September 20, 2020

# Abel 2020 Prize for Mathematics of “Mix and Shake”

Every year, when the Nobel Prizes are awarded in October, someone misses the presence of a “mathematics Nobel”. There are even legends about why it doesn’t exist, but let us talk today about science, not folklore. Mathematics also has its big days, and one of them always comes in March, when the Norwegian Academy of Sciences and Letters announces the winners of the Abel Prize. The award is named after a titan of the business, Norwegian Niels Henrik Abel (1802-1829), and is awarded each year to mathematicians, usually veterans, who have made unusual contributions to the field. It is accompanied, in addition to the honrilla, of more than 600,000 euros.

And without a doubt, this year’s winners are extraordinary figures. The Israeli-American Harry Furstenberg and the Russian-American Grisha Margulis are two of the exponents of a little-known field, the ergodic theory, but which thanks to them has spread far beyond its limits and has become a tool for solve problems in other branches of mathematics. This ability to “cross borders” is one of the most fascinating qualities of mathematics, and serves as a reminder that there is only a mathematics, but we express it in many different languages.

## A theory of mixing and shaking

Ergodic theory is an old acquaintance of physicists and, indeed, of any of us as well. We have all once added sugar to a glass of milk, a coffee or a cup of tea, right? We add a tablespoon, or two, or three, depending on how sweet we like, and then stir, so that it mixes. We know that, from the beginning, the amount of sugar we have added is correct: we know our glass, and we have the proportion under control. But stirring is necessary, because otherwise it will remain in the background. This is the ergodic theory: finding efficient ways to mix systems, and comparing the properties of the mixed and unmixed system.

More formally, what we do in that theory is to study finite systems, like the glass of milk. We look at a property of those systems; let’s say, in the sugar ratio. The ratio is the same in the stirred glass and without stirring, that does not change. And we look for processes, things that we can do to the system, that do not change neither the quantity nor the nature of the components. That is, we do not want to change part of the milk for water: that is not ergodic. We don’t want to add more sugar, that’s not ergodic either. Stir, however, is fine: we are not changing anything, we are just rearranging the things inside the glass.

An important result of ergodic theory is that if a process efficiently mixes the components of the system (and there are very precise ways of defining what “mixing efficiently” means), then just by applying that process many times all the properties of the system will tend to their value medium at all points in the system. In the case of our glass of milk: if we stir the sugar well, it will end up distributed evenly throughout the glass. Of course, in the case of a glass of milk this is a no-brainer and there is no need to use bombastic words like “ergodic”. But the power of these mathematics is that they define very precisely what processes are ergodic, and allow us to “go through” the systems and find out properties that would otherwise be off limits.

Let’s look at some of those “hidden” properties taking advantage of the curriculum of one of this year’s winners.

## Shaking the numbers

In 1936 the Hungarian mathematicians Paul Erdős and Pál Turán were analyzing the properties of sequences arithmetic of whole numbers. These sequences are essentially those in which each number differs from the next by the same amount as the previous one. For example, {2, 4, 6, 8} would be an arithmetic sequence in which the difference would be 2, and {0, 5, 10, 15} would be another in which the difference would be 5. These sequences are very abundant in integers, but Erdős and Turán asked each other a question: will we continue to find them if we eliminate any of the integers? For example, we could remove all ends in 5, or all multiples of 2.

Erdős and Turán thought that as long as we don’t remove “too many” numbers (and gave a strict definition of what this means) we will continue to find arithmetic sequences. In fact, we can find them as long as we want, although depending on what numbers we remove they may not be infinite. Erdős and Turán were unable to prove this claim, so they left it as a conjecture. Forty years later, in 1975, another Hungarian, Endre Szemerédi, proved the conjecture, which has since been called Szemerédi’s theorem.

Szemerédi’s demonstration was prodigious, extraordinarily complex, and required a powerful combinatorial charge, which is the hallmark of Szemerédi. Only two years later Harry Furstenberg presented an alternative demonstration using ergodic theory. A demonstration that, at first glance, seems insultingly simple.

What Furstenberg showed was that if we have any collection of integers and an ergodic mixing process we are always going to find that the initial set is going to have common elements with the mixed set once, or the mixed set twice, or with the whole mixed three times. What’s more, these mixed sets are also going to have elements in common with each other, and in some cases all of them they will have common elements. The truth statement, of course, is a bit more complicated, but the idea is simple: if the process is ergodic there will always be elements that “come back” after a sufficient number of mixes.

And what does this have to do with Szemerédi and arithmetic sequences? Well, very simple: one of the processes that Furstenberg properties fulfill is, simply, “let’s add 1 to each number in our collection”. The result of Furstenberg implies that, whatever the set we have, we only have to add 1 to its numbers enough times to end up finding some of the numbers that we had at the beginning. This is precisely the definition of arithmetic sequence.

Let’s see it with an example. Let’s take the sequence {2,4,6,8}. If we add 1 to each element we have {3,5,7,9}. Wow, they don’t have any elements in common, but don’t panic: Furstenberg’s theorem says you have to mix enough times. Let’s do it again: we get {4, 6, 8, 10}. Now yes: 4, 6 and 8 have “returned” after applying the process twice. Logical, because it was a difference sequence 2. Some of them will come back more times: by applying the process 4 and 6 times. All multiples of 2, in good logic.

Thus, a result on “mixing things” has turned out to give us information about relationships within a set of numbers. This was not the only foray into ergodic theory outside its comfort region: a result due to Margulis and also published in 1977 allowed to better understand the properties of a discrete group immersed within a larger group thanks to the fact that the elements of the groups are “mixed” ergodically.

Furstenberg and Margulis’ careers are full of extraordinary work, and what we’ve pointed out here is just a quick look at some deep and beautiful math, showing us that seemingly far-flung concepts may be related. This is, in short, the epic of mathematics: the search for new angles to understand what we do not know, but also to re-understand what we already believed we knew.

## DON’T COLLECT IT

• The Abel Prize is not “the Nobel of mathematics”. Although it resembles the Nobel in several respects (for example, it is awarded every year, and its endowment is high) it is still a young award, as it was first awarded in 2003. The most prestigious award in the field is probably The Fields Medal, but it is awarded every four years and is only awarded to mathematicians under the age of 40, so it doesn’t look too much like the Nobel.
• There is a curious story according to which the Nobel Prize for Mathematics does not exist due to a mess of skirts that a mathematician had with a woman with whom Alfred Nobel was in love. There is no historical evidence that such a thing happened, and therefore must be considered an urban legend.
• Throughout this article we have used very little rigorous language that will surely make all mathematicians stand on end. We recommend that the interested reader browse the references, where they will find much more precise descriptions of the works of Furstenberg and Margulis.