Reports from the Heart of the Machine
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The reviewers commented: "Chen-Cheng's groundbreaking work is extremely original and technically difficult, not only solving major problems in Keller geometry, but also providing profound insights into such nonlinear equations. It is foreseeable that this series of papers will become classics in the field of geometry and partial differential equations."
Just now, the University of Science and Technology of China announced that Professor Chen Xiuxiong, founding director of the Center for Geometric Physics of the university, and collaborator Cheng Jingrui have achieved "landmark results" in the field of partial differential equations and complex geometry, successfully demonstrating the core conjectures of two international mathematical communities that have been pending for more than 60 years.
Founding Directors of the Center for Geometric Physics, Prof. Chen Xiuxiong (middle) and Cheng Jingrui (left).
Specifically, they solved a fourth-order fully nonlinear elliptic equation that solved several well-known problems about constant scalar curvature measures and Calabi extreme value metrics on Keller manifolds, including long-pending mandatory conjectures and geodetic stability conjectures, which would have important implications for the study of geometric and partial differential equations. Two papers have been published in the internationally renowned journal Journal of the American Mathematical Society.
Fellow of the Royal Academy of Sciences, winner of the 1986 Fields Medal and inaugural Mathematical Breakthrough Prize, Simon Sir Donaldson believes that their work has provided numerous new examples of constant scalar curvature Keller metrics that will undoubtedly form the basis for a full understanding of the problem.
American Academy of Sciences Fellow Brian Professor Lawson said Chen and Cheng's recent series of papers were amazing and a substantial breakthrough in the field.
What problem did they solve?
In mathematics , a Kähler manifold is a manifold with a unitary structure ( a U (n ) - structure ) that satisfies an integrability condition. In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, and the three structures are compatible with each other. The existence of constant scalar curvature measures on Keller manifolds has been one of the central problems in geometry over the past six decades. There are three well-known conjectures about its existence – the stability conjecture, the coercive conjecture, and the geodetic stability conjecture.
The stability conjecture, which is limited to the Keller-Einstein measurement, is known as the Yau Chengtong conjecture, which was proposed by Chengtong Yau in the 1990s and pioneered by Chen Xiuxiong, Donaldson, and Sun Song. Their proof was approved by the academic community, and they won the Vibron Prize in Geometry (the highest award in the field of geometry and topology).
The necessity of mandatory conjectures and geodetic stability conjectures has become completely clear after the work of many famous mathematicians in the last two decades. However, proof of its sufficiency was considered unreachable until Chen-Cheng's work.
Solving a class of fourth-order completely nonlinear elliptic equations proves the existence of a constant scalar curvature measure. Chen-Cheng's job is precisely to prove the existence of such equation solutions under the assumption that K-energy is mandatory or geodetic stability. The study of such equations is extremely difficult, and for a long time industry experts generally did not believe that there would be a satisfactory theory of existence. Before Chen-Cheng's work, there were few suitable tools for dealing with such equations. The most important breakthrough of chen-cheng is to give a priori estimates of such equations and a strategy for successfully implementing the new continuous parameters proposed by Professor Chen Xiuxiong.
Experts believe that solving a class of fourth-order completely nonlinear elliptic equations, previously like an invisible curtain wall in front of mathematicians, Chen-Cheng's job is to "dig a hole" in the curtain wall, find a breakthrough without warning, not only find the solution of the equation, but also establish a set of methods for systematically studying such equations, providing a new tool for exploring the unknown mathematical world.
In addition, there are many other breakthrough results in Chen-Cheng's article. For example, they gave proof of the stability conjecture on ring-symmetric Keller manifolds, generalizing Donaldson's classical theorem on ring-symmetric Keller surfaces to higher dimensions. Regarding the proof of the general stability conjecture, the two authors present a series of profound problems and possible solutions in the article. Although there are still many difficulties to overcome, experts believe that a complete solution to the stability conjecture has become possible. In the two years since the preprint of the article was made public, a series of important developments have taken place.
Co-author profile
Professor Chen Xiuxiong is the "Wu Wenjun Chair Professor" of the University of Science and Technology of China, an internationally renowned geometric analyst, and became the founding director of the Institute of Mathematical Sciences of ShanghaiTech University in 2018. He graduated from the Department of Mathematics of the University of Science and Technology of China in 1987 and subsequently studied at the Graduate School of the Chinese Academy of Sciences with a master's degree. In 1989, he was sent by the State Security Agency to the University of Pennsylvania to pursue a doctorate, under the supervision of the famous geometer Calabi, and was funded by the National Science Foundation.
Professor Chen Xiuxiong has made a series of important progress in the field of partial differential equations and complex geometry in recent years. He and Simon The stability conjecture on Fano manifolds, demonstrated by Donaldson and Sun Song, is regarded by academics as the most significant breakthrough in differential geometry since Perelman solved the Poincaré hypothesis. In addition to the important results of this completion, he cooperated with Wang Bing to prove the weak tightness of the limit of the Fano Keller Ridge Flow, and then cooperated with Sun Song and Wang Bing to prove the uniqueness of the limit, and gave a new proof based on the Keller Ridge flow of Yau Chengtong's conjecture.
In January 2019, the American Mathematical Society awarded Xiuxiong Chen, Simon S. Donaldson, Son Song Oswald · The Oswald Veblen Geometry Prize, which recognizes their proof of a long-pending conjecture on Fano manifolds. In June, the Simmons Foundation was named Chen Xiuxiong's 2019 Simmons Scholar. In November of the same year, Professor Chen Xiuxiong was awarded the title of Principal Professor by the State University of New York Board of Trustees.
It is worth mentioning that Professor Chen's achievements in educating people are also very remarkable. The above-mentioned Sun Song and Chen Gao, who won the Green Orange Award some time ago, are both disciples of Professor Chen. In 2015, Chen Xiuxiong and Chen Gao collaborated to solve the "gravitational instantaneous" problem proposed by Hawking in 1977. In February 2021, Chen Gao's paper "The J Equation and the Deformation of the Supercritical Ermitt-Yang Zhenning-Mills Equation" was published in "New Advances in Mathematics", one of the four highest journals in the mathematical community, which aroused the attention of the international mathematical community and was first cited by Lawson, an academician of the American Academy of Sciences.
Professor Chen's collaborator, Cheng Jingrui, graduated from Tsinghua University with a bachelor's degree, received his Ph.D. in mathematics from the University of Wisconsin-Madison in 2018, and is now an assistant professor in the Department of Mathematics at the State University of New York at Stony Brook. Cheng Jingrui completed this work at a time when he had not yet graduated with a Ph.D., and his research potential is evident.
Thesis Link:
https://www.ams.org/journals/jams/2021-34-04/S0894-0347-2021-00967-0/home.html
https://www.ams.org/journals/jams/2021-34-04/S0894-0347-2021-00966-9/home.html
http://news.ustc.edu.cn/info/1048/77323.htm
https://mp.weixin.qq.com/s/jVIZkKmv_PhL8a9bm_yiuw