Two renowned faculty members speak at International Congress of Mathematicians

PROVIDENCE, R.I. [Brown University] - For a mathematician, being invited to speak at the International Congress of Mathematicians (ICM) is a career-defining honor. Held every four years, the conference is a bit like the World Cup of mathematics - the best of the best.

At this year's conference, held in Philadelphia, two Brown University professors got that rare opportunity.

Georgios Daskalopoulos, a professor of mathematics, and Benoît Pausader, a professor and chair of the Department of Mathematics, presented their work on the second day of the conference, which runs from July 23 to 30. The pair joins three other Brown mathematicians who have spoken at ICM in recent years. Professors Kavita Ramanan, Richard Schwartz and Joseph Silverman spoke in 2022.

"Being invited to speak at the International Congress of Mathematicians is a great honor, and it also makes me think about how fortunate I have been in my life," Daskalopoulos said. "I don't think of this invitation as something I achieved entirely on my own. I have benefited enormously from the people who taught me, encouraged me and gave me opportunities along the way."

Among those people were Daskalopoulos' father, who was also a mathematician, and his Ph.D. adviser Karen Uhlenbeck at the University of Chicago. From his father, Daskalopoulos says he got an early interest in mathematics. From Uhlenbeck, he learned a "very unique way of approaching mathematics."

"I feel very grateful for those [people]," he said, "and very fortunate to have the opportunity to speak at the ICM."

His talk, co-presented with Johns Hopkins Professor Chikako Mese, was titled "Harmonic Maps in Singular Geometry and Rigidity." Much of geometry involves smooth surfaces, curves and familiar shapes, but there are also sharp corners, singular points and other kinds of irregularities. Daskalopoulos' work uses mathematical devices known as harmonic maps to study the intersection of smooth surfaces and irregular points.

"The surprising thing is that, with some modifications, many powerful ideas from the analysis and geometry of smooth spaces continue to work even when the spaces are not smooth," Daskalopoulos said. "In particular, harmonic maps can help us understand when a space is so rigid that its geometry is essentially forced to be unique."

The ideas connect to some deep questions in geometry, Daskalopoulos says, including the geometry of Teichmüller space, "which describes the collection of all possible shapes of a surface."

Pausader's ICM talk, titled "Stability Problems in Collisionless Kinetic Theory," deals with questions that arise in systems where individual objects exert forces upon each other without physically colliding. Dynamics like this happen among stars in a galaxy or charged particles in a plasma. In both cases, objects interact across distances through long-range forces - stars through gravity and plasma particles through electrostatic force.

In these many-body systems, the dynamics are dizzyingly complex, and mathematicians are working to understand how these systems evolve over long periods of time. Pausader's work focuses on several key phenomena in these systems, including Landau damping, a process where waves of disturbance in an electrostatic field naturally fade over time as energy transfers to the moving particles. Pausader's work helps to better understand and predict the nature of that decay.

Ultimately, research like this provides fundamental mathematical backing to help scientists better understand how plasma systems work, while shedding new light on how structure in the universe evolves over time.

Pausader said the ICM experience has been extra special for him, as it was announced that his collaborator, Yu Deng, was awarded the Fields Medal - the highest honor in mathematics. Deng and Pausader have worked together on partial differential equations (PDEs), which are used to describe how wave patterns evolve over time.

"It is very nice to see how far he has gone and so gratifying to see the vitality of PDEs recognized at the highest level," Pausader said of his collaborator. "As for ICM, I'd say that it is a very humbling experience for me to be able to participate."