Computers, Geometry and Einstein

ELI5/TLDR

A mathematician at Oxford explains how computers help solve geometry problems that no human could crack alone, particularly finding special curved shapes called “Einstein metrics” on high-dimensional spheres. The key result: the only known way to find an Einstein metric on a 12-dimensional sphere is to let a computer numerically approximate a solution, then prove a real solution exists nearby. Computers have been quietly essential to mathematics since the 1970s, and the partnership is only deepening.

The Full Story

Computers in mathematics: older than you think

Lotay opens by surveying the recent AI-in-math headlines — DeepMind and OpenAI winning gold at the International Mathematical Olympiad, computer-verified proofs of Fields Medal work, AI-generated progress on the Navier-Stokes Millennium Prize problem — then gently notes that none of this is actually new. Computers and mathematics have been entangled since Turing. The four-color theorem was proved by computer in 1976, and fifty years later there is still no purely human proof. Mandelbrot’s fractal visualizations in the 1980s revived an entire dormant branch of complex dynamics simply by making it possible to see the objects.

“It depends who you ask maybe about this whether this is proved or not or proved to their satisfaction if it’s just proved by a computer.”

The quiet implication: mathematicians have been having an identity crisis about computational proof for half a century, and they are no closer to resolving it.

Three ways computers serve geometry

Lotay identifies three roles:

Visualization. Geometry is the study of shapes, but shapes in four or more dimensions are impossible to picture unaided. Computers render objects like the Clifford torus (a four-dimensional surface projected into 3D) and the tesseract (the four-dimensional cube) so that mathematicians can develop intuition about things they will never directly see.

Calculation. A physicist at Stanford is shown mid-equation, covering an entire wall. The equation is not finished. A computer handles this without complaint. Lotay notes an irony from the IMO: everyone expected AI to struggle with geometry problems, since geometry seems visual and intuitive. Instead, AI crushed them — by ignoring the pictures entirely, converting everything to coordinates, and solving algebra. This is, incidentally, exactly what research mathematicians do too. The AI just does it faster and without existential fatigue.

Numerical approximation. Want the surface area of an irregular blob? Break it into tiny triangles, compute each one, add them up. This “triangulation” trick is not just a computational shortcut — it is a foundational concept in topology. Count the corners, edges, and faces of your triangulation, and you can determine whether a shape has a hole in it without ever looking at it.

Einstein, Riemann, and the geometry of gravity

Einstein’s 1916 general relativity describes gravity not as a force but as the curvature of spacetime — a four-dimensional object combining the three dimensions of space with time. To build this theory, Einstein needed Riemannian geometry, a subject that Riemann invented in an 1854 lecture. That lecture was given for his habilitation (roughly, a tenure talk). Riemann decided to casually invent an entire new branch of mathematics for the occasion. Nobody paid attention. Riemann died. The lecture was published posthumously in 1868. Still nobody paid attention. It was only around 1900 that Ricci and Levi-Civita developed the framework into something usable — just 16 years before Einstein needed it. The timing was almost absurdly tight.

The trampoline analogy: place something heavy on a trampoline, and it dips. Walk into the garden, see the dip, and you know something heavy is there even if you cannot see it. Mass bends spacetime the same way. The bending changes what “shortest path” means — light no longer travels in straight lines near massive objects. Near a black hole, light bends so dramatically that you can see objects behind the hole. This is gravitational lensing.

Einstein metrics: a simple equation, a brutal problem

An “Einstein metric” is a shape where the average curvature (called Ricci curvature) is constant everywhere. The equation is compact: Ric = lambda times g, where lambda is a constant. This is elegant. It is also connected to physics: Einstein metrics describe vacuum solutions to Einstein’s field equations — the geometry of empty space around, say, a black hole. The constant lambda turns out to be the cosmological constant, the quantity whose small positive value is one reason physicists propose dark energy.

In two dimensions, finding Einstein metrics is straightforward. A sphere gives you positive lambda. A torus (donut) gives you zero. A double torus gives you negative lambda. Clean, tidy, done.

But here is the catch: in dimensions three and below, Einstein metrics must have constant curvature. They are, as Lotay puts it, “kind of boring.” Einstein metrics only become mathematically interesting in dimension four and above — which happens to be exactly the dimension of spacetime. Coincidence or not, it makes the subject physically relevant in precisely the regime where it becomes mathematically hard.

The dimension-by-dimension hunt on spheres

Lotay then runs the audience through a dimension-by-dimension survey of non-trivial Einstein metrics on spheres with positive cosmological constant. It plays out like a game show with no discernible pattern:

Dimension 4: None known. Nobody has found one. Nobody has proved there cannot be one.
Dimension 5 through 9: Infinitely many, proved by Christoph Bohm in 1998.
Dimension 10: Exactly three, proved by Nikonorov, Hau, and Wink — published only in 2024/2025.
Dimension 11: Infinitely many (Lu, Sano, and Tasin, 2024). “Infinitely infinite,” as Lotay puts it.
Dimension 12: Exactly one. Published this year. And this is the headline result.
Dimension 13 and all odd dimensions: Infinitely many.
Dimension 14: Unknown. Open problem. Lotay invites the audience to go home and try.

The pattern, such as it is, looks random. The gap between “infinitely many” and “we have no idea” can be a single dimension.

The computer proof in dimension 12

The strategy for the 12-dimensional sphere works like this:

Assume the metric has enormous symmetry. This simplifies Einstein’s equations (a ferociously difficult system of partial differential equations) into ordinary differential equations — still hard, but tractable for a computer.
Feed those ODEs into a computer. The computer produces a numerical approximation.
Prove, by hand (with mathematical rigor), that a genuine Einstein metric exists near the approximation.

“Most of the time this strategy fails miserably, but it happens to work in dimension 12.”

The computer does not prove anything. It finds a candidate. The human proves the candidate is real. Neither could do the other’s job. Lotay’s PhD student, Kushi Wang, has since used the same approach to find new Einstein metrics in four dimensions with negative cosmological constant — bringing the technique closer to physically relevant gravity.

Claude’s Take

This is a genuinely good public lecture. Lotay manages to explain Einstein metrics to a general audience without dumbing the subject down to meaninglessness, which is harder than it sounds when the punchline involves 12-dimensional spheres. The audience participation segment — guessing how many Einstein metrics exist on spheres of each dimension — is a clever pedagogical move. It demonstrates the sheer unpredictability of the answers better than any slide could.

The substance is solid. The historical claims check out: the four-color theorem’s computer proof in 1976, the Scholze liquid vector spaces formalization challenge, Viazovska’s sphere packing in dimension 8, DeepMind’s Navier-Stokes progress. The dimension-by-dimension survey of Einstein metrics on spheres reflects the actual published record, with the recent results by Bohm, Nikonorov-Hau-Wink, and Lu-Sano-Tasin being correctly attributed.

One thing worth noting: Lotay is careful to distinguish between “none known” and “none exist” for the 4-dimensional sphere. This is an important distinction that popular science often flattens. The fact that we cannot find non-trivial Einstein metrics on S4 does not mean they are not there — it means the problem is open. The lecture treats open problems as genuinely open, which is refreshing.

The computer-assisted proof strategy he describes — numerical approximation followed by rigorous existence proof — is a well-established technique in nonlinear analysis (sometimes called “computer-assisted proof” or “validated numerics”). It is not the same as saying “a computer proved a theorem.” The computer narrows the search space; the human closes the deal. Lotay explains this distinction clearly, which matters because the popular narrative around AI and mathematics tends to elide it entirely.

If there is a weakness, it is that the talk is more survey than depth. You leave knowing that an Einstein metric was found on S12 via computer, but not much about how the symmetry reduction works or what makes dimension 12 special compared to 14. That is a fair trade for a public lecture, but anyone hoping for technical insight will need to read the papers.