Just as music fans get excited when their favourite band goes on tour, so do we get excited when we can see a mathematical hero of ours at a conference. This week we were in for such a treat four times over. The opening session of this year's virtual Heidelberg Laureate Forum (HLF) featured interviews with four people we were lucky enough to interview back in 2018, when they won prestigious prizes at the International Congress of Mathematicians: Caucher Birkar, Alessio Figalli, and Peter Scholze, who all won Fields Medals in 2018, and Constantinos Daskalakis, who won the Rolf Nevanlinna Prize.

Back in 2018 we reported extensively on the work of these four laureates (see the links to articles, podcasts and videos below) and yesterday we were able to find out where their mathematical journey has taken them since then. Here is what they told their online audience.

Making machine learning fair

Constantinos Daskalakis won the 2018 Rolf Nevanlinna Prize, which is awarded every four years for outstanding contributions in mathematical aspects of information sciences.

One of the problems Daskalakis worked on affects many of us as we go about our everyday lives. When people analyse or design systems to be used by many people — such as road networks, real or online markets, or even dating apps — they often use the mathematics of game theory. An important notion in this context is that of a Nash equilibrium. When you throw together a collection of agents (people, cars, etc) in a strategic environment, they will probably start by trying out all sorts of different ways of behaving — all sorts of different strategies. Eventually, though, they all might settle on the single strategy that suits them best in the sense that no other strategy can serve them better. This situation, when nobody has an incentive to change, is called a Nash equilibrium.

John Nash, after whom the equilibrium is named, proved in 1950 that a Nash equilibrium always exists in a game theoretical system. If you think that this sounds too neat to be true, you are right, at least in practice. "What we showed is that [while] an equilibrium may exist, it may not be attainable. The best supercomputers may not be able to find it," Daskalakis explained when we interviewed him for the Nevanlinna Prize in 2018, awarded in part for this work on the Nash equilibrium. You can find out more in this article, this video, or this podcast.

Daskalakis' recent work is also highly relevant to people's lives, but this time in another area: machine learning. This area of artificial intelligence has recently made a lot of progress in replicating cognitive tasks only humans used to be able to do in the past, however there can be problems when it comes to actually using it. Daskalakis is running a research programme designed to address these problems.

"For example, machine learning models are trained using data that you collect in the real world," Daskalakis explained. "The problem is that data we have available is often biassed." The reason for this could be to do with the way the training data is collected, but also with the fact that the society we live in is biassed. "One of the challenges we are encountering is how to train an unbiassed system even though you are providing it with data that is biassed. This connects to classical problems in statistics and econometrics but adapting them to the high dimensionality of the data that is encountered in machine learning."

The problem of bias in machine learning will also be explored in another HLF session we are hoping to report on. To find out more about Daskalakis' research programme, see the video below.

Between water and ice

Alessio Figalli was awarded the Fields Medal in 2018 for "his contributions to the theory of optimal transport and his applications in partial differential equations, metric geometry and probability".

As the name suggests, optimal transport theory is about finding the best way of moving a distribution of things from one place to another when there's a choice to be made as to how exactly to do this. This may sound mundane, but optimal transport theory actually poses deep problems in the theory of mathematical functions. It was for his work on such problems, and applying the tools he developed to other situations such as weather forecasting, that Figalli received the Fields Medal. You can find out more in this article, our video featuring Figalli, and our podcast.

In the last four years Figalli has been working on something that at first appears quite different: the phenomenon of a phase transition where a physical system undergoes a dramatic, sudden change. Examples are water freezing to ice, or turning into steam. Both optimal transport problems and phase transitions, however, can be modelled with the same type of equations: differential equations. "There are classical equations that describe [phase transitions]," Figalli said at the HLF session. "In general you have two things at play. You have an equation modelling the temperature of the water, for example, but you also have an equation for what is called the free boundary — that's the interface separating the water and the ice. So you have a system with two unknowns that interact and you want to understand the properties of [both solutions] at the same time."

Although the problem and the equations modelling it have a long tradition, they are still difficult to understand. "There is a lot of beautiful mathematics that enters [this field] and I am really enjoying working on this problem," Figalli said. "But, you know, mathematics is rich, there are a lot of problems [to work on], and I also like to change and have new challenges."

A neat tweak

Peter Scholze was awarded a Fields Medal in 2018 for transforming an area of maths called arithmetic algebraic geometry. To get a loose idea about the motivation of Scholze's work, think of the equation $$2x=1.$$ This equation has the solution $x=1/2,$ which is a rational number. If we stipulated that we are only interested in solutions that are integers, then there would be no solution at all. In other words, whether the equation has a solution depends on the collection of numbers we are interested in, or over which the equation is defined as mathematicians would say.

In a similar vein, Scholze wanted to answer questions about equations over different types of number fields. In order to do this, he came up with so-called perfectoid spaces. Although he only introduced the notion of such spaces in 2011 (when he was just 23) it quickly made a great impact on mathematics, providing a growing new area of research, solving open problems, and opening up new avenues for research. It's this work that Scholze was honoured for when he received a Fields Medal. You can find out more in this article.

When it comes to his more recent work Scholze highlighted what is intriguingly called condensed mathematics. It's to do with the fact that some mathematical objects are more geometric in nature than they might at first seem. The real numbers are an example: at first sight they are just numbers, but on second sight you realise that together they form a line, which is a geometric object. In a similar sense, mathematicians sometimes want to equip algebraic objects with a topology. This doesn't come with precise notions of distance, area, angle or volume, but it gives you a general flavour of a geometric shape. For example, in topology a wonky orange still counts as a sphere and a wobbly rubber band as a circle.

But as Scholze explains, putting a topology on an algebraic object can lead to problems. "A lot of techniques we have in algebra break down when you also have a topology in place. But you can do a small tweaking to this notion of a topological space that has been around for more than a hundred years — this is what you call a condensed set. And if you do this small tweak, which in practice doesn't make much of a difference, suddenly all the good categorical [algebraic] properties are still there. So there's a thing called condensed mathematics where you can do algebra with some kind of a topology and it all works beautifully well. Suddenly you can do all sorts of things — it's quite fascinating!"

Classifying mathematical beasts

In 2018 Caucher Birkar was awarded a Fields Medal for his contributions to an area of maths which goes the other way around from what we just described: algebraic geometry starts with geometric objects and uses algebra to describe them. It's an idea you might be familiar with from school. For example, the equation

$y=2x+1$ describes a straight line, and the equation $x^2+y^2=1$

describes a circle. The circle and the line are both examples of algebraic curves, but you can also consider algebraic surfaces, or even higher-dimensional geometric objects, defined by more complex equations. Generally, these go by the name of algebraic varieties. There's an infinite zoo of of algebraic varieties, tempting mathematicians to classify them into species, just as a biologist might be tempted to classify butterflies. It is for such classification work that Birkar received his Fields Medal in 2018. You can find out more about this in the article we published at the time, or watch Birkar talk about his work in this video.

In his work since receiving the Fields Medal, Birkar has continued to be inspired by the classification of algebraic varieties. "This thing has evolved into a rather more ambitious and more inclusive kind of programme," he said at yesterday's HLF session. An example of the kind of object that is central to the field are so-called Calabi-Yau varieties. Interestingly, these varieties are important in physics. Physicists are currently hotly pursuing a "theory of everything" which can explain all the fundamental forces and particles of nature in a single mathematical framework. String theory is a candidate for such a theory, and it postulates that there are more than the three dimensions of space we can see. The idea, loosely speaking, is that the extra dimensions are rolled up so small that we can't perceive them. A Calabi-Yau manifold is a geometric object which can accommodate these tiny dimensions (you can find out more in this article).

But Calabi-Yau varieties, and related objects called log Calabi-Yau fibrations, also play an important role in mathematics — not just in algebraic geometry, but also in arithmetic geometry and differential geometry. It is this mathematical aspect that continues to fascinate Birkar and that he has been focussing some of his work on. "The nice thing about [the project about log Calabi-Yau fibrations] is that it unifies so many central notions in algebraic geometry," he said at the HLF session. "It's amazing that it relates to so many different things."

As Birkar pointed out at the HLF session (see the video below) the research described here is just one part of what he has been doing lately. But flexibility is one of the beautiful things about being a great mathematician. "I am just woking and seeing where these things are taking me. I am not fixed on what I should be doing in the next five or ten years."

Our four laureates were interviewed at the HLF by Carlos Kenig, President of the International Mathematical Union. Here's the video of the interview:

What is the HLF?

The Heidelberg Laureate Forum is an annual networking event at which young researchers get the chance to mingle with some of the best minds in mathematics and computer science. This year the HLF is taking place entirely online. You can see videos of the talks on the HLF YouTube Channel. To find out more about HLF in general, see the HLF website. And to see our all our coverage of this and past HLFs see here.