#503 – AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory

This felt like an emergency gathering: AI systems had released claimed solutions or major advances across problems from Navier–Stokes to unique games and quasi-Riemann. I wanted to know whether mathematics had hit a crisis, a golden age, or some unnerving mix of both.

I wouldn’t call it only a crisis. There’s real chaos here, and people can feel hope and dread at once. For millennia, we were carrying a candle through the woods; now we may have a flashlight, and mathematical knowledge could be entering an extraordinary era.

When the results landed, rumors had already circulated about unique games and related problems, so my first thought was: wow, it’s actually out. I’m thrilled by a tool that can teach me other fields in my own language, but the old roadmap for mentoring students has suddenly become much less clear.

I’m mostly hopeful because I’m an engineer: better mathematics could improve hearts, aircraft, energy, and all sorts of human technology. Still, the uncertainty is enormous—either we get an astonishing partner for curiosity, or we become dwarfed—and people who love formal rules are not naturally comfortable with that.

The route matters as much as arriving. But if another country announced teleportation, we would not shrug and say the journey is spoiled; we’d race to understand it. Knowing something can be done can also make us more curious, and much faster.

“Solving math” and “destroying math” are different claims. Even under a hypothetical solved axiom—anything a human-plus-AI team can do, AI can do alone—I don’t think pure math disappears, because a theorem sitting in a data center is not the whole value; understanding has to live in human minds.

We shouldn’t optimize for one visible output, namely a proved theorem. I can hold only so many ideas at once, so human work often produces abstractions and connections that travel to nearby problems. We use sloppy metaphors, new viewpoints, and structures—not merely formal certificates.

I like the doorman fallacy: reduce a complicated role to its most obvious task, automate that task, and imagine the person is replaced. A mathematician proves theorems, sure, but also builds theories, frames questions, makes concepts portable, and thinks deeply. If routine answers get cheap, those other dimensions become more valuable.

I’d put asking good questions one step before solving them. A worthwhile question has to be interesting, tractable, and interpretable. Humans are still choosing what deserves attention, and simplicity plus generality remain a gold standard—even as AI begins searching for compact laws in data.

The current proofs may combine known moves in breathtakingly long chains. Calling that a mere convex hull sounds dismissive; I prefer the semigroup image, where familiar tiles can be placed again and again until they reach surprisingly far. Still, creating a genuinely new definition feels like stretching the balloon’s boundary.

Even a result built from existing ingredients changes the landscape. The three-sum advance, for example, forced fine-grained complexity theory to reconsider assumptions that had organized a whole family of problems. Once a theorem arrives, young researchers inherit different questions; there’s no restoring the old map.

Formal verification changes what publication could be. Lean checks each logical step, while mathlib is the shared library that prevents everyone from rebuilding basic machinery. Natural-language AI proofs are not enough to trust, but a formal proof can be checked down to axioms—even if we still need humans to explain the story.

That distinction matters: engineering a proof and communicating it pull in different directions. In my analysis course, I begin with calculus intuition and only construct the real numbers when a theorem makes students need that machinery. Future papers may tell the path, the stuck point, and the key turn, alongside a formal artifact that confirms the details.

Quasi-RH stunned me because primes, the building blocks of multiplication, are governed through the zeros of an apparently remote complex function. The new result gives a whole zero-free vertical strip, but the full Riemann hypothesis needs the boundary pushed all the way to one-half; the present method cannot simply be extended there.

I was at the National Museum of Mathematics with whiskey in hand, refreshing my phone, when I saw quasi-RH. Then formalized Fermat’s Last Theorem arrived: thirteen million lines, generated with little apparent domain guidance. It was not clean, canonical library-quality mathematics, but it was a checked proof, and wildly beyond my expectations.

Navier–Stokes is momentum conservation for fluids, but a simple principle becomes ferocious when every tiny region interacts with every other one. The new three-dimensional construction uses smooth forcing to create a stretching, thinning vortex with finite energy yet unbounded local vorticity. That is a major theorem, though it doesn’t suddenly make ordinary fluid engineering easy.

The trillion-dollar challenge is still inverse design: I may want a wing with less drag or an artificial heart with better flow, but not know which shape produces it. Turbulence has important structure across many scales, and we still lack a small, useful description analogous to temperature, pressure, and density for gases.

Nature may offer clues. A moth controls complicated air without solving our equations, and robust strategies can work from partial information rather than perfect models. In theoretical computer science, that points beyond worst-case analysis: the hard part is often figuring out which real-world structure makes a practical problem easier than its adversarial version.

For students, I’d say curiosity is necessary but not sufficient; do things for reasons internal to you. The old career playbook may fail. Pretend you’ve already cleared the next status hurdle, then ask what you would genuinely build, learn, or contribute instead of flooding the world with proofs just to pad a résumé.

Keep the fundamentals: linear algebra, calculus, physics, history, and the ability to tell when an answer needs more scrutiny. AI can assist learning, but intelligence is not morality or wisdom. Also, boredom has value; sometimes curiosity and sustained focus show up only after the distractions quiet down.

My advice to math PhD students is still to learn the hardest, most abstract material they can manage. Use AI to interrogate a confusing paragraph, test yourself, and reach the next step of understanding—not to bypass the work. A calculator never made multiplication knowledge useless, and neither should these systems.

I hope we discover that, despite what machines can do, we still prefer doing important things with people. Research is meaningful, but it is not the foundation of my life. Going home to my kids, throwing a ball, being with people I care about—that is the foundation; the rest is cherry on top.

That’s a beautiful place to land. Intelligence may become cheaper, but curiosity, deep understanding, good judgment, weirdness, community, and love don’t become irrelevant. I’m grateful we could sit together in person and try to make sense of this strange, thrilling moment.