How a Fried-Chicken Question Helped a Mathematician Cut Through Time (and Win a Fields Medal)

Andrei Mihai

In 2021, inside a Korean fried-chicken stall in Providence, Rhode Island, mathematician Yu Deng found himself thinking about time. He and Zaher Hani had spent years studying turbulent waves, whose interactions multiply into increasingly tangled histories. The longer the waves evolved, the more unmanageable the mathematics became.

Then, in the fried-chicken stall, Deng asked a question that sounded almost too simple to be of any use: “What if we try to divide long time intervals into short time intervals?” he recalled to University of Chicago News.

Yet this simple approach paved the way for great things to happen.

Deng and Hani first made that strategy work for interacting waves. Later, with mathematician Xiao Ma, they rebuilt it for an ideal gas of colliding hard spheres. The resulting theorem strengthened a century-old bridge between Newton’s laws, Ludwig Boltzmann’s statistical theory of gases, and the equations used to describe fluids.

The Waves Came First

Wave turbulence appears in many physical systems, particularly in systems containing huge numbers of weakly interacting waves, from ocean surfaces to some quantum systems. It’s virtually impossible to follow every oscillation in such complexity, so instead, physicists use a wave kinetic equation to describe how energy is statistically redistributed among different wave modes.

But the equation itself needed a rigorous foundation. In the underlying model, waves can be separated into mathematical modes, but those modes do not evolve independently. Nonlinear interactions continually link their histories and expanding the dynamics produces increasingly elaborate diagrams, with each new interaction adding further branches and pairings. 

Over short periods, Deng and Hani had learned to control these expansions. Over longer ones, the number and complexity of the possible interaction histories threatened to overwhelm their estimates.

Dividing time was useful only if the proof could preserve the correlations crossing from one interval into the next. Otherwise, each short interval would falsely treat the system as though it had begun again with independent waves.

Deng and Hani still had to work out how to carry information from one interval into the next, identify cancellations among vast families of tree-like diagrams and cut the remaining diagrams into pieces they could estimate. The method they eventually developed allowed them to justify the wave kinetic equation throughout the lifespan of its regular solution – the first long-time derivation of its kind in a nonlinear collisional kinetic limit. Later, its overall architecture helped guide their “attack” on colliding particles.

Running a Gas Backwards

A gas is a state of matter where particles have large spaces between them and move quickly in random directions. This collection of atoms or molecules moves freely through space, far enough apart that they spend most of their time traveling rather than touching. Unlike particles in a solid, which remain locked near fixed positions, or those in a liquid, which stay close together, gas particles spread to fill whatever container holds them.

This large-scale behavior emerges from an enormous number of tiny motions and collisions. In 1900, German mathematician David Hilbert held a now-legendary speech at the International Congress of Mathematicians in which he challenged mathematicians to reconcile the large-scale behavior of fluids with the fact that they’re made from many small particles. In Deng, Hani and Ma’s mathematical model, each molecule becomes an identical hard sphere that moves in a straight line until it strikes another.

Hard spheres are simpler objects than turbulent waves. In the mathematical model, they travel freely until they collide, then rebound elastically, conserving energy and momentum. They do not rotate, vibrate, react chemically, or radiate as real molecules might.

Yet even this simplified gas is puzzling.

Film two hard spheres colliding and play the recording backwards. The reversed collision still obeys the same mechanical laws. Reverse every velocity in an entire gas with perfect precision and, in principle, all the spheres retrace their paths.

Do that same thing with a gas, and it hardly seems to make sense. A gas released from one side of a box ordinarily spreads throughout it. If every particle’s velocity were then reversed exactly, the gas would retrace its history and gather back into the original region. But that seems unimaginable in a practical sense. Boltzmann’s equation captures the familiar tendency towards equilibrium – but if microscopic mechanics allows both a trajectory and its reverse, why does only one direction describe normal macroscopic behaviour? That is the puzzle.

The answer requires moving between three levels of description. Newtonian mechanics tracks every sphere’s exact position and velocity. Boltzmann’s equation tracks a distribution: How many particles are likely to be found at different positions and moving at different velocities. Fluid equations go further, treating the gas as a continuous material described by quantities such as density, flow and temperature.

Mathematicians had long wanted to prove that Boltzmann’s statistical picture really follows from Newtonian collisions. Some 50 years ago, Oscar Lanford achieved a landmark result for a diluted hard-sphere gas – but only over a small fraction of the average time between collisions. The more collisions that happen, the more complex the problem gets. A particle strikes another, which later hits a third. Branching histories grow and some spheres meet again, creating recollisions and loops.

Cutting Time Without Erasing History

This is where Deng’s idea came in. He divided a long time interval into short layers, within which expansions could be controlled. But they maintained a structured record of the collisions and overlaps inherited from earlier layers.

To organise that record, they turned collision histories into graphs they called “molecules.” These are not the actual, physical molecules, but more like bookkeeping devices whose parts encode particles and the relationships created by collisions, free motion, and repeated encounters.

The most problematic graphs contained the collision histories that had to be reconnected. The researchers devised a cutting algorithm that separates a large molecule (the graph) into elementary components whose mathematical contributions can be estimated. The cuts are chosen so that the proof extracts enough control from the graph’s structure, particularly from the recollisions that made it difficult in the first place.

This idea may not seem all that complex, but behind it, there lies an almost 200-page paper with a dazzling amount of practical combinatorics. One of its central quantities is the cumulant, which, loosely speaking, measures the correlations that prevent a collection of particles from behaving independently. The proof carries these cumulants through time and shows that they remain sufficiently controlled in the limit.

So in essence, this novel algorithm finds safe places to cut the record of the past.

It also helps clarify the backwards-gas paradox. A dispersed gas whose velocities have all been perfectly reversed is not an ordinary, freshly prepared gas. Its particles possess fantastically precise correlations encoding their previous collision history. Those correlations guide them back towards the original arrangement.

So you can, in theory, reverse that gas. But you need an ungodly amount of molecular coordination. Change the reversed velocities, even slightly, and the resulting motion is no longer the exact reverse of the original trajectory.

This is still a simplified version of reality, but it shows, within limits, how Boltzmann’s irreversible-looking statistical evolution can remain valid over long periods despite reversible microscopic dynamics.

The Fields Medal

In July 2026, Deng received the Fields Medal, mathematics’ most prestigious award. The International Mathematical Union cited his rigorous derivation of the Boltzmann equation from hard-sphere dynamics, his derivation of wave kinetic equations, and his probabilistic work on nonlinear wave equations. The citation joined the two sides of this story: the techniques developed for turbulent waves and the long-sought bridge from colliding particles to the statistical behaviour of gases.

In a recent statement, Deng noted that this is a major achievement “for my work with collaborators and for the field that I am representing.” While Deng got the medal, the work and breakthroughs were collaborative. Deng and Hani developed the long-time strategy for waves. Deng, Hani, and Ma transformed it for hard spheres. Their proof also rests on decades of work by mathematicians who built the route from Boltzmann’s equation to fluid dynamics.

Deng himself was, perhaps unsurprisingly, always drawn to puzzles. He also came within a few games of becoming a pro Go player, but after he lost in a key tournament, he decided to focus on math. It was probably one of those moments that put him on a path for mathematical success.

The moment in the fried-chicken stall may have been one of those turning points. By itself, it solved nothing. But it suggested a way to attack a problem that had resisted mathematicians for decades. Years of collaborative work later, that simple question helped lead Deng and colleagues get one step closer to explaining how the smooth, irreversible world we observe can emerge from countless reversible microscopic motions.

The post How a Fried-Chicken Question Helped a Mathematician Cut Through Time (and Win a Fields Medal) originally appeared on the HLFF SciLogs blog.