ICM: "Mathematics has a surprising logic, both global and interconnected"

The ICM is the major congress for global mathematics. Held every 4 years in connection with the International Mathematical Union (IMU), it is also the occasion for awarding the Fields Medal, one of the most prestigious distinctions in mathematics, along with the Abel Prize. Russia's invasion of Ukraine made it impossible to hold the congress in Saint Petersburg as originally planned; the decision was ultimately made to hold it virtually from 6-14 July. Only two events proceeded in person in Helsinki (Finland): the IMU General Assembly, which is scheduled before each ICM, and the awarding of the Fields Medals and other prizes on 5 July as part of this occasion.

The ICM, a snapshot of the mathematical landscape

Held every 4 years, the 2022 edition of the ICM is the 29th since it began in 1897. It is marked by plenary conferences, as well as side conferences and events, Marie-France VIGNERAS, Professor at l’Université de Paris.  and aims to promote international cooperation in mathematics. Building on its success, registration to virtually attend the congress reached maximum capacity well before the event began. "The ICM is a fine opportunity for the mathematics community to meet, and offers a snapshot of the best and most dynamic work being done in mathematics. It is an occasion to follow mathematics that is distant from one’s own research field," recounts Alice Guionnet, a CNRS Senior Researcher at the Unit for Pure and Applied Mathematics at l'ENS Lyon,1  which she directed from 2016-2021.

"We must think of the people who will be listening"

Alice Guionnet is among the 34 plenary speakers invited by the ICM. This represents a genuine "honour" and an acknowledgment of the mathematics she helped develop—random matrices in particular—when, during her time as a student, probability "did not enjoy the same recognition as a field" as it does today. Much of her research focuses on large random matrices. This mathematical object appeared in statistics nearly a century ago in the work of the statistician John Wishart to analyse charts of noisy data. As this data is quite voluminous, this amounts to studying a random matrix with a dimension "tending to infinity."  Since then, large random matrices have been used in numerous domains of mathematics and physics, ranging from number theory and nuclear physics to statistical learning and neural networks. Guionnet is trying to better understand this object and to expand its theory, for instance by studying the properties in its spectrum, or by developing applications for it in other fields, such as operator algebra and statistical learning.

Another plenary speaker is Laure Saint-Raymond, who is a permanent Professor at the renowned Institute for Advanced Study, and a member of the Alexandre Grothendieck Laboratory.2  "The ICM is a genuine exercise in teaching. We must think of the people who will be listening, instead of showcasing those who are speaking," stresses the specialist in partial differential equations. She is interested in the mathematical modelling of gases and fluids, and is notably trying to understand how certain details can be ignored (typically on the molecular level), all while preserving a clear description of their dynamic on our scale of observation. "The fact that we can predict this dynamic at large scale is nothing short of miraculous when we consider the complexity of microscopic phenomena." During the conference she will present an overview of familiar results, but also numerous questions that remain unresolved: "Even though we understand certain things, much work remains to be done." These ICM plenary speakers must serve as ambassadors for their fields of research. "I will discuss random matrices, explain their beginnings, and the key ideas that enabled the topic to emerge, along with certain recent developments and open problems," Guionnet explains.

Supporting international cooperation

The congress will also provide an opportunity to discover different fields of mathematical research. "Mathematics has an overall coherence. We must not shut out areas of research that, on the face of it, seem far removed, as they could potentially nourish my own work. It is important to leave room for the unexpected," explains Saint-Raymond. For mathematics emerge first and foremost from collective effort and the confrontation of ideas, with their universality being their strength. Also present from the CNRS will be François LÊ, Associate Professor at l’Université Claude Bernard Lyon 1, who will lead the plenary session for the Montucla Prize awarded by the International Commission on the History of Mathematics, and Marie-France VIGNERAS, Professor at l’Université de Paris, Emmy Noether plenary session.

"What I enjoy in mathematical research is that we have great freedom with respect to our subjects. Meetings and creativity, along with a good deal of persistence, are important drivers for our research," affirms Guionnet, who as a student planned a future as an engineer, before discovering her passion for research at l’École normale supérieure. For Saint-Raymond, mathematics can be compared to an art, and requires curiosity and creativity: "There is elegance in mathematics. I like simple things, when a mathematical demonstration transforms into an image of the spirit.

Programme https://icm2022.abstractserver.com/program/#/program/1/horizontal

Streaming: https://www.youtube.com/c/InternationalMathematicalUnion

• Marie-France VIGNERAS, Professor at l’Université de Paris.
• 1UMPA/ENS de Lyon/ CNRS.
• 2CNRS/IHES.