Director’s Main Achievements

丘成桐,一位伟大的数学家

 

季理真

密歇根大学

 

 

在过去半个世纪,几何分析已发展成为数学中最活跃和最重要的领域之一。数学与物理的密切联系,解决了许多看似无法逾越的问题,开启了新的研究领域,改变了人们对于数学和物理的看法。丘成桐是这些振奋人心的发展中最活跃的领军人物和实践者。他的贡献,他的引领和他的著述,使得这些学科领域拥有今天的面貌,永久影响了数学的发展。

 

虽然20世纪是数学发展史中成果最为辉煌的世纪,但是并不确定的是,有多少人,哪些理论和定理会永久成为数学史的一部分。另一方面,毫无疑问,作为数学家,丘成桐拥有伟大的原创性,宏大的视野和强有力的技巧。他是20世纪最杰出的数学家之一。他所发展的几何分析学科,他对于卡拉比猜想的解,卡拉比—丘成桐流形将会是未来几代数学家和物理学家手中的基本工具。

 

原创性与技巧

 

丘成桐率先提出并强调的一个观点是,偏微分方程需与几何相结合(包括微分几何,复几何与代数几何)。他与合作者的许多深刻工作表明,只有通过几何,才能透彻理解诸如蒙日—安培方程等复杂的非线性方程。另一方面,可以利用非线性方程,以及指标定理的线性理论来有效构造几何结构。这也几乎是仅知的构造几何结构的方法。

 

也许最惊人的例子,是丘成桐关于凯勒—爱因斯坦度量存在性的卡拉比猜想的证明,并在其基础上解决了代数几何中几个长期悬而未决的难题。卡—丘流形对数学物理,特别是弦论,有关键的贡献。其他的例子包括汉密尔顿的里奇流在三维流形几何拓扑中的应用,Taubes Uhlenbeck 关于四维流形上反自对偶联络和反自对偶度量存在性的粘合技巧,以及Uhlenbeck和丘成桐关于高维复流形Hermitian Yang-Mills 联络存在性的证明。

 

也许并不为人所熟知的是,Hermitian Yang-Mills 联络的存在性在超对称杂化弦论中至关重要,并对代数几何和复几何也有重要影响。比如,Uhlenbeck和丘成桐的证明是目前仅知的证明,在Higgs 场和Hodge结构的变分理论中有重要应用。

 

在丘成桐1970年代几何分析工作问世之前,微分方程的发展与微分几何并无关联。很难想象非线性方程可以应用于复几何或代数几何。丘成桐和他的朋友们(比如Rick Schoen, Leon Simon, 郑绍远, Karen Uhlenbeck, Peter Li, Richard Hamilton, Cliff Taubes, Simon Donaldson)的工作使得几何分析发展成为过去50年中几何与拓扑学中最强大的学科。

 

丘成桐在线性微分方程的工作也极具原创性和影响力。比如,调和函数的梯度估计已经成为几何分析中的标准技巧,热方程的李伟光—丘成桐估计是研究所有抛物方程的基本工具,并启发了汉密尔顿里奇流中的估计技巧,被 Perelman 用于证明庞加莱猜想。

 

关于丘成桐的伟大原创性和技巧的另一个例子,是他关于调和映照和极小曲面的应用。Meeks 和丘成桐证明极小曲面方程的Douglas解是嵌入的这一经典问题,并用之解决了等变Dehn 引理。这是关于三维球面群作用的著名的Smith 猜想的证明中的关键一步。

 

Schoen 与丘成桐应用调和映照理论证明了著名的关于广义相对论中质量为正的爱因斯坦猜想。为广义相对论提供了重要的佐证。(如果质量为负,那么整个系统会变得不稳定,分崩离析。)这个正质量定理在几何中有许多重要的推论。

 

在他的学术生涯中,丘成桐总是持续高产极富原创的工作。比如最近他与合作者解决了经典广义相对论中的一个长期悬而未决的难题——给出了拟局部质量的正确定义。由爱因斯坦等效原理,引理不存在质量和能量密度的概念。大多数先前关于质量和能量概念的理解局限于孤立系统(比如最著名的当属ADM质量和Bondi质量),其中质量和能量需要在无穷远处估算。不过,有一个关于质量和能量的拟局部的描述会极为有用,因为大多数物理系统建构于非孤立的有限区域上。

 

视野

 

丘成桐提出过许多有着深远影响的猜测和公开问题。比如,弦理论中的镜对称在代数几何中有许多意想不到的应用。解释镜对称成为理论物理和数学的一大挑战课题。Strominger, 丘成桐和Zaslow1996年提出了镜流形的新构造,被称为SYZ猜想。它给出了镜对称的几何实现与解释,成为当今指引镜对称研究的纲领。

 

1982年,丘成桐在普林斯顿高等研究院组织了微分几何学术年,并提出了120个公开问题。它们覆盖了极为广泛的专题。许多问题至今仍极具影响,变革了整个课题的面貌。比如,第65个问题问道,如何定义非正曲率流形的秩,推广局部对称空间的情形,以及是否高秩度量是局部对称度量。

 

这个问题由Ballmann Burns-Spatzier 完全解决。他们证明中所发展的技巧,完全变革了非正曲率流形理论,导致了CAT(0)空间和CAT(0)群概念的诞生。吸引了许多学者投身非正曲率流形的几何,拓扑与分析研究。

 

在最初的120个问题获得巨大反响后,丘成桐又在1992年提出了另100个公开问题,也产生了巨大的影响。比如,问题之一是证明第一陈类为正的紧致凯勒流形容许凯勒—爱因斯坦度量当且仅当这个流形在几何不变量意义下是稳定的,即切丛稳定且自同构群可约。

 

这个问题最近引发了世界顶尖复几何和代数几何专家们的大量工作。丘成桐还在其他论文和场合提出过许多问题。所有这些问题,体现了丘成桐对于数学的全局洞察力。最重要的是,几何学家们更深刻的认识到,发掘不同数学分支之间联系的重要性。

 

著述,编辑和指导学生

 

除了数百篇论文,丘成桐也写了许多综述文章和著作。他的文章《微分几何中的微分方程综述》(Survey on partial differential equations in differential geometry, in Seminar on Differential Geometry, pp. 3--71, 1982 是第一篇关于几何分析的全面综述。从某种意义上,这是一位先驱在学科初创时期的第一手记录。他还有许多关于微分几何的精彩综述。

 

他与Rick Schoen的著作《微分几何讲义》(Lectures on differential geometry)开门见山的直达几何分析最活跃的课题,指引读者很快进入研究,培育了好几代微分几何学家,也是今后许多年几何学的必备参考书。

 

丘成桐至今指导了超过67位博士生。他通过建立数学研究中心,组织数学竞赛和学术活动,与中学生,大学生和海内外众多数学家有着许多交流,积极推动中国数学的发展,提升中国数学的影响力。

 

除了研究,教学,著述,会议和管理数学中心,丘成桐还发起和编辑多本纯粹和应用数学的杂志。比如在他的努力下,《微分几何杂志》(Journal of Differential Geometry)成为数学的顶尖期刊,其中发表过Simon Donaldson, Michael Friedman, Richard Hamilton, Rick Schoen 等数学家最好的工作。

 

结论

 

简而言之,丘成桐为数学而生。他不仅是一位几何学家,一位分析学家,或者一位数学物理学家。他是一位真正的最广泛意义上的数学家。他总是致力于创造和发展数学,慷慨分享他的思想。可以说,如果没有他的贡献,他的视野和他的著述,几何分析学科不可能会像今天这样成熟和丰富多彩。所以,丘成桐属于整个数学史上最杰出的数学家之列,比肩Euler, Gauss, Riemann, Poincare, Hilbert Weyl

 

Shing-Tung Yau, a great mathematician

 

 

Lizhen Ji, University  of Michigan

 

 

 

In the past half-century,  the field of geometric analysis has risen and became one of the most active and important subjects  in mathematics, and close integration between mathe- matics and physics has solved many seemingly insurmountable problems, opened new fields and changed people’s perspectives on both mathematics and physics.  Shing-Tung Yau is one of the most active leaders  and  practitioners in these exciting developments.  His contribu- tions,  his leadership  and his writings have made these subjects  what  they  are now, and he has permanently  affected the mathematical  landscape.

Though  the  twentieth century  is the  most  productive  century  in the  history  of mathe- matics,  it is not clear how many  people, theories  and  theorems  will become a part  of the permanent history  of mathematics.  On the other  hand,  there  is no question  that Yau is a mathematician of great  originality,  grand  vision and awesome technical  power, and that  he is one of the most prominent  mathematicians in the 20th century.   Furthermore, the broad subject of geometric  analysis  developed by Yau,  his solution  of the Calabi  conjecture,  and Calabi-Yau  manifolds  will be used by mathematicians and  physicists  for many  decades  to come.

 

Originality and Technical Power.  Yau proposed  and  emphasized  the  philosophy  that the  subject  of nonlinear  partial  differential  equations  should  be combined  with  geometry (including differential geometry, complex geometry and algebraic geometry).  Many deep re- sults by him and his co-authors have shown that  only through geometry, one can understand thoroughly  nonlinear  differential  equations  such as the  Monge-Amp`ere equation,  and  that geometric  structures can  be constructed effectively by nonlinear  partial  differential  equa- tions coupled with the linear theory of Index theorems,  which are still the only way to build geometric structures.

Probably  the most striking  example is his solution of the Calabi conjecture  on the exis- tence of Kahler-Einstein metrics  which lead to immediate  solution of several long standing open problems in algebraic geometry.  Calabi-Yau  manifolds have also contributed crucially to the extensive development of mathematical physics, in particular string theory.  Other ex- amples constructed in this way include the geometric structures on 3-dimensional manifolds by Hamilton’s  Ricci flow, the  gluing arguments  by Taubes  and  Uhlenbeck for existence of anti  self-dual connections and anti  self-dual metrics on four manifolds, and the existence of Hermitian  Yang-Mills connections  on higher dimensional complex manifolds by Uhlenbeck- Yau.

It  might not  be well-known that  the  existence of Hermitian  Yang-Mills connections  is absolutely  essential  for the construction of supersymmetric  heterotic  string and has a huge impact  in algebraic geometry and complex geometry.  For example,  the  proof of Uhlenback and  Yau  is still the  only proof that  can be used for applications  such as Higgs fields and variations  of Hodge structures.

Before Yau’s work in geometric analysis in 1970s, theories  of differential  equations  were largely separately  from the differential  geometry,  and  applications  of nonlinear  differential equations  to complex geometry  and  algebraic  geometry  could not  be imagined.   Works of Yau with his friends such as Rick Schoen, Leon Simon, Shiu-Yuen Cheng, Karen Uhlenbeck, Peter Li, Richard Hamilton, Cliff Taubes, Simon Donaldson and others have made geometric analysis the most powerful subject in geometry and topology in the past  50 years.

Yau’s work on linear differential equations is also very original and influential.  For exam- ple, gradient estimates for harmonic functions has become a standard technique in geometric analysis, and the Li-Yau estimate for the heat equation is fundamental to all parabolic equa- tions and has motivated  the major estimate  in Ricci flow, which has been used to prove the Poincar´e conjecture by Perelman.

Another example of great originality and technique of Yau is the application  of harmonic maps and minimal surfaces.  Meeks and Yau proved the famous classical problem of embed- ding of the Douglas solution of the minimal surface equation,  which was turned  around  to solve the equivariant Dehn’s Lemma, which was a crucial step in the solution of the famous Smith conjecture for group actions on the three sphere.

Schoen and Yau used the theory of harmonic map to prove the famous Einstein conjecture on the positivity of the mass in general  relativity  and  provides one of the  most  important justifications of the theory of general relativity.  (If the mass were negative, the whole system will be unstable and  fall apart!)   Many important consequences of the  solution  of positive mass in geometry have been developed.

Throughout his career, Yau has always been very active and been producing highly orig- inal work. For example, most recently, Yau and his collaborators  resolved one of the longest standing  problems in classical general relativity:  giving the correct  definition of “quasi-local mass.   By Einsteins  equivalence principle,  there  is no well-defined concept  of mass/energy density  for gravitation.  Most prior understanding of the  notion  of mass/energy  (most  no- tably,  ADM mass and  Bondi mass)  is limited  to isolated  systems  where the  mass/energy is evaluated  at  infinity.   However, it  is extremely  useful to  have  a quasi-local description of mass/energy, as the majority  of physically observable systems are modeled on a finitely extended  region that  may not be isolated.

 

Vision. Yau has formulated  conjectures  and proposed many open problems which have far reaching  consequences.  For example,  the  mirror  symmetry  in string  theory  has had many unexpected  applications  in  algebraic  geometry,  and  understanding the  mirror  symmetry conjecture  has  been a major  challenge in theoretical  physics  and  mathematics.   In  1996, Strominger,  Yau  and  Zaslow proposed  a new construction  of mirror  manifolds,  called the SYZ conjecture.   It gives a geometrical realization  and explanation  of the mirror symmetry in string theory, and has been the guiding principle for a whole generation  of people working in the mirror symmetry.

In 1982, Yau proposed a list of 120 open problems when he organized a special year on differential geometry at IAS, Princeton.   They cover a broad range of topics.  Some of them have been very influential  and have changed  the  subject.   For example,  Problem  65 called for a notion  of rank  of nonpositively  curved  manifolds  which extends  the  one for locally symmetric spaces, and asked whether higher rank metrics are locally symmetric metrics.

This  problem  was completely  solved by Ballmann,  and  Burns-Spatzier.  The  efforts to solve this problem have completely changed the subject of manifolds of nonpositive curvature which led to  the  notion  of CAT(0)-spaces  and  CAT(0)-groups,  and  have  attracted many people to work to understand geometry,  topology and analysis of manifolds of nonpositive curvature.

After the success of this list of open problems, Yau proposed another  100 open problems in 1992. This list has also had a huge influence on topics covered. For example, one problem can be stated  as follows: Prove  that  a compact  Kahler  manifold  with positive first Chern class admits  a K¨ahler metric  if and only if the manifold is stable in the sense of geometric invariant theory,  the tangent  bundle is stable as a bundle and  the automorphism  group is reductive.

This problem has also inspired a huge amount of the most recent work in complex geom- etry and complex algebraic geometry by the top experts around  the world.

Yau also raised many problems in other papers and contexts.  All these problems show the global perspective of Yau towards mathematics. More importantly, they teach generations of geometers (in the broadest  sense) the importance  to understand and appreciate  connections between different parts  of mathematics.

 

Writing, Editing and Supervising. Besides several hundred  research  papers,  Yau has written  many influential survey papers and books. The paper Survey on partial  differential equations in differential geometry, in Seminar  on Differential Geometry, pp.  3–71, 1982, was the first major survey on geometric analysis.  In some sense, it is the record of the first hand experience of this emerging subject by one of the most important practitioners. He has kept on writing many comprehensive surveys on differential geometry in the broadest  sense.

His joint book with Rick Schoen, Lectures  on differential  geometry goes directly  to the most  important and  active  topics  of the  subject  and  hence guides the  reader  to  get  into research quickly.  It has educated  generations  of differential geometers and will continue to be an indispensable  book for many years to come.

It is important to note that  Yau has supervised over 67 Ph.D. students up to now, and has reached out to mathematics students and mathematicians at various levels starting  at high school through  many mathematics centers,  mathematics competitions  and other  initiatives organized by him.

On  top  of all these  research,  teaching,  writing,  conferences and  mathematics centers, Yau has also launched and edited many journals both in pure and applied mathematics. For example, he turned  the Journal  of Differential Geometry  into a leading global journal which has published  the  best  works of Simon Donaldson,  Michael Friedman,  Richard  Hamilton, Rick Schoen etc.

 

Conclusion. To put it simply, Yau is a man of mathematics. He is not only a geometer, an analyst,  or mathematical physicist.  He is a mathematician in the true  and broadest  sense. He is always devoted  to  creating  and  developing  mathematics, and  generously shares  his ideas with others.  It is safe to say that  without  his contributions, his vision and his writings, the  broad subject of geometric  analysis  would not  be in the  current mature  and  satisfying state.  Therefore, Yau belongs to the distinguished list of universal mathematicians in history such as Euler,  Gauss,  Riemann,  Poincare, Hilbert, and Weyl.