Prof. Rafał Latała, a specialist in probability theory from the University of Warsaw, has developed tools that have allowed for the proof of Talagrand’s hypothesis concerning Bernoulli processes. In practice, this means responding to questions about stochastic processes crucial for the development of machine learning, statistics, and probability theory. The mathematician from the University of Warsaw has just been awarded the FNP 2023 Prize in the field of mathematical-physical and engineering sciences.
The Bernoulli process is a type of random process that refers to situations where we are examining events associated with repeatedly performing a single random experiment with two possible, equally likely outcomes, such as flipping a coin. In the Bernoulli process, we can determine, for example, the probability that exactly three tails will occur in five coin tosses. The fundamental question for years remained what is the maximum value of such a process. At the end of the 90s, French mathematician Michel Talagrand formulated a hypothesis according to which there would only be two ways to estimate the upper limit of this process. The problem remained to conduct a proof of this hypothesis, something accomplished by Polish mathematician Rafał Latała, together with Prof. Witold Bednorz.
“This was a hypothesis about stochastic processes, that is, variables changing over time,” explains Prof. Rafał Latała, mathematician from the University of Warsaw, in an interview for Newseria Innowacje.
Stochastic processes are part of probability theory, a branch of mathematics that deals with random events. To prove the Bernoulli hypothesis, the Polish mathematician applied and combined many sophisticated methods, including majorizing measures and associated partition constructions, concentrative inequalities, Sudakov-type minorizing estimations, “greedy” induction algorithms, and maximal inequalities for random vectors.
“The result itself is outstanding, not only the conducted proof but also the very idea of how it was carried out. The idea of proof and the mathematical tools developed by Professor Rafał Latała for this purpose are nontrivial and may prove useful in solving many problems that we cannot even predict today,” reads the justification for the FNP 2023 Prize.
Michel Talagrand earmarked a $5,000 prize for anyone who could prove his hypothesis. This sum went to the scientists from Warsaw. Their results were reported in 2014 in the “Annals of Mathematics”. According to the justification for the FNP Prize, the question raised in Talagrand’s hypothesis, and found by Prof. Rafał Latała, is crucial for probability theory, statistics, and machine learning. But the significance of this achievement goes far beyond the field in which it was accomplished and affects vast areas of pure and applied mathematics, as well as many other, very different scientific fields.
“It would not surprise me if people conducting research in these areas used the results I obtained with colleagues,” says Professor Rafał Latała. “It is always difficult to answer the question of how the results of mathematical research can be used because my field is basic science, a very theoretical discipline. On the other hand, stochastic processes are not suspended in the air, they model some world. For example, the temperature on Earth, the river level, the position of a particle – these are all random variables changing over time. It is very natural to ask what is the maximum possible position, such as how high should a bridge be built or how high should embankments be so that the river does not exceed this level. In response to the question about applications, I can broadly say that my research concerns processes that also appear in everyday life, in practical applications.”
Prof. Rafał Latała specializes in probability theory and currently works in the field of random matrices.
The FNP Prize is often referred to as the “Polish Nobel Prize”. It is awarded to distinguished scientists for outstanding achievements and scientific discoveries that push the boundaries of knowledge and open new cognitive perspectives. The prize amounts to PLN 200,000.
