Random processes take place all around us. It rains one day but not the next; stocks and bonds gain and lose value; traffic jams coalesce and disappear. Because they’re governed by numerous factors that interact with one another in complicated ways, it’s impossible to predict the exact behavior of such systems. Instead, we think about them in terms of probabilities, characterizing outcomes as likely or rare.
Today, the French probability theorist Michel Talagrand was awarded the Abel Prize, one of the highest honors in mathematics, for developing a deep and sophisticated understanding of such processes. The prize, presented by the king of Norway, is modeled on the Nobel and comes with 7.5 million Norwegian kroner (about $700,000). When he was told he had won, “my mind went blank,” Talagrand said. “The type of mathematics I do was not fashionable at all when I started. It was considered inferior mathematics. The fact that I was given this award is absolute proof this is not the case.”
Other mathematicians agree. Talagrand’s work “changed the way I view the world,” said Assaf Naor of Princeton University. Today, added Helge Holden, the chair of the Abel prize committee, “it is becoming very popular to describe and model real-world events by random processes. Talagrand’s toolbox comes up immediately.”
Talagrand views his own life as a chain of unlikely events. He barely passed grade school in Lyon: Though he was interested in science, he didn’t like to study. When he was 5 years old, he lost sight in his right eye after his retina detached; at age 15, he suffered three retinal detachments in his other eye, forcing him to spend a month in the hospital, eyes bandaged, fearing he’d go blind. His father, a mathematics professor, visited him every day, keeping his mind busy by teaching him math. “This is how I learned the power of abstraction,” Talagrand wrote in 2019 after winning the Shaw Prize, another major math award that comes with a $1.2 million bounty. (Talagrand is using some of this money, along with his Abel winnings, to found a prize of his own, “recognizing the achievements of young researchers in the areas to which I have devoted my life.”)
He missed half a year of school while he recovered, but he was inspired to start focusing on his studies. He excelled in mathematics, and after graduating from college in 1974, he was hired by the French National Center for Scientific Research, Europe’s largest research institute, where he worked until his retirement in 2017. During that time, he obtained his doctorate; fell in love with his future wife, a statistician, at first sight (he proposed to her three days after meeting her); and gradually developed an interest in probability, publishing hundreds of papers on the topic.
That wasn’t preordained. Talagrand began his career studying high-dimensional geometric spaces. “For 10 years, I had not discovered what I was good at,” he said. But he does not regret this detour. It eventually led him to probability theory, where “I had this other viewpoint … that gave me a way to look at things differently,” he said. It enabled him to examine random processes through the lens of high-dimensional geometry.
“He brings in his geometric intuition to solve purely probabilistic questions,” Naor said.
A random process is a collection of events whose outcomes vary according to chance in a way that can be modeled — like a sequence of coin flips, or the trajectories of atoms in a gas, or daily rainfall totals. Mathematicians want to understand the relationship between individual outcomes and aggregate behavior. How many times do you have to flip a coin to figure out whether it’s fair? Will a river overflow its banks?
Talagrand focused on processes whose outcomes are distributed according to a bell-shaped curve called a Gaussian. Such distributions are common in nature and have a number of desirable mathematical properties. He wanted to know what can be said with certainty about extreme outcomes in these situations. So he proved a set of inequalities that put tight upper and lower bounds on possible outcomes. “To obtain a good inequality is a piece of art,” Holden said. That art is useful: Talagrand’s methods can give an optimal estimate of, say, the highest level a river might rise to in the next 10 years, or the magnitude of the strongest potential earthquake.
When we’re dealing with complex, high-dimensional data, finding such maximum values can be tough.
Say you want to assess the risk of a river flooding — which will depend on factors like rainfall, wind and temperature. You can model the river’s height as a random process. Talagrand spent 15 years developing a technique called generic chaining that allowed him to create a high-dimensional geometric space related to such a random process. His method “gives you a way to read the maximum from the geometry,” Naor said.
The technique is very general and therefore widely applicable. Say you want to analyze a massive, high-dimensional data set that depends on thousands of parameters. To draw a meaningful conclusion, you want to preserve the data set’s most important features while characterizing it in terms of just a few parameters. (For example, this is one way to analyze and compare the complicated structures of different proteins.) Many state-of-the-art methods achieve this simplification by applying a random operation that maps the high-dimensional data to a lower-dimensional space. Mathematicians can use Talagrand’s generic chaining method to determine the maximal amount of error that this process introduces — allowing them to determine the chances that some important feature isn’t preserved in the simplified data set.
Talagrand’s work wasn’t just limited to analyzing the best and worst possible outcomes of a random process. He also studied what happens in the average case.
In many processes, random individual events can, in aggregate, lead to highly deterministic outcomes. If measurements are independent, then the totals become very predictable, even if each individual event is impossible to predict. For instance, flip a fair coin. You can’t say anything in advance about what will happen. Flip it 10 times, and you’ll get four, five or six heads — close to the expected value of five heads — about 66% of the time. But flip the coin 1,000 times, and you’ll get between 450 and 550 heads 99.7% of the time, a result that’s even more concentrated around the expected value of 500. “It is exceptionally sharp around the mean,” Holden said.
“Even though something has so much randomness, the randomness cancels itself out,” Naor said. “What initially seemed like a horrible mess is actually organized.”
This phenomenon, known as concentration of measure, occurs in much more complicated random processes, too. Talagrand came up with a collection of inequalities that make it possible to quantify that concentration, and proved that it arises in many different contexts. His techniques marked a departure from previous work in the area. Proving the first such inequality, he wrote in his 2019 essay, was “a magical experience.” He was “in a state of constant elation.”
He’s particularly proud of one of his subsequent concentration inequalities. “It’s not easy to get a result which tries to think about the universe and that at the same time has a one-page proof that’s easy to explain,” he said. (He recalls with delight that he once used a cab service whose owner recognized his name, having learned the inequality during a probability class in business school. “That was extraordinary,” he said.)
Like his generic chaining method, Talagrand’s concentration inequalities appear all over mathematics. “It’s amazing how far it goes,” Naor said. “Talagrand inequalities are the screws that hold things together.”
Consider an optimization problem where you have to sort items of different sizes into bins — a model of resource allocation. When you have a lot of items, it’s very difficult to figure out the smallest number of bins you’ll need. But Talagrand’s inequalities can tell you how many bins you’re likely to need if the items’ sizes are random.
Similar methods have been used to prove concentration phenomena in combinatorics, physics, computer science, statistics and other settings.
More recently, Talagrand applied his understanding of random processes to prove an important conjecture about spin glasses, disordered magnetic materials created by random, often conflicting interactions. Talagrand was frustrated that, though spin glasses are mathematically well defined, physicists understood them better than mathematicians. “It was a thorn in our foot,” he said. He proved a result — about the so-called free energy of spin glasses — that provided a foundation for a more mathematical theory.
Throughout his career, Talagrand’s research has been marked by “this ability to just step back and find the general principles that are reusable everywhere,” Naor said. “He revisits and revisits, and thinks about something from all kinds of perspectives. And eventually he puts out an insight that becomes a workhorse, that everyone is using.”
“I like to understand simple things very well, because my brain is very slow,” Talagrand said. “So I think about them for a very, very long time.” He’s driven, he said, by the desire to “understand something deeply, in a pure way, which makes the theory much easier. Then the next generation can start from there and make progress on their own terms.”
Over the past decade, he has achieved this by writing textbooks — not just about random processes and spin glasses, but also about an area he doesn’t work in at all, quantum field theory. He had wanted to learn about it, but realized that all the textbooks he could find were written by and for physicists, not mathematicians. So he wrote one himself. “After you can no longer invent things, you can explain them,” he said.