This year the Abel Prize has been awarded to the Frenchman Michel Talagrand. The Norwegian Academy of Sciences announced this today. The 72-year-old mathematician, affiliated with the Center national de la recherche scientifique (CNRS), will receive the prize – an amount of 7.5 million Norwegian krone (650,000 euros) – for his “pioneering contributions to probability theory and functional analysis, and the applications in mathematical physics and statistics,” said the jury of the unofficial Nobel Prize for Mathematics.
The announcement of the Abel Prize laureate could be followed via a livestream. “If I had heard that an alien mothership had landed in front of the city hall, I don’t think I would have been more surprised,” was Talagrand’s first reaction. “It was a shock. But a pleasant shock, of course.”
According to Eric Cator, professor of probability and statistics at Radboud University Nijmegen, Talagrand has “done brilliant things.” One of Talagrand’s achievements is his theoretical work on methods to create order where chaos reigns. Around 1980, the Italian physicist Giorgio Parisi worked on models for ‘spin glass’, a magnetic alloy of copper and iron that behaves chaotically on an atomic scale: the molecular magnetic elements do not hang together as in an ordinary magnet, but move according to a chance principle.
Spin glass is notoriously difficult to analyze. Parisi discovered a structure in the apparently chaotic way in which the magnetic fields of iron atoms orient themselves and was awarded the Nobel Prize for Physics in 2021. Parisi’s physical theory was not perfect, because it lacked rigorous mathematical proof. This aspect seemed far beyond what was feasible, but years later Talagrand succeeded. He was the first to provide Parisi’s discovery with a complete mathematical foundation. “That was really a tour de force, a great piece of work,” says Cator.
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Blind in one eye
At the age of five, Talagrand lost the sight in his right eye because his retina detached. Ten years later the same thing threatened to happen to his left eye, but doctors were able to save his eye. Months of hospital visits caused him to miss many school classes. But his father, who was a mathematician, taught him every day about his profession. His disadvantage at school in many subjects was more than compensated for by the large advantage Talagrand gained over his classmates in mathematics, although he once said that he was “never really good” at geometry.
When asked during the livestream about the achievement he is most proud of, Talagrand mentioned his research into Gaussian processes. The normal distribution, also called Gaussian distribution, describes an important continuous probability distribution. The well-known symmetrical ‘bell curve’, with higher concentrated values in the middle, has many applications: fish weights and errors made by measuring equipment are examples of normally distributed quantities.
If you consider large numbers of normally distributed variables at the same time, which may also be dependent on each other in a certain way (think of properties of large numbers of atoms), then the results of these variables appear to be very concentrated: the spread around the mean is smaller than you might initially think. Talagrand developed ingenious methods to prove this concentration, which can explain global properties of a collection of particles.
Normally distributed quantities
In the early part of the last century, probability theory was perfected with the introduction of a new idea from analysis, the concept of a ‘measure’. Talagrand’s contributions to measure theory concern the Gauss measure, which is used to describe normally distributed quantities. In a 2019 interview with the Gazette from the Société Mathématique de France, Talagrand was asked about his favorite results. From his younger years he called his research into ‘tau regularity of Gauss measures’. “I estimate the number of people who know both the definition of a Gauss measure and that of tau regularity at three, including myself,” he said. “The article has been cited exactly once… but of course there is always the silent hope that the ideas will one day become useful.”
Cator also acknowledges this: “Talagrand often walked in front of the troops. His work is considered very difficult. The impact of his work on others may therefore be less significant than was the case with previous Abel Prize winners.”