Yu Deng — In-Depth Study

The Problem: The Boltzmann equation, formulated in 1872, describes how gases behave at a macroscopic level — how temperature, pressure, and flow emerge from countless molecular collisions. It's one of the most important equations in physics. But mathematicians have long sought to prove that this macroscopic equation actually follows from the microscopic laws governing individual particles — to derive the Boltzmann equation from first principles.

The Challenge: Gases contain astronomically many particles (roughly 10²³ molecules in a single breath). Tracking every collision is impossible. The mathematical question is whether the collective behavior of these particles, under reasonable assumptions, converges to the Boltzmann equation as the number of particles grows. This is a problem of connecting two different scales of description — microscopic chaos to macroscopic order.

Deng's Breakthrough: Deng and his collaborators cracked a longstanding problem within this effort. They proved rigorous mathematical results connecting the equations governing gas motion from the smallest to the largest scales — a derivation that had eluded mathematicians for decades. As Benoît Pausader (Brown University, a collaborator) put it: "People thought this was, I don't know, a decade in the future."

Why It Matters: This work places a cornerstone of statistical physics — the idea that macroscopic laws emerge from microscopic dynamics — on firmer mathematical ground. It validates a fundamental assumption in physics: that we can describe gases without tracking every molecule.

The Problem: When many waves interact — think of ocean waves, light in a fiber-optic cable, or plasma waves — they exchange energy in complex ways. Wave turbulence theory describes how energy cascades through these interacting wave systems, moving from large waves to small waves (similar to how energy cascades in fluid turbulence). The wave kinetic equation is the mathematical equation that captures this process.

The Challenge: For decades, the wave kinetic equation was used by physicists as a phenomenological tool — it worked in practice, but mathematicians had not rigorously proved that it actually emerges from the fundamental equations governing individual waves (nonlinear dispersive equations). The gap between the physical theory and mathematical proof was significant.

Deng's Breakthrough: Deng and his collaborator established the long-time justification of wave turbulence theory, rigorously deriving the wave kinetic equation from nonlinear dispersive systems over extended time intervals. This work, published in Inventiones Mathematicae in 2023, showed that the wave kinetic equation is not just a useful approximation — it genuinely follows from the underlying mathematics of interacting waves.

Why It Matters: Wave turbulence appears across physics — oceanography, optics, plasma physics, and even quantum field theory. Having a rigorous mathematical foundation means physicists can trust the theory in new regimes, and mathematicians have a new framework for studying complex wave interactions.

The Problem: The nonlinear Schrödinger equation (NLSE) describes how wave packets evolve in nonlinear media — from light pulses in optical fibers to Bose-Einstein condensates. The random data problem asks: what happens when the initial state of the wave system is not precisely known, but is instead random? Real-world systems always have some randomness, so this question is physically natural — but mathematically extremely difficult.

The Challenge: Standard PDE theory assumes you know the exact initial state. When the initial state is random, traditional methods break down. Mathematicians needed new tools to handle the interplay between the deterministic wave equation and the probabilistic initial conditions — balancing, as Quanta Magazine described Deng's work, "randomness and order."

Deng's Breakthrough: With Andrea Nahmod and Haitian Yue, Deng developed two innovative probabilistic tools: the random averaging operator and random tensor theories. These yielded new insights into the long-time behavior of nonlinear Schrödinger equations, the propagation of randomness through wave systems, and stability properties of nonlinear dispersive PDEs. The IMU specifically cited these as bringing "new insights into long-time behavior, propagation of randomness, and stability."

Why It Matters: This work brings the flexible power of probability into the rigid, structured world of wave equations. It opens a new mathematical framework for studying systems where randomness is not noise to be eliminated, but a fundamental feature to be understood — with applications from quantum mechanics to optical communications.