Celebrating Michel Talagrand
About five years ago, I was in conversation with a respected acquaintance who worked in an organization housed in the same building as ours. We were discussing Philip Tetlock, the political scientist whose groundbreaking work on improving the accuracy of probability judgments of high-stakes, real-world events has been lauded. I asked if it was ever possible to predict chaos. Random events are generally chaotic because there is seldom any pattern to them. If you live in Lagos, consider how traffic jams coalesce and disappear. One moment, everything is at a standstill, and the next, you cannot even discern what caused it and where the gridlock disappeared to. That’s how random events work; it rains today, and there’s no guarantee it will rain tomorrow.
This individual asked if I knew a man called Michel Talagrand. I said I had never heard of him. He told me that Talagrand is a French probability theorist who has dedicated his life to developing a deep and sophisticated understanding of such processes. I was intrigued and thereafter spent time reading up on his work on probability theory and its applications. It left me wondering why more people do not know about this brilliant mathematician. I chalked it up to individuals like him being poor promoters of themselves.
Imagine my delight when I received a Google Alert today informing me that Talagrand has been awarded the Abel Prize.
The Abel Prize is a significant accolade in the field of mathematics. It is the equivalent of the Nobel Prize for mathematics. In fact, the Abel Prize is directly modeled after the Nobel Prizes and is awarded annually by the King of Norway to outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and comes with a monetary award of $700,000.
So, what did Talagrand do to deserve this year’s award? It’s difficult for me to describe in a way that could be easily understood, but think about the way stocks and bonds gain and lose value. Today, Reddit puts its shares up for sale to the public for the first time. It priced them at $34 at a valuation of $748 million. But already, Reuters is reporting that it is set to open up to 53% above the initial public offering (IPO) price. Don’t be surprised if tomorrow, it drops to $25 a share. Of course, it is impossible to predict the exact behavior of such a system. However, how can we model them in terms of probabilities, characterizing outcomes as likely or rare?
That’s what Talagrand is an expert in. For instance, 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. By simulating the behavior of the river over extended periods, the effectiveness of different mitigation strategies can be assessed to evaluate their impact on reducing flood risk. 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. This allows him to preserve the dataset’s most important features while characterizing it in terms of just a few parameters.
Okay, it’s difficult to understand for a layman, but I promise you that other mathematicians and those in other fields find it very valuable, especially because the technique he developed is very general and therefore widely applicable. It is useful in Machine Learning and Data Mining, Finance and Economics, and even in Genomics and Drug Discovery, where gene expression data or molecular structures involve dealing with high-dimensional datasets. Talagrand’s generic chaining method is widely used.
But let me attempt to describe something else he was instrumental in which you may be able to identify with: If you flip a coin, you can’t predict in advance 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 (it’s been mathematically calculated). 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.
Even though flipping a coin is so random, that randomness cancels itself out. What seemed like a horrible mess is actually organized.
This phenomenon, known as the concentration of measure, occurs in much more complicated random processes, too. Talagrand came up with a collection of inequalities, a powerful mathematical tool, that make it possible to quantify that concentration and he further proved that it arises in many different contexts, outside of coin flipping.
Like his generic chaining method, Talagrand’s concentration inequalities appear all over mathematics. Consider an optimization problem where you have to sort items of different sizes into bins. 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. It is used in urban transportation systems, and internet traffic. That’s how much reception it has received in combinatorics, physics, computer science, and statistics.
These are two of his works I somewhat understand. He has become a respected figure in his field. Talagrand’s work “changed the way I view the world,” said Assaf Naor of Princeton University. 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.”
Interestingly, 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.
After two years of university, he had to choose between mathematics and physics. At the time, physics had no jobs for the foreseeable future so he chose mathematics. Even after he graduated, Talagrand could not decide what to focus on and kept dabbling in different fields. “For 10 years, I had not discovered what I was good at,” he said. But as soon he got into probability theory, he knew that was where he belonged. Since then, he has published hundreds of papers on the topic.
In the interview he had after being awarded, he confessed, and that could bring some comfort to some of us mortals, that mathematics did not come easily to him:
“I’m not able to learn mathematics easily. I have to work. It takes a very long time and I have a terrible memory. I forget things. So I try to work, despite handicaps, and the way I worked was trying to understand really well the simple things. Really, really well, in complete detail. And that turned out to be a successful approach.”
As a gesture to the field of mathematics, which has given him much, Talagrand has announced he would be using some of the money won through the Abel Prize and the $1.2 million he received when he won the Shaw Prize, another major math award in 2019, to found a prize of his own, “recognizing the achievements of young researchers in the areas to which I have devoted my life.”
