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. In 2025, he and collaborators demonstrated that Hilbert's tenth problem has a negative answer over the ring of integers of every algebraic number field. On July 19, 2026, Alpöge presented an explicit counterexample to the Jacobian conjecture in three-dimensional space, stating that it was discovered using the Claude Fable 5 AI model. 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. In 2010, he was a finalist in the Intel Science Talent Search for a computer-science project on the algorithmic analysis of MRI scans of blood vessels. His earliest publication, on a CUDA-optimized vessel-detection method, dates from the same period. He earned a Barry Goldwater Scholarship. The main-belt asteroid 25898 Alpöge is named after him. 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. His undergraduate thesis, The average elliptic curve has few integral points, was supervised by Jacob Tsimerman. 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. He received his PhD from Princeton University in 2020 with a dissertation titled Points on Curves, supervised by Manjul Bhargava.
Career After completing his doctorate, Alpöge held an NSF Postdoctoral Fellowship affiliated with Columbia University and Harvard from 2020, and was a Junior Fellow in the Harvard Society of Fellows from 2021 to 2025. He subsequently joined the artificial-intelligence company Anthropic as a member of the technical staff.
Research Alpöge works in number theory and arithmetic statistics, including the study of rational and integral points on elliptic and higher-genus curves, Selmer groups, and the distribution of arithmetic invariants of number fields.
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.
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 K / F {\displaystyle K/F} there exists an abelian variety over F {\displaystyle F} of positive rank whose rank does not grow after base change to K {\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. 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.
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. 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 C ∞ {\displaystyle C^{\infty }} 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. Proposed formalizations by Dean Cureton and by Boris Alexeev were subsequently posted on GitHub, and Philip Engel published some notes about the proof. Manon Bischoff reported on the finding in Spektrum der Wissenschaft, the German language edition of Scientific American. Alpöge published a 100-page document containing a proof of the positive solution.
Awards and honors Morgan Prize (2015) Sophia Freund Prize, Harvard University (2014) Captain Jonathan Fay Prize, Harvard Radcliffe Institute (2014)
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