谢尔盖·彼得罗维奇·诺维科夫Sergei Novikov

谢尔盖·彼得罗维奇·诺维科夫的父亲是彼得·诺维科夫，本档案馆中有他的传记。诺维科夫的母亲Ludmila Vsevolodovna Keldysh也是一位杰出的数学家，她成为数学教授，并对集合论和几何拓扑学做出了重要贡献。诺维科夫的父母双方都来自具有非凡数学天赋的家族，还有其他几位成员也值得注意。我们应特别提到诺维科夫的叔叔姆斯季斯拉夫·克尔德什，他对复变函数论、微分方程及其在空气动力学中的应用做出了重大贡献。姆斯季斯拉夫·克尔德什是一位杰出的苏联科学家，曾多年担任苏联科学院院长。

来自这样具有数学天赋的家族，得知诺维科夫并非彼得和柳德米拉的五个孩子中唯一从事数学科学职业的人，也就不足为奇了。他们的长子列昂尼德后来成为固体物理学领域的国际领军人物。他们的次子安德烈成为algebraic number theorist，而他们的第三个孩子就是本传记的主人公诺维科夫。诺维科夫有两个妹妹，她们都选择了数学以外的职业。

诺维科夫在数学环境中长大，这不仅是由于周围家庭成员的数学兴趣，还因为成立了一个特殊的社团，让各位数学家的孩子接受额外的指导。尽管他与数学有着密切的联系，或者更可能是因为这种联系，诺维科夫多年来一直不确定自己是否想从事这一学科的职业。他在13岁和14岁时参加了数学奥林匹克竞赛，因此他和他的学校老师都充分意识到他的杰出才能。

当诺维科夫年满十七岁时，他终于决定要从事数学职业。1955年离开学校后，他进入莫斯科大学数学力学系。在那里，他从学习一开始就接触到了高强度的数学。安德雷·柯尔莫哥洛夫的学生V A Uspenskii在诺维科夫大学第一年组织了一个讨论班，研究集合论、数理逻辑和实变函数的问题。在开始第二年的学习之前，诺维科夫必须选择一个专业课题和一位导师。他决定研究代数拓扑学，他的工作由M M Postnikov指导。

此时，莫斯科大学的力学与数学系是实分析研究的世界领先中心，安德雷·柯尔莫哥洛夫是主要影响者。安德雷·柯尔莫哥洛夫在1957年完全解决了大卫·希尔伯特的第十三问题，他周围有一个非常活跃的研究小组。另一方面，就莫斯科大学而言，拓扑学在当时并不是那么重要的课题，所以当Postnikov在1958-59学年去中国时，诺维科夫就没有导师了。在这个学年里，他研究了弗兰克·亚当斯和勒内·托姆的工作，并找到了自己的研究问题。约翰·米尔诺在1957年引入了一种称为诺曼·斯廷罗德代数的海因茨·霍普夫代数，次年，约翰·柯西·亚当斯研究了诺曼·斯廷罗德代数的上同调。诺维科夫在1959年发表的第一篇重要出版物Cohomology of the Steenrod algebra，发表在Doklady Akademii Nauk SSSR上，进一步发展了约翰·柯西·亚当斯的方法和结果。

另一篇重要论文《与勒内·托姆空间理论相关的流形拓扑学中的一些问题》由诺维科夫于1960年发表。它宣布了对应于“稳定Thom复形”的配边理论的结果。他的几何方法在约翰·柯西·亚当斯的一篇评论中被描述为：-

……简单、优雅且自然。

诺维科夫于1960年获得第一个学位，随后成为莫斯科弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所的研究生。1960-61年间，他[5]：-

……研读了哈斯勒·惠特尼、列夫·庞特里亚金、勒内·托姆和约翰·米尔诺的著作，这些著作都写得极为清晰。这些论文中证明的完备性是在不损害理解且没有任何人为形式化的情况下实现的。

阅读顶尖研究者的论文对任何年轻数学家都很重要，但个人接触也可能极为宝贵。在这方面，诺维科夫很幸运，因为约翰·米尔诺、弗里德里希·希策布鲁赫和斯蒂芬·斯梅尔都在1961年夏天访问了苏联，参加各种会议。通过与他们的个人会面，诺维科夫了解到了拓扑学中正在研究的主要方向和问题[5]：-

在弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所与斯蒂芬·斯梅尔会面后，当时我是那里的研究生，我的本地导师开始把我视为一位严肃的科学家。

受1961年夏天会面的启发，诺维科夫在那年秋天解决了一个重大问题。他对单连通流形分类问题的贡献最终于1964年以《同伦等价的光滑流形》为题发表。到那时，威廉·布劳德已独立发现了与诺维科夫所发展的类似技术。诺维科夫因这项工作于1964年获得苏联科学院的奖项，并于同年获得博士学位。

1963年，诺维科夫被任命为弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所的工作人员，次年，他还被任命为莫斯科大学微分几何系的成员。诺维科夫于1971年成为苏联科学院理论物理列夫·朗道研究所数学部的负责人。

诺维科夫还在1983年成为莫斯科大学高等几何与拓扑学系的负责人，次年他成为苏联科学院数学研究所几何与拓扑学系的负责人。他于1985年被任命为莫斯科数学会的主席，接替安德雷·柯尔莫哥洛夫，并担任此职位直到1996年。

自1996年以来，他一直在美国马里兰大学工作，但与俄罗斯保持着密切联系，在莫斯科大学、朗道理论物理研究所以及弗拉基米尔·安德烈耶维奇·斯捷克洛夫研究所的几何与拓扑学研究小组担任研究职务。

诺维科夫直到1971年的工作是关于代数微分拓扑；特别是他研究了计算稳定homotopy groups以及对维度大于4的平滑单连通流形进行分类。他还研究了有理的列夫·庞特里亚金类的拓扑不变性。

1965年，诺维科夫证明了关于列夫·庞特里亚金类不变性的著名定理，并提出了现在称为诺维科夫猜想的猜想，该猜想涉及流形的列夫·庞特里亚金类中某些多项式的同伦不变性，这些多项式源自基本群。这是拓扑学中最基本的问题之一。诺维科夫最初的动机是单连通情形下的威廉·布劳德-诺维科夫和Wall理论，这导致了高维流形的分类。诺维科夫在1970年于尼斯举行的国际数学家大会上作报告讨论了他的猜想，并在那里获得了菲尔兹奖。然而，他未能亲自出席在尼斯的颁奖典礼，因为苏联当局想惩罚他，因为他支持那些因公开反对政权而被逮捕并送入精神病院的人。

1971年后，诺维科夫对数学物理和动力系统产生了兴趣。他研究了数学的广泛应用，如齐次宇宙学模型理论中的动力系统、孤子理论、线性算子的谱理论、量子场论和弦理论。1982年，他对金属物理理论中出现的拓扑问题产生了兴趣。

诺维科夫因其杰出工作获得了许多荣誉。其中最重要的奖项或许是上文提到的约翰·查尔斯·菲尔兹奖章，他于1970年获得该奖章。1981年，他成为苏联科学院的正式成员，并于同年获得该科学院的罗巴切夫斯基奖。

许多学会授予诺维科夫会员荣誉，如1987年的伦敦数学会和1996年的宗座科学院。1994年，他当选为美国国家科学院。在许多其他荣誉中，我们还应提到他获得了雅典大学和特拉维夫大学的荣誉博士学位。

2005年，诺维科夫被授予沃尔夫奖：-

……因其对拓扑学和数学物理的根本性和开创性贡献。

American Mathematical Society的Notices2005年5月部分的一篇文章解释了导致该奖项的工作：-

他在代数拓扑和微分拓扑方面的早期工作包括诸如计算协边环和稳定同伦群、证明有理Pontrjagin类的拓扑不变性、提出关于更高签名不变量的“诺维科夫猜想”，以及证明3球面二维叶状结构中闭叶的存在性等里程碑。

20世纪70年代初，诺维科夫将注意力转向数学物理，最初对广义相对论和金属导电性做出了贡献。他在流形和环路空间上构造了莫尔斯理论的整体版本，这对量子场论（多值作用泛函）有新颖的应用。他在数学物理方面最重要的成就源于他将代数几何方法引入完全可积系统的研究。这些包括对二维可积系统有限隙解的系统研究、KP方程代数几何解的分类与波恩哈德·黎曼曲面共形分类等价性的表述，以及（与Krichever合作）关于弦理论和矩阵模型中出现的“几乎对易”算子的工作（“Krichever-诺维科夫代数”，现广泛用于物理学）。

诺维科夫 在数学的两个不同领域做出了根本性的、引人注目的贡献，同时他是那些罕见的数学家之一，能将深刻、关键的数学思想应用于物理学中困难的枢纽问题，其方式令数学家和物理学家都感到惊叹和信服。

Sergei Novikov's father was Petr Sergeevich Novikov who has a biography in this archive. Sergei's mother, Ludmila Vsevolodovna Keldysh, was also an outstanding mathematician who became a professor of mathematics and made important contributions to set theory and geometric topology. Both of Sergei's parents came from families with remarkable mathematical talents and several other members were noteworthy. We should mention especially Sergei's uncle Mstislav Keldysh who made major contributions to complex function theory, differential equations and applications to aerodynamics. Mstislav Keldysh was a leading Soviet scientist who was President of the USSR Academy of Sciences for many years.

Coming from such mathematically talented families, it will come as no surprise to learn that Sergei was not the only one of Petr and Ludmila's five children to follow a career in the mathematical sciences. Their oldest child, Leonid, went on to be a leading international figure in solid state physics. Their second child, Andrei, became an algebraic number theorist while their third child was Sergei, the subject of this biography. Sergei had two younger sisters who both chose careers outside mathematics.

Sergei grew up in a mathematical environment, not only the result of the mathematical interests of family members around him but also since a special society was formed where the children of various mathematicians received additional instruction. Despite his high involvement with mathematics, or more likely because of it, Sergei was uncertain for many years whether he wanted to follow a career in the subject. He took part in Mathematical Olympiad Competitions when aged 13 and 14 and both he and his school teachers were thus fully aware of his outstanding talents.

When he reached the age of seventeen Sergei finally decided that he wanted to follow a career in mathematics. On leaving school in 1955, he entered the Faculty of Mathematics and Mechanics of Moscow University. There he became involved in high powered mathematics from the very beginning of his studies. V A Uspenskii, a pupil of Kolmogorov, organised a seminar during Novikov's first year as a student in which problems in set theory, mathematical logic, and functions of a real variable were studied. Before beginning his second year of study Novikov had to choose a specialist topic and a supervisor. He decided to work on algebraic topology and his work was supervised by M M Postnikov.

At this time the Faculty of Mathematics and Mechanics of Moscow University was a world leading centre for research in real analysis, with Kolmogorov the major influence. Kolmogorov had completely solved Hilbert's Thirteenth Problem in 1957 and he was surrounded by a remarkably active research group. Topology, on the other hand, was not such an important topic at that time as far as Moscow University was concerned, so when Postnikov went to China for the academic year 1958-59, Novikov was left without a supervisor. During this academic year he studied the work of Frank Adams and René Thom, and found his own research problems. Milnor had introduced a type of Hopf algebra called a Steenrod algebra in 1957 and in the following year the cohomology of the Steenrod algebra had been investigated by Adams. Novikov's first important publication in 1959 Cohomology of the Steenrod algebra, published in Doklady Akademii Nauk SSSR, developed further Adams's methods and results.

Another important paper Some problems in the topology of manifolds connected with the theory of Thom spaces was published by Novikov in 1960. It announced results on the cobordism theories corresponding to "stable Thom complexes". His geometrical approach was described by Adams in a review as:-

... simple, elegant and natural.

Novikov obtained his first degree in 1960 and then became a research student at the Steklov Institute of Mathematics in Moscow. In 1960-61 he [5]:-

... studied the writings of Whitney, Pontryagin, Thom and Milnor, all of which are written with great clarity. The completeness of the proofs in these papers was achieved with no detriment to understanding and without any artificial formalisation.

Reading the papers of the leading researchers is important for any young mathematician, but personal contact can also be invaluable. In this respect Novikov was fortunate since Milnor, Hirzebruch and Smale all visited the USSR during the summer of 1961 to attend various conferences. From personal meeting with them Novikov learnt about the major directions and problems which were being studied in topology [5]:-

After my meeting with Smale at the Steklov Institute of Mathematics, where I was a postgraduate, my local supervisors began to regard me as a serious scientist.

Inspired by his meetings in the summer of 1961, Novikov solved a major problem in the autumn of that year. His contributions to the classification problem for simply connected manifolds was eventually published as Homotopically equivalent smooth manifolds in 1964. By that time Browder had, independently, discovered similar techniques to those Novikov had developed. Novikov received an award from the USSR Academy of Sciences in 1964 for this work and he was awarded his doctorate in the same year.

In 1963 Novikov had been appointed to the staff of the Steklov Institute of Mathematics and, the following year, he was also appointed to the Department of Differential Geometry at Moscow University. Novikov became head of the Mathematics Division at the L D Landau Institute for Theoretical Physics of the USSR Academy of Sciences in 1971.

Novikov also became head of the Department of Higher Geometry and Topology of Moscow University in 1983 and, the following year he became head of the Department of Geometry and Topology of the Mathematical Institute of the USSR Academy of Sciences. He was appointed as president of the Moscow Mathematical Society in 1985, when he succeeded Kolmogorov, and held this position until 1996.

Since 1996 he has been working at the University of Maryland in the United States but retains close links with Russia with a research appointments in Moscow University, in the Landau Institute for Theoretical Physics, and as Head of the Geometry and Topology research groups at the Steklov Institute.

Novikov's work up to 1971 was on algebraic and differential topology; in particular he studied calculating stable homotopy groups and classifying smooth simply-connected manifolds of dimension greater than 4. He also studied the topological invariance of rational Pontryagin classes.

In 1965 Novikov proved his famous theorem on the invariance of Pontryagin classes and stated the conjecture, now known as the Novikov conjecture, concerning the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group. It is one of the most fundamental problems in topology. Novikov's original motivation was the theory, in the simply connected case, of Browder-Novikov and Wall, which led to the classification of manifolds in high dimensions. Novikov discussed his conjecture in a lecture given at the 1970 International Congress of Mathematicians in Nice where he received a Fields Medal. However, he was not allowed to attend the award ceremony in Nice in person, since the Soviet authorities wanted to punish him for the support he had given to those arrested and sent to mental institutions for speaking out against the regime.

After 1971 Novikov became interested in mathematical physics and dynamical systems. He studied a wide variety of applications of mathematics such as dynamical systems in the theory of homogeneous cosmological models, the theory of solitons, the spectral theory of linear operators, quantum field theory and string theory. In 1982 he became interested in topological problems which arise in the physical theory of metals.

Novikov has received many honours for his outstanding work. Perhaps the most important of these awards has been the Fields Medal, referred to above, which he received in 1970. In 1981 he became a full member of the USSR Academy of Sciences, receiving the Lobachevsky Prize of the Academy in the same year.

Many societies have honoured Novikov with membership such as the London Mathematical Society in 1987 and the Pontifical Academy of Sciences in 1996. He was elected to the National Academy of Sciences of United States in 1994. Among many other honours, we should mention that he received honorary doctorates from the universities of Athens and Tel Aviv.

In 2005 Novikov was awarded the Wolf Prize:-

... for his fundamental and pioneering contributions to topology and to mathematical physics.

An article in the May 2005 part of the Notices of the American Mathematical Society explains the work which led to the award:-

His early work in algebraic and differential topology includes such milestones as the calculation of cobordism rings and stable homotopy groups, proof of the topological invariance of rational Pontrjagin classes, formulation of the "Novikov Conjecture" on higher signature invariants, and proof of the existence of closed leaves in two-dimensional foliations of the 3-sphere.

In the early 1970s Novikov turned his attention to mathematical physics, initially contributing to general relativity and conductivity of metals. He constructed a global version of Morse theory on manifolds and loop spaces that had novel applications to quantum field theory (multivalued action functionals). His most significant achievements in mathematical physics flow from his introduction of algebraic-geometric methods to the study of completely integrable systems. These include a systematic study of finite-gap solutions of two-dimensional integrable systems, formulation of the equivalence of the classification of algebraic-geometric solutions of the KP equation with the conformal classification of Riemann surfaces, and work (with Krichever) on "almost commuting" operators that appear in string theory and matrix models ("Krichever-Novikov algebras", now widely used in physics).

Novikov made a fundamental and striking contribution to two separate fields in mathematics, while he is one of those rare mathematicians who brings deep, key mathematical ideas to bear on difficult pivotal problems of physics, in ways that are stunning and compelling for both mathematicians and physicists.