# Sergei Petrovich Novikov

### Quick Info

Gorky (now Nizhny Novgorod), Russia

**Sergei Novikov**is a Russian mathematician who worked in algebraic topology and soliton theory. He won a Fields medal in 1970 and a Wolf prize in 2005.

### Biography

**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.

### References (show)

- Academician S P Novikov (on the occasion of his 50th birthday) (Russian),
*Vestnik Akad. Nauk SSSR*(9) (1988), 133-134. - M F Atiyah, On the work of Serge Novikov,
*Actes du Congrès International des Mathématiciens***1**(Paris, 1971), 11-13. - V M Buchstaber and S P Novikov, The S P Novikov Seminar, in
*Topics in topology and mathematical physics*(Providence, RI, 1995), 1-7. - S C Ferry, A Ranickiand J Rosenberg, A history and survey of the Novikov conjecture, in
*Novikov conjectures, index theorems and rigidity***1**(Cambridge, 1995), 7-66. - Interview with Sergy P. Novikov,
*European Mathematical Society Newsletter***42**(December 2001), 17-20. - Sergei Petrovich Novikov (on the occasion of his fiftieth birthday) (Russian),
*Uspekhi Mat. Nauk***43**4(262) (1988), 3-9. - Sergei Petrovich Novikov (on the occasion of his fiftieth birthday),
*Russian Math. Surveys***43**(4) (1988), 1-10. - S Weinberger, Aspects of the Novikov conjecture, in
*Geometric and topological invariants of elliptic operators*(Providence, RI, 1990), 281-297.

### Additional Resources (show)

Other pages about Sergei Novikov:

### Honours (show)

Honours awarded to Sergei Novikov

Written by J J O'Connor and E F Robertson

Last Update September 2009

Last Update September 2009