## Plus Blog

September 23, 2020

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.

Constantinos Daskalakis (third from the left) receiving the Nevanlinna Prize in 2018.

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

This equation has the solution 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.

Peter Scholze

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

Caucher Birkar with his Fields Medal in 2018. (PHOTO MARCOS ARCOVERDE/ICM 2018)

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

describes a straight line, and the equation

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.

September 11, 2020

The mathematician Martin Hairer has been awarded the 2021 Breakthrough Prize for Mathematics which, apart from the accolade, comes with \$3 million in prize money.

Hairer received the prize for "transformative contributions to the theory of stochastic analysis, particularly the theory of regularity structures in stochastic partial differential equations". You can find out more about what this means in the following article and podcast, which we produced after interviewing Hairer about receiving the Fields Medal, one of the highest honours in maths, in 2014.

Martin Hairer: At the interface — A brief look at Hairer's groundbreaking work, which involves the maths of change and chance.

Interview with Martin Hairer: The podcast — Listen to Hairer talk about his research in this podcast, recorded a day before he received the Fields Medal in 2014.

Hairer has also written us a series of articles about the maths of randomness. These are based on a lecture he gave at the Heidelberg Laureate Forum 2017, where we also recorded a video of Hairer talking about issue surrounding randomness. The video is shown below.

The maths of randomness — Randomness is surprisingly hard to define. But as Hairer explains in this article, the mathematical language we use to describe it is beautifully succinct.

The maths of randomness: Symmetry — In this article Hairer explains why symmetry is one of the two guiding principles in understanding probabilities.

The maths of randomness: Universality — The second guiding principle of the maths of randomness is universality, which Hairer explains in this article.

The two envelopes problem — Here's a brain teaser Hairer presented at the Heidelberg Laureate Forum 2018. See if you can crack it!

November 19, 2019

Do you like the number e? Then you might like this curious fact discovered by Aziz Inan of the University of Portland.

Every year on February 7 maths enthusiasts celebrate e day. That's because the first two digits of the number e are 2 and 7, and because February 7, written the American way with the month first, is 2/7.

The year 2018 saw a very special e day: February 7, 2018 is written as 2/7/18, and 2, 7, 1, and 8 represent the first four digits of e. This coincidence occurs only once a century!

But there is more: the calendar date that follows 2/7/18, namely 2/8/18, coincides with the next four digits of e. So 2/7/18 and 2/8/18 put side by side as 27182818 constitute the first eight digits of e. This property makes the once-a-century special e Day 2/7/18 even more special!

To find out more about the number e and some interesting properties of its digits, see this article by Aziz Inan. You may also want to read the following Plus articles.

Maths in a minute: Compound interest and e — Compound interest is the curse of debt and the blessing of saving. Find out how it works and what it has to do with

Maths in a minute: Euler's identity —Here's a quick introduction to the beauty queen amongst mathematical formulas, which involves .

The making of the logarithm —The natural logarithm is intimately related to the number and that's how we learn about it at school. When it was first invented, though, people hadn't even heard of and they weren't thinking about exponentiation either. How is that possible?

Have we caught your interest? —This is a longer and more detailed article about compound interest and , complete with some history.

Polar power —Like spirals and flowers? Then you'll love polar coordinates and the pretty pictures they allow you to draw. One of those is intimately related to .

Radioactive decay and exponential laws —Arguably, the exponential function crops up more than any other when using mathematics to describe the physical world. This article looks at radioactive decay and exponential laws.

Light attenuation and exponential laws —This is another article that explores the appearance of exponential laws in nature. It explores light attenuation: the way in which light decreases in intensity as it passes through a medium. the a

September 20, 2019

One of our favourite authors, Wim Hordijk, recently sent us a digital postcard from the beautiful city of Vienna, where he had traced the steps of the eminent mathematician Kurt Gödel. Here is what he discovered.

Kurt Gödel.

Kurt Gödel was one of the greatest mathematicians of the 20th century. He made many important contributions to mathematical logic and philosophy, but is best known for his incompleteness theorems. Loosely speaking, the theorem states that the dream of phrasing all of mathematics in terms of a formal system, based on a set of axioms and the rules of logic, is bound to fail: there will always be statements that are true but whose truth cannot be proved within the axiomatic system itself. This abruptly ended a longstanding quest by some mathematicians to construct a set of axioms sufficient for all of mathematics (you can find out more in this Plus article).

Gödel was born in 1906 in Brünn, then part of the Austro-Hungarian empire (now Brno, Czech Republic). At the age of 18 he moved to Vienna, where he studied and worked from 1924 until 1940, and also took part in the famous Vienna Circle. He completed his PhD dissertation at the age of 23, and became a lecturer at the University of Vienna a few years later. In January of 1940, after the start of World War II, Gödel and his wife left Europe for good to start working at the Institute for Advanced Study in Princeton, USA, where he became close friends with Albert Einstein. He died in Princeton from self-imposed starvation in 1978.

The Kurt Gödel Research Center for Mathematical Logic (KGRC) of the University of Vienna was named after and in honor of Gödel, who proved his completeness and incompleteness theorems in Vienna in the years 1929-1931. For a long time the KGRC was housed in a beautiful building called the Josephinum on the Währingerstrasse, close to the main university building, and just a few doors down from one of the buildings where Gödel lived for a while.

The Josephinum on the Währingerstrasse in Vienna, where the KGRC was located. Image: Wim Hordijk.

In fact, Gödel lived in quite a few places in Vienna. During the roughly 15 years that he studied and worked there, he took up residence in seven different places throughout the city. On the KGRC website you can find a list of addresses where he lived (and when), together with a map indicating these locations. Moreover, each building where Gödel lived has a commemorative plaque next to its entrance.

The seven plaques at the buildings in Vienna where Gödel lived. Top row, left to right: Florianigasse, Frankgasse, Währingerstrasse, Lange Gasse. Bottom row, left to right: Josefstädterstrasse, Himmelstrasse, Hegelgasse. Images: Wim Hordijk.

With Vienna's excellent public transportation, it is actually possible to visit all of these places in one day, making Gödel's story come to life. But if you need more time, the building at Währingerstrasse 33 (just a few steps from the Josephinum) is now a hotel. So if you want to make it even more real, it is possible to spend a couple of nights in one of the places where Gödel once lived.

The building at Währingerstrasse 33, where Gödel lived as a student, is now a hotel. Image: Wim Hordijk.

The beautiful city of Vienna is already worth a visit in itself, but for mathematics aficionados the visible legacy of Kurt Gödel makes it even more worthwhile. The information presented here will hopefully serve as an inspiration for others to also experience some real maths history first-hand.

Wim Hordijk is an independent and interdisciplinary scientist and popular science writer. He has worked on many research and computing projects all over the world, mostly focusing on questions related to evolution and the origin of life. More information about his research can be found on his website.

March 28, 2019

Have you ever wondered what the world would be like without mathematics? And who are the people who make new mathematics and how they do it?

Who is your favourite mathematician of all time?

This competition, organised by the British Society for the History of Mathematics, is your chance to explore how mathematics has developed and achieved its status and who were the most important mathematicians in history who contributed to it. This year we would like you to concentrate on choosing one mathematician who has, in your opinion, been the most important person, your favourite, and to make the case for your choice — to explain his/her mathematics and to show their importance or what you think was special about it and them.

The British Society for the History of Mathematics (BSHM) believes that understanding where mathematics comes from and who has contributed to the development of mathematical ideas is an important part of understanding mathematics today. BSHM, working with Plus, invites secondary school students to explore this question and communicate their findings for a wide audience (age 16 upwards).

You could write an article (maximum 1500 words), make a short video (maximum ten minutes) or a multi-media project (maximum ten minutes).

The competition is open to all young people aged 11 to 15 and 16 to 19 who are in secondary education. A number of monetary prizes will be awarded, depending upon the quality and the number of entries. The maximum prize will be £100.

The deadline for entries is Friday, 1st September 2019. All the info about how to submit your entry and where to ask questions is on the BSHM website.

Winners will be notified to collect their prizes in London, at the Society's Gresham College meeting on the 23rd October 2019, and the recording of this will be posted on the BSHM website, with a link given also from Plus.

Good luck!