Fields medallist Yu Deng on randomness and the future of mathematics

Mathematician Yu Deng, Professor at the University of Chicago, visited Bilbao last week to take part of the conference celebrating 40 years of the Analysis and PDE seminar, organised by the APDE group at EHU and BCAM. It was his first public appearance in Spain since being awarded the Fields Medal on 23 July 2026 at the International Congress of Mathematicians in Philadelphia. Speaking on the sidelines of the conference, Deng reflected on the work behind the award, the unexpected convergence of two research programmes, and the growing role of artificial intelligence in mathematical research.

For Deng, the medal recognises both his work and the fields of partial differential equations and probability that he represents. He is still adjusting to its consequences, but he already sees greater freedom in the way he can work. "I think it's good that I can now convey some of my opinions," he said, adding that he hopes to work with less stress and choose problems more freely.

Deng was recognised for a rigorous derivation of the Boltzmann equation, which describes the large-scale behaviour of a gas, from the microscopic dynamics of colliding hard spheres. The central difficulty is that the laws governing individual particles are reversible in time, while the Boltzmann equation is irreversible. "You need to find what becomes irreversible in this limit," Deng explained, "which is also what makes our proof difficult." Earlier work established the connection over short times. Extending it to long times required dividing the evolution into small intervals and tracing how irreversibility emerges at each stage.

The route to that result began with a different problem. In 2018, while studying Gibbs measures and random-data theory for the Schrödinger equation, Deng recognised the same diagrams and structures that appear in wave turbulence, the study of how energy spreads through interacting waves. That observation led to a research programme on wave turbulence and nonlinear dispersive equations in 2019 and 2020. Deng and his collaborators later realised that the framework could also be applied to systems of colliding particles. There is, he said, "to some extent, a one-to-one matching between the objects and methods". Although the technical details and limiting regimes differ, the overall scheme is "pretty much the same".

At the centre of both projects is a way to control diagrams of arbitrarily large size by balancing the growth of the quantities under study against the diagrams' combinatorial complexity. Deng expects this general approach to apply beyond kinetic theory and nonlinear dispersive equations. He pointed to quantum field theory as another area where similar problems arise.

Deng also discussed artificial intelligence's growing role in his work. "Even in my own research, I've been benefiting from this a lot," he said. He sees an emerging pattern in which researchers provide the ideas, guidance and overall framework, while AI works through technical details. In his view, this could bring "pretty significant acceleration" to mathematical research over the next few years.

His own plans remain focused on probability and partial differential equations. One major open problem asks whether introducing random initial data can yield almost-global well-posedness for supercritical equations. The techniques developed in Deng's recent work do not directly resolve the question, particularly over the long periods in which the propagation of randomness is poorly understood. "I really want to see if there are any good new ideas that help us with this particular problem," he said. For young researchers entering the field, he added, the abundance of open questions and the acceleration offered by AI make this an especially exciting time.

Asked what mathematicians in 2066 might say mattered most about the mathematics of 2026, Deng highlighted two directions he expects to shape the future of partial differential equations: a deeper connection with randomness and the development of computer-assisted proofs. "I think randomness is one of the main things that's important," he said. His interest in poetry offers a more personal echo of that outlook. Deng described moments of mathematical discovery that reminded him of particular poems. The connection, he said, is philosophical rather than direct, but the similarity is real.