Xinyi Yuan

Peking University

Work description: 

Xinyi Yuan has made fundamental contributions to arithmetic geometry. His pioneering work on arithmetic bigness in Arakelov geometry, first developed in his Ph.D. thesis and culminating in a landmark publication, has transformed the study of small points on algebraic varieties. More recently, he established the uniform Bogomolov conjecture over global fields, and further proved a striking quantitative form of the uniform Mordell conjecture originally proposed by Barry Mazur.

Arithmetic geometry studies algebraic equations over the integers using methods from algebraic geometry. It is a modern development of a subject dating back to Diophantus in the third century. Landmark achievements in the field include the proof of Fermat's Last Theorem by Andrew Wiles in 1994, the proof of the Mordell conjecture by Gerd Faltings in 1983 (for which he received the Fields Medal in 1986), and the proof of the Bogomolov conjecture by Emmanuel Ullmo and Shou-Wu Zhang in 1998.

Yuan developed the theory of arithmetic bigness. Generalizing the foundational work of Szpiro, Ullmo, and Zhang, he introduced powerful new techniques for studying the equidistribution of small points on arithmetic varieties. His theory has become a fundamental tool in both arithmetic geometry and arithmetic dynamics. He proved the uniform Bogomolov conjecture over global fields by his theory of arithmetic bigness. Furthermore, in a collaboration with Jiawei Yu and Shengxuan Zhou, he obtained a quantitative and effective version of the uniform Mordell conjecture proposed by Barry Mazur in the 1980s. While Dimitrov, Gao, Habegger, and Kühne established Mazur's conjecture in 2021, Yuan's works provided a powerful geometric perspective and proved explicit quantitative bounds, representing a major advance in Diophantine geometry.

Beyond his influential works mentioned above, Yuan has made major collaborative contributions with Shou-Wu Zhang and Wei Zhang, including extensions of the Gross–Zagier formula and the proof of the averaged Colmez conjecture. These results have played important roles in subsequent breakthroughs on special cases of the Birch and Swinnerton-Dyer conjecture and the André–Oort conjecture.

Xinyi Yuan is a Professor of Mathematics at Peking University. He was born in 1981 in Macheng, Hubei Province, China, received his B.S. from Peking University in 2003, and received his Ph.D. from Columbia University in 2008.