Dorothy Lee '26

In conversational language, “randomness” describes unexpected, out-of-the-ordinary occurrences. Similarly, in mathematics, a random set of numbers refers to a set that is unpredictable and lacks a distinguishable pattern (Cepelewicz, 2024). While studying the indescribable may appear counterintuitive, mathematicians have made significant breakthroughs in understanding random processes through the lens of probability. 

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One such scholar, French mathematician Michel Talagrand, was awarded the Abel Prize 2024 on March 20th for his work in three areas: suprema of stochastic processes, concentration of measures, and spin glass. Talagrand focuses his studies on the Guassian distribution, more commonly known as “the normal distribution” or “the bell curve” (The Abel Prize, 2024). 

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When random processes follow a normal distribution, the values closer to the mean tend to be more common, while those further away are less common. Through a series of proofs, Talagrand determined several inequalities that placed limitations on the extremes values in a normal distribution. With these discoveries, scholars can now, for instance, calculate the minimum and maximum heights a river may possibly rise to and place safety measures to prevent the water overflowing its banks (Cepelewicz, 2024). In order to apply these methods to high-dimensional data sets as well, Talagrand devised a general chaining method that simplifies the parameters affecting the data, while simultaneously accounting for the maximum margin of error resulting from this simplification process (Cepelewicz, 2024).

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In addition to being bounded, random events, despite their individually unpredictable nature, can follow distinct patterns when viewed from a broader perspective. Known as the concentration of measure, this principle holds significant value to studies in probability and combinatorics (Cepelewicz, 2024). Consider a simple dice roll; although the outcome of a singular roll is indeterminable, after a thousand rolls, there will be a relatively even distribution among the six different sides, each consisting of around one-sixth of the total number of outcomes. Talagrand’s work involved determining inequalities for these concentrations, which have since been acclaimed for their widespread applications throughout various mathematical fields (Cepelewicz, 2024). 

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Talagrand’s final major achievement connects mathematics with physics, regarding the study of “spin glass,” or disordered magnetic matter, whose arrangement initially surprised physicists (The Abel Prize, 2024; Cepelewicz, 2024). Drawing from probability concepts, the mathematician determined and proved limits to spin glass behavior, completing a proof by Nobel Prize recipient Giorgio Parisi in the process. 

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According to an interview with Quanta Magazine, Talagrand’s works have been inspired by the desire to “understand something deeply, in a pure way, which makes the theory much easier.” He appreciates the increasing recognition for the area of mathematics he works in and hopes that “the next generation can start from there and make progress on their own terms” (Cepelewicz, 2024). As part of his efforts to enlighten students in mathematics, Talagrand authors textbooks on probability and quantum field theory and presents a series of mathematical questions and puzzles on his website (Cepelewicz, 2024; The Abel Prize, 2024). His career has transformed research and application of probability theory, and will continue to challenge future generations of mathematicians.

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References

Cepelewicz, Jordana. “Michel Talagrand Wins Abel Prize for Work Wrangling Randomness.” Quanta Magazine, 20 Mar. 2024, 

www.quantamagazine.org/michel-talagrand-wins-abel-prize-for-work-wrangling-randomness-20240320/. 

The Abel Prize. “Michel Talagrand Awarded the 2024 Abel Prize.” The Abel Prize, Mar. 2024, abelprize.no/article/2024/michel-talagrand-awarded-2024-abel-prize.