Levent Alpöge

Sums of two rational cubes

With Manjul Bhargava and Ari Shnidman, Alpöge established the first proven bounds on the proportion of integers that can be written as a sum of two cubes of rational numbers. Their 2022 work showed that at most about five-sixths of integers are so expressible, combining geometry of numbers techniques with the circle method.^[17] The upper bound for the density of integers that are so expressible has been subsequently improved to 68.05% by Peter Koymans and Alexander Smith. ^[18]

Hilbert's tenth problem over number fields

In a paper published in Inventiones Mathematicae in 2025, Alpöge, Bhargava, Wei Ho, and Ari Shnidman proved that for any quadratic extension of number fields ${\displaystyle K/F}$ there exists an abelian variety over ${\displaystyle F}$ of positive rank whose rank does not grow after base change to ${\displaystyle K}$. This implies that Hilbert's tenth problem has a negative solution over the ring of integers of every algebraic number field—that is, there is no algorithm to decide whether a polynomial equation has solutions in that ring.^[3] The result gave an alternative completion of a program relating Hilbert's tenth problem to the rank stability of elliptic curves and abelian varieties initiated by Bjorn Poonen, Barry Mazur, Karl Rubin, and Alexandra Shlapentokh; the Mazur--Rubin-Poonen-Shlapentokh program had been completed in 2024 by Peter Koymans and Carlo Pagano, using instead elliptic curves. ^[4][19]

Jacobian conjecture counterexample

In July 2026, Alpöge announced on the social-media platform X that, using Anthropic's Claude Fable 5 large language model, he had found a counterexample to the Jacobian conjecture, a problem in algebraic geometry begun by Ludwig Kraus in 1884 and formally posed by Ott-Heinrich Keller in 1939.^[20][21] The example is a polynomial map from three-dimensional complex space to itself whose Jacobian determinant is the nonzero constant −2 but which is not injective, disproving the conjecture in dimensions three and above. The two-variable case of the conjecture remains open. The announcement was made outside of formal peer review, and the arithmetic of the example was independently checked by other mathematicians in the days that followed.^[2]

Hadamard matrices

On August 12, 2026, Alpöge announced on X that, together with Philippe Voinov and Saul Reynolds-Haertle and using Anthropic's Claude, he had constructed Hadamard matrices for all twelve orders below 2000 for which no such matrix had previously been known: 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948 and 1964.^[22][23] Order 668 had been the smallest open case of the Hadamard conjecture since 2005.^[22]

Carathéodory conjecture counterexample

On August 19, 2026, he announced an explicit ${\displaystyle C^{\infty }}$ counterexample for the Carathéodory conjecture checked with John-Paul Smith and Anthropic's artificial-intelligence system Claude.^[24] Once again the result was announced in a tweet^[25], but on this occasion Alpoge offered a more comprehensive backstory, asserting that "i literally wanted to just read about it but i did see the complicated literature situation and so in hopes i could help asked him if it was ok to look at with a model" before confirming that "i did wanna have it recorded somewhere public though" and then giving the example.

Hopf's problem

Alpöge announced an answer to Hopf's problem on X on August 23, 2026, stating that he constructed a complex manifold structure on a 6-dimensional sphere using one of Anthropic's models.^[26][7] The same day Alpöge published a 100-page document containing a proof of the positive solution on his personal website.^[27].

Proposed formalizations by Dean Cureton^[28] and by Boris Alexeev^[29] were subsequently posted on GitHub, and Philip Engel published some notes about the proof.^[30] Manon Bischoff [de] reported on the finding in Spektrum der Wissenschaft, the German language edition of Scientific American.^[31]

Navier–Stokes existence and smoothness

Alpöge worked with New York University mathematician Tristan Buckmaster [de] on the Navier–Stokes existence and smoothness problem, one of the seven Millennium Prize Problems. On August 15, 2026, the pair proved that the Euler equations, the frictionless counterpart of Navier–Stokes, can "blow up" in finite time, extending a forcing technique developed by Diego Córdoba [eu] and Luis Martínez-Zoroa [eu].^[8] After verifying the proof in Lean, the pair published a preprint establishing finite-time blow-up with smooth forcing for the incompressible porous medium equation, the 2D Boussinesq system, and the 3D incompressible Euler equations, using Anthropic's Claude and OpenAI's Codex in the work.^[8]

After word of the pair's progress reached OpenAI, a team led by OpenAI scientist Sébastien Bubeck used an internal model to find a solution for Navier–Stokes equations.^[32] On September 8, 2026, OpenAI announced a full solution to the problem that resembled the approach used by the pair.^[9] Buckmaster suggested OpenAI researchers may have accessed private work he had stored in a Codex session before publication.^{[8][33][9][34]}