A few years ago, when I first started talking about mathematics in public, a professor of algebraic number theory pulled me to one side and gave me some sharp words of warning: “The world fears mathematicians, Hannah. Let’s keep it that way.” He was being, to put it mildly, a bit of a snob. Mathshad an intimidating mystique to maintain, and I was apparently jeopardising it by titting around on the internet making videos and podcasts, rather than focusing on a perfectly respectable research career.
I have thought a lot about that comment since. Partly because I have spent the intervening years doing everything I can to completely ignore him (and getting a _New Scientist_ column feels like a rather lovely reward for doing so). But buried beneath his snobbery, I have come around to the idea that he was right about something.
And I have never felt that more than in the past few weeks, since a swarm of AI agents have been busy solving our most famous long-standing problems. I’m talking, of course, about OpenAI’s announcements over the past few weeks that it has found proofs for hundreds of mathematics problems that had previously resisted human attempts to solve them. The furore began, most notably, with a proof (by way of finding a counterexample) that the Navier-Stokes equations can break apart.
This particular gem was such a big deal to the mathematical community that at the turn of the millennium, the Clay Mathematics Institute promised they’d give $1 million to whoever managed to solve it. Those Navier-Stokes equations are what I did my PhD on. They are basically Isaac Newton’s second law of motion: F=ma, but applied to fluids.
They are, in my not entirely unbiased opinion, staggeringly beautiful, able to articulate everything from the drip of a water droplet on a leaf and the churn of an entire atmosphere to the gases swirling around in a galaxy in the sky. The thing is, these equations have a tricky little bit that pops out of the acceleration part, which contains a feedback loop within it. The faster fluid flows, the faster the gradient can get between two adjacent streams, and so the faster fluid flows.
It’s a bit like the fluid version of putting a microphone next to a speaker. The feedback loop can get louder and louder until everything ends up breaking. Or, in the case of the Navier-Stokes equations, the speed of the flow heads off to infinity.
Extract — continue reading at the source.
