Levent Alpöge

Levent Hasan Ali Alpöge (born April 1, 1992) is an American and Turkish mathematician. He is known for his work in number theory and arithmetic statistics. He was a Junior Fellow in the Harvard Society of Fellows and later joined the artificial-intelligence company Anthropic as a member of the technical staff. He announced a counterexample to the Carathéodory conjecture on August 19, 2026 using the model. Four days later, he announced a positive answer to Hopf's problem concerning the existence of a complex structure on a 6-dimensional sphere.

Early life and education Alpöge was born on April 1, 1992, and grew up on Long Island, New York, where he attended Half Hollow Hills High School West in Dix Hills, New York. He holds dual United States and Turkish citizenship.

Alpöge studied at Harvard University, where he received a Bachelor of Arts in mathematics and a Master of Arts in physics in 2014, graduating summa cum laude.

After earning a graduate fellowship from the National Science Foundation, Alpöge studied Part III of the Mathematical Tripos at the University of Cambridge as a Churchill Scholar, earning a Master of Advanced Studies degree in 2015.

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 there exists an abelian variety over of positive rank whose rank does not grow after base change to . 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. The result completed 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; a parallel unconditional proof was obtained independently by Peter Koymans and Carlo Pagano. 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.

Carathéodory conjecture counterexample On August 19, 2026, he announced an explicit counterexample for the Carathéodory conjecture checked with John-Paul Smith and Anthropic's artificial-intelligence system Claude.

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. The same day Alpöge published a 100-page document containing a proof of the positive solution on his personal website. .

Proposed formalizations by Dean Cureton and by Boris Alexeev were subsequently posted on GitHub, and Philip Engel published some notes about the proof. reported on the finding in Spektrum der Wissenschaft, the German language edition of Scientific American.

Awards and honors Morgan Prize (2015) Sophia Freund Prize, Harvard University (2014) Captain Jonathan Fay Prize, Harvard Radcliffe Institute (2014)

Selected publications