# Abel Prize in mathematics 2024: the improbability of a coin tossed 1,000 times coming up heads in 600

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How can you earn 7.5 million Norwegian crowns (about €660,000) with mathematics?

If you are thinking about something like “discovering the fault of a roulette wheel” or “counting cards”, I am sorry to disappoint you: the probability in these cases is usually against us, no matter how much the cinema insists on stating the opposite.

Perhaps when you saw the “Norwegian crowns” thing you thought about the Nobel Prize, but it is given in “Swedish crowns” and furthermore, it seems that Alfred Nobel had little affection for pure mathematics and left no endowment to award it. We are talking, therefore, about another award, the Abel , awarded by the Norwegian Academy of Sciences and Letters in honor of the mathematician Niels Henrik Abel .

This year 2024, the Abel Prize has been awarded to the Frenchman Michel Talagrand “for his pioneering contributions to probability theory and functional analysis, with outstanding applications in mathematical physics and statistics.”

The description of these achievements, indicated in the awarding of the Shaw Prize to Talagrand himself in 2019 , was more precise: “for his work on concentration inequalities and supremes of stochastic processes and for his rigorous results for spin glasses.”

#### The probability of the coin in the works of Talagrand

These achievements have been highlighted in the press with a headline that we reproduce here: “The unlikelihood of a coin tossed 1,000 times turning out heads 600 times.” But how can this be deduced from the work for which Talagrand has been awarded?

Of the three achievements that were highlighted in the Shawn award and that are also part of the Abel award, two of them explicitly refer to Probability and Statistics, and the third is closely related. In particular, they all have to do with the study of so-called stochastic processes , a probabilistic concept that surrounds us without us noticing it.

For example, imagine that we are in a supermarket, in line at the checkout. Every 5 minutes we will count the number of people in it. This value varies at every moment, hence we call the “number of people in the queue” a “variable”. Furthermore, we use the adjective “random” because it has a behavior marked by chance behind it in which some values ​​will be repeated more than others.

But what interests us about this queue is not so much the number of people but understanding how its value evolves. Hence we talk about “process”. And it is very revealing to understand where the adjective that accompanies that term comes from: stochastic.

#### Able to guess

The word “stochastic” has its origin in the Greek “στοχαστικός” ( stokhastikós ), which means “able to guess.” An action, that of predicting or guessing, for which science has Statistics and Probability.

Returning to the supermarket, let’s suppose that our interest is to know how many people will be in line at a specific time and thus decide with some advance notice how many boxes should be open.

Another example that is often used a lot to talk about stochastic processes is the weather. Understanding how a storm evolves is essential to take appropriate measures to avoid disasters.

Some of these processes are easy to study, and are supported by basic rules such as those of the so-called Markovian processes (in honor of the Russian mathematician Andrèi Markov) , a type of process by which the state of the system only depends on what was happening. in the previous moment. In other words, the current number of people in the queue only depends on the number that was there 5 minutes ago and not on those that were there 10 or 15 minutes ago.

#### The moment of uncertainty

Other processes, however, are much more complex and depend on what we call dynamic systems. They involve mathematical equations that explain how the system changes (such as those we use to model epidemics ) in relation to many other variables that may have influence.

And be careful, because those variables can be very variable. That is to say, the conditions under which a storm develops, those that make a fruit grow, or those surrounding the toss of a coin can be extremely changeable and difficult to control, adding uncertainty to the system.

We now have all the ingredients to understand a little better what contributions Talagrand has received the Abel Prize for.

#### Spin glasses

Let’s start with the last and most complex: “his rigorous results for spin glasses” and his contributions to physics, according to the Abel Prize.

Spin glasses are a concept that appears within the theories of Italian physicist Giorgio Parisi, Nobel Prize in Physics in 2021 for the “ discovery of the interaction of disorder and fluctuations in physical systems from atomic to planetary scales.  .

It is a type of disordered physical system in which Parisi tried to understand the complex interaction between the atoms that make it up. Parisi’s intuition lacked mathematical formalization (and that makes people who dedicate themselves to this nervous). But Talagrand arrived with his knowledge of stochastic processes that could also be applied to this type of system and managed to formalize Parisi’s intuitions.

#### The maximum and minimum of a stochastic process

With respect to the “supreme of stochastic processes”, one of the great things that Talagrand has contributed has been the possibility of limiting how much a stochastic process can be worth – maximum and minimum value. In our examples, it can tell us what will be the maximum number of people in the queue or liters per square meter that a storm will leave us. Don’t tell me it’s not practical!

And we finally arrive at Talagrand’s first contribution: “concentration inequalities”, which have directly to do with the owner’s currency.

#### The moment of flipping the coin

Concentration inequalities help us understand how unlikely it is that, if we have varied data, we will diverge much from what is expected. By “we have varied data” what I mean is that our observations take into account all the variability that exists around a process.

In the case of the coin, each time we throw it we do it with a different force and, possibly, a different style. This means that if we expect to observe 500 heads in 1,000 tosses (which is why the coin has a 50% probability of heads), it is difficult for us to observe exactly 500. But how difficult?

What exactly these inequalities do is quantify how difficult it is to get away from that 500 and, specifically, with 1,000 launches.

Talagrand’s inequality states that the probability of getting more than 600 will be 1% divided by two million. This is because all the variability between the different forces or ways of tossing the coin is canceled, and the number of heads will be concentrated between 450 and 550, with a probability of 99.7%.

In short, probability can make money, even if the reason is far from what was expected.

Author Bio: Anabel Forte Deltell is Doctor in Mathematics and Professor at the Department of Statistics and Operations Research at Universitat de València

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