The 2019 Oswald Veblen Prize in Geometry will be awarded to Xiuxiong Chen, Simon Donaldson—both of Stony Brook University—and Song Sun, University of California, Berkeley, for their three-part series, Kähler-Einstein metrics on Fano manifolds, I, II and III, published in 2015 in the Journal of the American Mathematical Society, in which they proved a long-standing conjecture in differential geometry. (Photo, left to right: Xiuxiong Chen, Simon Donaldson, and Song Sun. Photo of Simon Donaldson by Nora Donaldson.)
In 1982 Shing-Tung Yau received the Fields Medal in part for his proof of the so-called Calabi Conjecture. He later conjectured that a solution in the case of Fano manifolds, i.e., those with positive first Chern class, would necessarily involve an algebro-geometric notion of stability. Seminal work of Gang Tian and then Donaldson clarified and generalized this idea. The resulting conjecture—that a Fano manifold admits a Kähler-Einstein metric if and only if it is K-stable—became one of the most active topics in geometry.
Chen, Donaldson, and Sun announced a complete solution of the conjecture for Fano manifolds in International Mathematics Research Notices in 2014, and full proofs followed in Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than $2\pi$, and Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches $2\pi$ and completion of the main proof, all published in 2015 in the Journal of the AMS.
As one nominator put it, This is perhaps the biggest breakthrough in differential geometry since Perelman’s work on the Poincaré conjecture. It is certainly the biggest result in Kähler geometry since Yau’s solution of the Calabi conjecture 35 years earlier. It is already having a huge impact that will only grow with time.
Biographical notes
XiuXiong Chen has been a professor of mathematics at Stony Brook University since 2009. He was an invited speaker at the International Congress of Mathematicians (ICM) 2002 in Beijing, a 2015 Fellow of the AMS, and a 2016 Simons Fellow in mathematics. Over his career, he has supervised around 20 PhD students in mathematics.
Simon Donaldson is a permanent member of the Simons Center for Geometry and Physics, Stony Brook University. Over his career he has supervised about 45 doctoral students, many of whom are now leading figures in mathematical research. Donaldson was awarded a Fields Medal in 1986 for his work on gauge theory and four-dimensional manifolds, and has made contributions to several other branches of differential geometry. He was an invited speaker at ICM 1983, 1986, 1998, and 2018, and was a member of the Inaugural Class of AMS Fellows in 2012.
Song Sun joined the faculty at University of California, Berkeley earlier this year. He received an Alfred P. Sloan Research Fellowship in 2014, and this summer was an invited speaker at ICM 2018 in Rio de Janeiro.
Response of the recipients
It is a great honour to be awarded the 2019 Oswald Veblen Prize for our work on Kähler -Einstein metrics. Our work builds on that of many others. In 1954, Calabi proposed his vision of far-reaching existence theorems for canonical metrics on Kähler manifolds—a vast extension of the classical theory for Riemann surfaces. The foundation for this vision came from the developments of complex differential geometry over the preceding decades by Kähler, Hodge, Chern and others. In its general formulation, involving “extremal” Kähler metrics, Calabi's problem remains to a large extent open, but in the case of Kähler-Einstein metrics, the existence theory is now in a relatively satisfactory state. A crucial breakthrough by S-T Yau, which famously dealt with the cases of negative or zero first Chern class, was recognized in the 1981 Veblen Prize. Many mathematicians have contributed to the understanding of the remaining “positive” case over the four decades since Yau's work. We feel very fortunate and privileged to have had the opportunity to play a part in this long story.
About the Veblen Prize
The Oswald Veblen Prize in Geometry is awarded every three years for a notable research work in geometry or topology that has appeared in the last six years. The work must be published in a recognized, peer-reviewed venue. The 2019 prize will be awarded Thursday, January 17 during the Joint Prize Session at the 2019 Joint Mathematics Meetings in Baltimore.
Find out more about the Veblen Prize and see previous recipients.
Full biography, response, and photo are available from the AMS Public Awareness Office.
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