Mathematicians and physicists are trying to explain why the extreme values of random matrices—grids of numbers used in data science—behave almost identically to the extreme values of L-functions, a class of mathematical objects central to number theory. This matters because the two systems appear to have nothing in common. Random matrices describe noise in complex systems, from stock markets to quantum physics. L-functions encode deep properties of prime numbers. If their extreme-value statistics are truly governed by the same hidden rules, that points to a universal principle underlying many complex systems—a kind of mathematical law of large fluctuations. The research is fundamental science with no immediate practical application. It aims to build new probabilistic and analytical tools that could eventually help predict rare, extreme events in any system with many interacting parts: financial crashes, network failures, or material phase transitions. Past work on similar universal patterns in random matrices, for example, unexpectedly became the mathematical backbone of modern wireless communications and data compression. A deeper understanding here could one day lead to similar surprises.
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Complex systems are ubiquitous in nature and society. A system is generally comprised of a large number of agents (e.g., particles, traders, or numbers in mathematics) that interact with each other in a specific way. These interactions often occur locally. However, the large number of interacting agents can lead to dramatic and intriguing behaviour at large scale (e.g., water freezing, a crash in the market, or singular behaviour of a function). An important area of contemporary research in physics and in mathematics is to precisely describe and predict the behaviour of complex systems, and to classify them according to the general phenomena they exhibit. The research proposal focuses on two types of complex systems: random matrices that are important objects in probability, physics and data sciences; and L-functions in mathematics. It has been conjectured and numerically observed that these two seemingly very distinct classes of complex systems present very similar behaviour in the statistics of large values they produce. This is a mysterious connection that most likely hides fundamental principles. The proposal designs specific objectives to unravel the connection between these objects. It relies on new connections between statistical mechanics in physics, branching processes in probability, and L-functions in number theory. It is expected to produce new probabilistic and analytical tools to describe a large class of complex systems, and as such, extend the current understanding of extreme values.
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