### THE MATHEMATICAL LIFE OF PROFESSOR STEPHEN S. T. YAU

Stephen is a top research mathematician working on a variety of subjects both in pure and applied mathematics. He was born in 1952 and went to Chinese University of Hong Kong for his undergraduate. Then he went to the State University of New York at Stony Brook in 1973. Working under the direction of Laufer for his PHD degree. He moved to Harvard University as Benjamin Pierre Assistant Professor working under the mentorship of Hironaka in 1977. After his postdoctoral training at Harvard, he was hired as a tenured faculty member at University of Illinois at Chicago and was promoted to the University Distinguished Professorship in 2005. He retired from UIC last year(2011) before moving to Tsinghua University as a senior Professorship.

Stephen's mathematical work is broad and has long lasting impact in many branches of mathematics. He is the author or coauthor of more than 275 scientific publications. Stephen is one of the pioneers in the study of CR (Cauchy-Riemann) Geometry from the global point of view. Among many of his important papers is his Annals paper published in 1980, in which he solved a long standing open question raised by Joseph Kohn concerning the interior regularity of complex analytic variety bounded by a smooth CR strongly pseudoconvex submanifold of codimension three in a complex Euclidean space. As a result, he solved the famous complex Plateau Problem for strongly pseudoconvex CR manifolds in complex Euclidean space. This work opened a new direction in CR geometry. Furthermore, Stephen has also obtained very exciting results on the relationship between the holomorphic De Rham cohomology and the interior singularities of CR manifolds. Most strikingly, for a quarter century it was believed that strongly pseudoconvex CR manifolds exhibit a certain type of boundary regularity (this assumption was even used in some published theorems). Stephen and Luk gave a counter-example to this belief, which relies on a "coming back phenomenon." This phenomenon is important in its own right and will open a new area in Complex Analysis. Yau has made a fundamental breakthrough in studying complex geometry of varieties with isolated singularities. He first introduced the Bergman function on any strongly pseudoconvex variety with isolated singularities. This Bergman function is a biholomorphic invariant of and vanishes precisely on the singularities of . This Bergman function not only can distinguish analytic structures of isolated singularities, but it can also distinguish the CR structures of the boundaries of . Yau has also proved a very surprising strong results on rigidity of CR morphisms between compact strongly pseudoconvex CR manifolds lying in . i.e., any non-constant CR morphism from to is necessarily a CR biholomorphism.

Stephen is also well-known as one of most distinguished specialists in singularity theory. He invented the notion of elliptic sequence in Elliptic singularities which allowed him to introduce a new theory of maximally elliptic singularities. He successfully classified topologically all maximally elliptic hypersurface singularities as well as all elliptic double points. This opened up a new research in surface singularities. Tomari introduced the Yau sequence, which formally generalizes Yau's elliptic sequence, to singularities of higher genera. Konno introduced the Yau cycle which plays a fundamental role in studying normal surface singularities. His other significant work includes a joint work with John Mather published in Inventiones in 1982 where a famous theorem states that two isolated hypersurface singularities are biholomorphically equivalent if and only if their moduli algebras are isomorphic. Thus, via their result, the biholomorphic classification of isolated hypersurface singularities is contained in the algebraic classification of commutative Artinian local algebras. In another important contribution, Yau was the first to systematically study the Lie algebras of derivations of these moduli algebras. He proved that these Lie algebras are solvable, and he showed in many important cases how they vary as the complex structures of the singularity vary. He also showed that the Lie algebra that he constructed can characterize whether the original singularity has a -action. Thus Yau has established a natural connection between the theory of singularities and the theory of finite dimensional solvable Lie algebras. These Lie algebras are called Yau algebras by A. Elashvili and G. Khimshiashvili. They defined the dimension of these Lie algebras the Yau numbers.

Yau has made a very significant contribution on computational number theory, especially on the sharp estimate of number of integral points in real simplex. His results are important subjects in number theory and singularity theory. It could be applied to finding large gaps between primes, to Waring's problem, to primality testing and factoring algorithms.

Since 1990, he got interested in many branches of applied mathematics, including filtering, control theory and Bioinformatics. In 1990, Stephen first developed a very general class of finite dimensional filters, which is later named as "Yau filters". The Yau filters include all previously well-known filters, such as Kalman-Bucy filters and Bene's filters as special cases. Yau filter is universal in the sense that it works for arbitrary initial distribution. Generically, all finite dimensional filters are Yau filters. Furthermore, the classification of finite-dimensional estimation algebras with maximal rank was completed by Yau. When it comes to the nonlinear filtering theory, Naval Research once asked ten years ago: How can one solve the nonlinear filtering problem if adequate amount of computational resources are provided. Stephen solves this problem recently, by using very delicate PDE technique and some idea from Quantum Mechanics. One important ingredient of Yau's algorithm is to solve the Kolmogorov equation at each time step. Yau constructed explicit integral represented convergent solution to that equation, which has tremendous impact on both Engineering and Applied Mathematics. In fact, his explicit solution to Kolmogorov equation makes him one of the top scientists in the world because this has been an important open problem in Engineering and Applied Science for more than half century.

Stephen works in the area of bioinformatics, concentrating on the creation and application of mathematics, statistics, and computer science to molecular biology, particularly to DNA and protein sequence data. He has made some revolutionary contributions to this field. Stephen first proposed a two-dimensional DNA graphical representation without degeneracy. This work was published in the famous journal "Nucleic Acids Research". Based on this graphical technique, two important contributions are the constructions of protein map and genome space. By using this protein map, protein sequences, protein domains, and even arbitrary amino acid sequences can be efficiently analyzed. This provides a new powerful tool for protein functional studies. Later he constructed a novel genome space with biological geometry. This genome space has very important biomedical significance. Furthermore, Stephen makes some breakthroughs in the direction of gene prediction. Based on the three-base periodicity of exon sequence, he developed some Fourier analysis approaches to predict gene regions. The results show that the proposed algorithms frequently outperform the best existing popular methods. In addition, Stephen developed some novel numerical characterizations of nucleic acids or amino acids, such as feature vector, distribution vector, and natural vector, to characterize the genetic sequences. The most remarkable achievement in this direction is the construction of natural vector. It produces a one-to-one correspondence between the DNA (protein) sequence and its natural vector. The distance between two DNA (protein) sequences is defined as the distance between their associated natural vectors. This creates a genome (protein) space with a biological distance which makes global comparison of genomes (proteins). The natural vector method provides not only high-speed, unique representation for genomes and proteins, but tools for accurate classification and prediction. Stephen plans to construct a genome database and a protein database based on natural vectors. Unlike the currently available genome or protein database, his database will support simultaneous comparative study for all available genomes or proteins. Nowadays, public accessible protein database contains more than 8 million records. Among the currently available methods, only the natural vector method could accomplish the "impossible mission". Stephen is building up a team of researchers at Tsinghua to work on this promising project.

Stephen's service to mathematical community is also tremendous. He is the founding editor of a highly respected mathematical research journal, Journal of Algebraic Geometry. He is also a co-founding editor of a more applied mathematical research oriented research journal, Communications in Information and Systems. He co-organized many important international academical conferences. He is very active in training PhD students and postdoctors. By now , there are 34 theses written under his direction.

Stephen is also a very successful and dedicated educator. In the past years, he mentored 4 talented high school students with lots of energy and effort, and with lots of success. For example, under Stephen's mentorship, All 4 high school students got in the Intel Prize semi-final level and one got in the Intel Prize final level. All of them had publications in good SCI journals. In particular, one of them in 2010 won the Gold Award of the S.T. Yau High School mathematics award.

Stephen's recognition includes University Distinguished Professor, University of Illinois at Chicago (2005-), IEEE Fellow (2003), Guggenheim Fellowship (2000-2001), C.M. Cha Fellow from Hong Kong Baptist University(1995), University Scholar of University of Illinois at Chicago (1987-1990), Alfred P.Sloan Research Fellowship (1980-1982), Biographical note in American Men And Women of Science, Biographical note in Who's Who in America Science and Engineering, in the Midwest, in American, among Asian Americans, in Sciences, Higher Education respectively.

### 丘成栋教授学术生涯

丘成栋教授是国际著名数学家， 他的研究领域非常广泛， 涉及众多数学分支。丘成栋教授出生于1952年，在香港中文大学获得学士学位, 于1973年到纽约州立大学石溪分校留学， 成为了著名数学家Laufer的学生并于1976年获得博士学位。博士毕业后，他在普林斯顿高等研究所工作学习了一年，之后于1977年被数学大师、菲尔兹奖获得者Hironaka聘为哈佛大学Benjamin Pierce助理教授。在完成哈佛大学的研究工作之后，他获得了伊利偌伊大学芝加哥分校的终身教职位并于2005被聘为伊利偌伊大学杰出教授。丘成栋教授于2011年接受了清华大学的邀请并辞去了伊利偌伊大学芝加哥分校职位，成为了清华大学的全职教授，开始了他在清华大学新的职业生涯。

丘成栋教授的数学研究工作非常广泛，在众多的数学分支都有取得了很好的成果， 具有很大的国际影响力。丘成栋教授是非常高产的数学家， 迄今为止，他共发表了超过275篇高质量的论文。他是整体CR几何前沿研究非常活跃的几个研究者之一。他在发表于顶尖杂志《Annals Math.》（1980）的一篇重要论文里面，解决了J. Kohn提出的关于复解析簇的内部刚性的一个困扰数学家很长时间的著名难题。 这个问题的解决直接导致他后来解决了著名的Plateau问题。 这些重要工作开辟了CR几何的新方向。除此以外， 在全纯De Rham 上同调和CR流形内部奇点的联系方面丘成栋教授也有重要的工作，特别要提出的是， 在很长一段时间里面，数学家们认为强拟凸CR流形具有某种边界刚性，丘成栋教授和他的合作者Luk颠覆了这个观念并给出了一个反例。 他们的成果开创了复变函数论的一个新方向。丘成栋教授在带有孤立奇点的解析簇的复几何的研究也有重大突破。他是第一个引入Bergman 函数来研究具有孤立奇点的解析簇的数学家。Bergman 函数是奇点的一个重要不变量且具有某些很好的特性， 它可以被用来区分奇点的解析结构。丘成栋教授用Bergman 函数的方法得到了一个令人惊讶的重要结果： 任意非常数的CR态射是CR双全纯态射。

丘成栋教授也因他在奇异性理论的杰出工作而著名。他创造性地提出了椭圆列概念，并基于椭圆列提出了一整套研究椭圆奇点的理论。他成功的给出了极大椭圆超曲面奇点完整的拓扑分类。这些研究开辟了曲面奇点新方向，这个方向目前仍是很活跃的研究分支，例如日本数学家Tomari称丘成栋教授引入的椭圆列为"丘列"并进一步的推广这个"丘列"到高亏格奇点；Konno引入了"丘闭链"来研究曲面奇点等。丘成栋教授的另外一项成名作是他和Mather合作的发表在顶尖杂志《Invent. Math.》（1982）的一个著名定理：两个孤立超曲面奇点是等价的当且仅当它们的Moduli 代数同构。这个重要定理表明，奇点的分类问题等同于相应的Moduli 代数的分类问题。除此以外，丘成栋教授还首次引入并系统研究了Moduli 代数的导子李代数。他证明了一个著名的结果：Moduli 代数的导子李代数是可解的。他还用此李代数来刻化奇点是否具有一个C* 作用。丘成栋教授的这些出色工作建立了奇点理论和李代数理论这两个截然不同领域深刻的联系。目前丘成栋教授引入的这个Moduli 代数的导子李代数被数学家同行Elashvili 和Khimshiashvili称为 "丘代数"， 这个"丘代数"的维数被称为"丘数"。

在计算数论方面，丘成栋教授也取得了许多重要成果。他的研究成果，特别是在多面体整数点的个数上界估计方面，在数论和奇点理论方面非常有应用价值。例如其结果可被运用到算素数差，华林问题，素数测试，以及大数分解算法等问题。

自1990年开始，丘成栋教授开始对应用数学的很多分支产生了浓厚的兴趣，包括滤波理论、控制论和生物信息学。1990年，丘成栋教授第一个推导出了最广泛的有限维滤波，后被称为"Yau filters"。Yau filters涵盖了迄今为止已知的几乎所有滤波模型，如著名的Kalman-Bucy filters和Bene's filters都是它的一个特例。之所以Yau filters被认为是最广泛的原因是它适用于任意的初始分布函数。一般来讲，所有的有限维滤波都是Yau filters。 丘成栋教授更进一步地利用代数工具将所有具有极大秩的有限维滤波完全分类。而后，丘成栋教授转向研究非线性滤波。谈到非线性滤波就不得不提到美国海军研究所在十年前提出了的一个问题：如果提供足够多的计算资源，能否很有效地解决非线性滤波问题？最近，丘成栋教授运用非常优美的偏微分方程技巧和量子力学中的思想解决了这个问题。解决这个问题中的一个核心步骤是在每个时间区间上求解Kolmogorov方程。丘成栋教授构造了一个显式积分表达的函数级数，并证明该级数收敛到Kolmogorov方程的解，使得一个在工程学和应用数学中长达半世纪之久的问题得以解决。这在工程学和应用数学领域产生了很大的影响。也正是这个发现使丘成栋教授跻身于顶尖科学家的行列。

在生物信息学方面，丘成栋教授的工作主要集中在利用数学，统计学和计算机科学的方法来解决分子水平上的生物问题，尤其是DNA和蛋白质序列的分析问题。在这方面，他做出了很多革新的工作。他首先提出来一种非退化性的二维DNA图形表示方法，这项工作发表在了著名杂志《Nucleic Acids Research》上。随后，基于该图形技术，他做出了两个重要贡献：蛋白质地图和基因组空间的构造。蛋白质地图可以帮助我们分析整个蛋白质序列，部分结构域序列，或者任意的氨基酸序列，这为蛋白质功能研究提供了一个有力的新工具。基因组空间是一个全基因组的模空间，具有生物几何意义，每一个基因组对应空间中的一个点。这为研究基因组的聚类和进化分析提供了新方向，有很高的生物医学价值。进而，丘教授在基因区域和剪切点位置预测方面，也做出了很多突破性工作。基于外显子序列的三基周期性，他采用傅立叶分析技术提出了几种基因区域和剪切点位置预测方法。结果显示，这些方法得到的预测准确率明显高于目前的其他方法。此外，丘教授还开发了多种数值表示向量法来刻画DNA和蛋白质序列，包括特征向量，分布向量，和自然向量。其中，最有效的方法当属刚刚提出的自然向量法。我们可以从数学上严格证明，一条DNA（或蛋白质）序列和它的自然向量之间是一一对应关系。因此，两条序列间的距离可以用它们相对应的自然向量间的距离来定义。这就创造了一个具有生物进化距离的基因组（或蛋白质）空间。自然向量不仅给出了基因组（或蛋白质）快速和唯一的表示，而且提供了准确的系统发育和分类关系研究工具。基于该方法，丘教授计划构造一个基因组数据库和一个蛋白质数据库。与当前的基因组或蛋白质数据库不同，他构造的新数据库将支持对所有已知的基因组或蛋白质进行同时的比较研究。公共蛋白质数据库记录了当前有超过八百万的蛋白质存在。在目前的方法中，只有自然向量法可以完成同时比较这个"不可能的任务"。丘教授在清华大学的生物信息研究团队正在开展这个科研项目。

除了研究工作外， 丘成栋教授还做了很多对数学的普及发展很有益的工作。他创办了一份具有国际影响的顶尖刊物《代数几何杂志》，以及和其它同行共同创办了应用数学刊物《信息系统传播杂志》。 他还组织了许多重要的国际学术会议。丘成栋教授培养了很多的博士生和博士后， 截至目前为止，在他的指导下，共有34篇博士论文顺利完成并通过答辩。

丘成栋教授还是个热心的，成功的教育专家。过去4年，他花费极大精力精心培养了4名优秀高中生 参加国际数学比赛并获得了很大成功：4个高中生全部进入国际极具盛名的Intel 半决赛，其中的一个还进入了Intel决赛，其中的另一个于2010获得国际丘成桐高中生数学竞赛金奖， 且4人在SCI杂志上都发表了至少一篇论文。

丘成栋教授曾经获得很多荣誉，如：伊利偌伊大学杰出教授(2005-)， IEEE 院士 (2003)，Guggenheim 奖 (2000-2001)， 香港Baptist 大学 C.M. Cha 奖(1995)， 伊利偌伊大学芝加哥分校大学者奖 (1987-1990)， Alfred P.Sloan 研究奖(1980-1982)，Biographical note in American Men And Women of Science奖， Biographical note in Who's Who in America Science and Engineering, in the Midwest, in American, among Asian Americans, in Sciences, Higher Education respectively奖。