea were the Cinderellas. Turkey had an easy path to the semi-finals, but the backbone of their team was the same side which beat Arsenal a few years earlier to win the UEFA cup. In 2006, Portugal reaching the semi-finals might be a Cinderella moment, or it might not. 2010, the Dutch could have been considered Cinderellas, except that they played so dirty in the final that it was a disgrace. Uruguay performed very well, but it turned out that they had Luis Suarez, Diego Forlan and Edinson Cavani. They also had the handball gamesmanship in the infamous Ghana game. Ghana did well to reach the quarters. There weren't any surprises amongst the semi-finallists in 2014. Costa Rica and Colombia achieved all time bests in reaching the quarters. The real surprise was the infamous Brazil – Germany match in the semis. In 2018, we had 3 dark horses in the semi-finals. England, Belgium and Croatia. Belgium had one of the best teams, so this was more like a golden generation who didn't finally go on to win it. England are so crap at World Cups that reaching the semi-finals at that time felt like a great achievement, although they should have found some way to get past Croatia. Croatia is one of the few teams that actually appeared on this list twice. Russia is a country that is a pariah state now, but the way they powered over Spain was at that time truly inspiring. In 2022, the Cinderella was Morocco, who ensured that an African country finally reached the World Cup semi-finals. There were Asian countries who did it before, but the don't count: Turkey had done it before, but they were considered European because they compete in UEFA and anyway Istanbul (which have their 3 great clubs) is in Europe. South Korea had done it before, but when they were knocking out Italy and Spain, the refereeing was so biased that nobody believed those games were fair.

There is more of a good record of Cinderellas winning tournaments in the Euros. Sweden and Denmark getting to the semi-finals would have been considered a success. Sweden proved in 1992 and 1994 that they had a golden generation. Denmark actually won the whole thing. During the final, there was an outrageous amount of time wasting and gamesmanship that took full advantage of being allowed to play the back pass for the last tournament before it was officially banned. In 1996, the Czechs were the dark horses, having knocked out France in the semifinals to get to the finals. The English getting to the semi-finals were over-performing, although they were agonisingly close to beating the Germans, and after that it wouldn't have been the hardest thing to beat the Czechs in front of their own fans. There were no dark horses in Euro 2000, which was celebrated as one of the great tournaments, because there were so many great teams there. Maybe Portugal could consider themselves moving ahead in life to get to the semis but that was the golden generation that had Rui Costa and Figo. In 2004, that was the mother of all Cinderella performances, when the Greeks knocked out France, the Czechs and finally the hosts Portugal to win their first and only international trophy. It was an era where defensive play could win you tournaments: Mourinho would introduce the concept of parking the bus to the world. But for a country that was really a minnow (they had never won a game before in a tournament) to beat 3 great nations on their way to the championship is a truly extraordinary achievement, so let's not quibble too much about how they did it. In 2008, Turkey reaching the semi-finals would have been a great achievement. Russia unearthed a few gems among their players, including Arshavin and Pavluchenko. But the star of the show was Spain, whose golden generation was starting to blossom. Finally, after having 2 of the greatest club sides in football, they broke their duck and won their first Euros in 44 years. They were such underdogs at that time that nobody thought they would win 3 tournaments in a row. No real surprises in Euro 2012, other than Italy reaching the finals. They were in their massively underachieving phase and Serie A was on a downward trajectory that Inter Milan winning the champions league in 2010 was unable to disguise. Euros 2016 would see a few dark horses. First was Iceland, who somehow managed to not just reach the knockout stages but knock out England. Then there was Wales, who went all the way to the semi-finals, although they had an easy path: they only had to beat Northern Ireland and Belgium. OK, Belgium were quite strong in 2016. The most significant thing about that tournament was that Portugal finally won and Cristiano Ronaldo had his continental trophy. He just had to beat Croatia, Poland and Wales to get to the finals, but they somehow managed to beat France. In Euro 2020 (held in 2021 because of the pandemic), the Cinderellas were Denmark, England and Italy. Denmark were lucky to get to the semi-finals, only having to beat Wales and the Czechs. But they may also have been spurred on by Christian Eriksen collapsing on the pitch. England got all the way to the final, having beaten Germany, Ukraine and Denmark to get there. The Germans were the Germans, but probably at a low ebb in their powers. Italy had a good squad, probably. Their road to the final was not easy: Austria, Belgium and Spain. And finally they got over England with Chiellini executing that shirt pull on Bukayo Saka to hold him from running away with the ball. No Cinderellas in Euro 2024, where England reached the final but were denied by Spain again. Their route to the final was relatively straightforward: Slovakia, Switzerland and the Netherlands.

What we can see is that the list of minnows rarely repeat themselves. Minnows who overperform usually don't do it again. Only a select few make it to becoming true world powers, like Croatia, who took on the mantle of the great football nation that the former Yugoslavia was threatening to become. (God knows why that didn't happen to Serbia). Like Spain who in retrospect were building a golden generation. Belgium were truly unlucky to have had Marc Wilmots and Roberto Martinez as coaches of their golden generation. They might have won something with a truly great coach. So we can praise Bulgaria, Romania, Turkey, Sweden, Colombia, Costa Rica, Greece, Denmark, the Czechs, Russia, Wales, Iceland, Morocco to the skies when they pull of something extraordinary. But it's rare that any of them can repeat that feat.

Anyway, who knows which nation can be cinderellas at the 2026 World Cup? Somehow, it seems that there's more potential for a power shift in the World Cup, because the African nations are rising and the Europeans – other than a few giants like England / France / Spain are declining.