2021 Cole Prize in Algebra to Chenyang Xu
November 10, 2020

The 2021 Frank Nelson Cole Prize in Algebra is awarded to Chenyang Xu. He leads a group developing an algebraic theory of moduli for K-stable Fano varieties and working out a radically new approach to the singularities of the minimal model program using K-stability. （photo by Allegra Boverman）

Xu’s（single-authored）paper “A minimizing valuation is quasi-monomial” proves a conjecture of Jonsson and Mustaţă on log canonical thresholds and also a conjecture of Li on normalized volumes. In a series of papers with various coauthors（in particular, the papers “Boundedness of moduli of varieties of general type” with Hacon and McKernan and “Uniqueness of K-polystable degenerations of Fano varieties” with Blum）, new powerful methods were introduced, and they established a general framework for moduli of K-stable Fano varieties and give detailed descriptions in many cases. This opens up a completely new area of moduli theory, which earlier was assumed to apply only to general type and Calabi-Yau varieties. This can be viewed as a vast expansion of the minimal model program initiated by Shigefumi Mori.

The prize is awarded for these particular papers:

[1]  C. Hacon, J. McKernan and C. Xu, “Boundedness of moduli of varieties of general type,” J. Euro. Math. Soc. 20（2018）, Issue 4, 865–901.

[2]  C. Li, X. Wang and C. Xu, “On the proper moduli spaces of smoothable Kähler-Einstein Fano varieties,” Duke Math. J. 168（2019）, 138–1459.

[3]  H. Blum and C. Xu, “Uniqueness of K-polystable degenerations of Fano varieties,” Annals of Math. 190（2019）, 609–656.

[4] C. Xu, “A minimizing valuation is quasi-monomial,” Annals of Math. 191（2020）, 1003–1030.

[5] J. Alper, H. Blum, D. Halpern-Leistner and C. Xu, “Reductivity of the automorphism group of K-polystable Fano varieties,” Invent. Math., to appear.

Response of Chenyang Xu:

Constructing moduli spaces to parametrize objects has always been one of the most powerful tools in algebraic geometry. D. Mumford settled the case in dimension one for curves. This was later vastly generalized to higher dimension to parametrize varieties with a negative Chern class, via the Kollár-Shepherd-Barron program, which has been an intellectual engine in higher-dimensional geometry since the late ‘80s, whose development intertwines with the minimal model program started by S. Mori.

It has been mysterious to algebraic geometers for a very long time how to construct a moduli space for varieties with a positive Chern class. Such varieties are called Fano varieties, named after the Italian mathematician G. Fano. After a period of experimental searching, it has eventually become clear that one can investigate the notion of K-stability defined by G. Tian and S. Donaldson to capture the existence of Kähler-Einstein metric, using the machinery from higher dimensional geometry, and build up a new field—the algebraic K-stability theory of Fano varieties. Then the moduli space of Fano varieties comes out of the theory, as the best reward. Under the local-to-global philosophy, one could expect a local K-stability theory for singularities, and such expectation is fulfilled by studying the geometry of the minimizer of the normalized volume function on the valuation space of a singularity, a picture far outside the scope of the traditional higher dimensional geometry.

I’m very glad that the committee recognized the field and I feel profoundly honored that they chose me to represent the area. It is still a relatively new area and I am very happy to see that there have been a number of young brilliant mathematicians working on it. I hope that the recognition by the Frank Nelson Cole Prize will spur further activities.

The Cole Prize gives me the precious opportunity to acknowledge the invaluable aid I have received from others. I would like to thank my advisor J. Kollár, who shaped my thinking of the moduli of higher dimensional varieties. I would like to thank C. Hacon and J. McKernan; through our collaborations I learned tremendously about the minimal model program. I also want to thank all my collaborators on this topic whom the recognition should also be associated with, especially C. Li and X. Wang, for suffering the pain together earlier when it was not clear to which direction the subject would move; as well as J. Alper, H. Blum, D. Halpern-Leistner, Y. Liu and Z. Zhuang, for their energy and ideas in our joint works. I want to thank Peking University, MIT and Princeton for providing me a wonderful environment to accomplish the research. Finally, I want to thank my family, especially my wife Xiaoyu, for her constant support.

Biographical sketch of Chenyang Xu:

Chenyang Xu was born in Chongqing, China, in 1981. He did his undergraduate studies at Peking University and his graduate studies at Princeton University, with János Kollár. He held a postdoctoral position at the Massachusetts Institute of Technology（MIT）. Xu was hired as a junior faculty at the University of Utah in 2011. He joined the Beijing International Center for Mathematical Research at Peking University in 2012 and was promoted to professor there in 2013. In 2018, he moved to MIT, and in 2020 he became a professor at Princeton University. Xu’s main research area is the birational geometry of higher dimensional algebraic varieties, and he likes to explore its connections to other fields.

About the prize:

The Cole Prize in Algebra is awarded every three years for a notable research memoir in algebra that has appeared during the previous six years. This prize and the Cole Prize in Number Theory were established in 1928 to honor Frank Nelson Cole（1861–1926）on the occasion of his retirement as secretary of the AMS after 25 years of service. He also served as editor-in-chief of the Bulletin for 21 years. The 2021 prize will be recognized during the virtual 2021 Joint Mathematics Meetings in January 2021.

数学快讯：许晨阳获得美国数学会科尔代数奖

当地时间2020年11月10日，美国数学会发布公告，北京大学数学科学学院院友许晨阳因在K-稳定性Fano簇的模理论以及使用K-稳定性研究极小模型纲领中的奇点方面的贡献，获得2021年弗兰克·尼尔森·科尔代数奖（The Frank Nelson Cole Prize）。该奖项将在2021年1月举行的联合数学会议上颁发。

许晨阳的论文“A minimizing valuation is quasi-monomial”（独立作者）证明了Jonsson和Mustaţă关于log-典范阈值的猜想，以及2000级院友李驰的关于体积函数的猜想。在一系列与他人合作论文中（特别是与Hacon、McKernan合作的“Boundedness of moduli of varieties of general type”和与Blum合作的“Uniqueness of K-polystable degenerations of Fano varieties”），他们提出了新的有力方法，建立了一个关于K-稳定的Fano簇的模空间的通用框架，并对很多情况给出了详细描述。这些成果为模空间理论开辟了一个全新的领域，因为此前专家们认为模空间理论只适用于一般型的代数簇和Calabi-Yau簇。这可以被视为由Shigefumi Mori首创的极小模型纲领的一个巨大扩展。

许晨阳1999年保送进入北京大学数学科学学院学习，2002年、2004年分别获学士和硕士学位；2008年获美国普林斯顿大学博士学位；2012年回到北京国际数学研究中心任教；2018年秋任美国麻省理工学院教授；2020年任美国普林斯顿大学数学系教授。曾获2013年求是基金会杰出青年科学家奖、第十三届中国青年科技奖、2016年ICTP拉马努金奖、2017/2018庞加莱讲座教席、2017年未来科学大奖之“数学与计算机奖”以及2019年科学突破奖—数学新视野奖；2018年8月受邀在国际数学家大会上作题为“Interaction between singularity theory and the minimal model program”的报告，回顾了关于奇点理论的最新研究，重点是它们与极小模型纲领的关系，包括了对偶复形的构造与性质，对数典范阈值上升链猜想的证明以及关于klt奇点的“局部稳定性理论”的最新进展。

2020年北大数学捷报频传，姚珧（03级本科）、黄皓（03级本科）、马临全（04级本科）、苏炜杰（07级本科）获斯隆研究奖；何旭华（97级本科）、刘钢（05级硕士）获第二届科学探索奖；恽之玮（00级本科）获西蒙斯学者奖；30名北大数学人获阿里巴巴数学大奖……百余年来，北大数学星光璀璨，人才辈出，一代代数学家怀着对科学研究的热情、专注和坚持，前赴后继，永攀科学的高峰，不断得到国际数学界的认可。以黄金一代为代表的青年数学家辛勤耕耘，成为国际数学研究舞台的生力军，也点亮了一批批年轻北大数学人的科学梦想。我们期待更多北大数学人在专业领域持续探索，实现新的突破！

科尔代数奖（Cole Prize in Algebra）

科尔代数奖是美国数学会在代数方面的最高奖，每三年颁发一次，奖励过去六年出现的杰出的代数研究成果。院友许晨阳是第一位获得科尔代数奖的中国人。