Sergei Novikov (mathematician)

Career

In 1966, Novikov became a corresponding member of the Academy of Sciences of the Soviet Union. In 1971, he became head of the Mathematics Division of the Landau Institute for Theoretical Physics of the USSR Academy of Sciences. In 1983, Novikov was also appointed the head of the Department of Higher Geometry and Topology at Moscow State University. He became President of the Moscow Mathematical Society in 1985 and remained in that role until 1996, when he moved to the University of Maryland College of Computer, Mathematical, and Natural Sciences at the University of Maryland, College Park. He continued to maintain research appointments at the Landau Institute for Theoretical Physics, Moscow State University, and the Department of Geometry and Topology at the Steklov Mathematical Institute after his move to Maryland.

Research

Novikov's early work was in cobordism theory, in relative isolation. Among other advances he showed how the Adams spectral sequence, a powerful tool for proceeding from homology theory to the calculation of homotopy groups, could be adapted to the new (at that time) cohomology theory typified by cobordism and K-theory. This required the development of the idea of cohomology operations in the general setting, since the basis of the spectral sequence is the initial data of Ext functors taken with respect to a ring of such operations, generalising the Steenrod algebra. The resulting Adams–Novikov spectral sequence is now a basic tool in stable homotopy theory.

Novikov also carried out important research in geometric topology, being one of the pioneers with William Browder, Dennis Sullivan, and C. T. C. Wall of the surgery theory method for classifying high-dimensional manifolds. He proved the topological invariance of the rational Pontryagin classes, and posed the Novikov conjecture. From about 1971, he moved to work in the field of isospectral flows, with connections to the theory of theta functions. Novikov's conjecture about the Riemann–Schottky problem (characterizing principally polarized abelian varieties that are the Jacobian of some algebraic curve) stated, essentially, that this was the case if and only if the corresponding theta function provided a solution to the Kadomtsev–Petviashvili equation of soliton theory. This was proved by Takahiro Shiota (1986), following earlier work by Enrico Arbarello and Corrado de Concini (1984), and by Motohico Mulase (1984).

Writings

• Novikov, S. P.; Fomenko, A. T. (1990). Basic Elements of Differential Geometry and Topology. Mathematics and Its Applications. Vol. 60. Dordrecht: Springer Netherlands. doi:10.1007/978-94-015-7895-0. ISBN 978-90-481-4080-0.
• Novikov, S. P.; Manakov, S. V.; Pitaevskii, L. P.; Zakharov, V. E. (1984). Theory of solitons: the inverse scattering method. New York: Consultants Bureau. ISBN 0-306-10977-8. OCLC 10071941.
• with Dubrovin and Fomenko: Modern geometry- methods and applications, Vol.1-3, Springer, Graduate Texts in Mathematics (originally 1984, 1988, 1990, V.1 The geometry of surfaces and transformation groups, V.2 The geometry and topology of manifolds, V.3 Introduction to homology theory)
• Topics in Topology and mathematical physics, AMS (American Mathematical Society) 1995
• Integrable systems – selected papers, Cambridge University Press 1981 (London Math. Society Lecture notes)
• Novikov, S. P.; Taimanov, I. A. (2007). Topological Library: Part 1: Cobordisms and Their Applications. Series on Knots and Everything. Vol. 39. Translated by Manturov, V. O. World Scientific. doi:10.1142/6379. ISBN 978-981-270-559-4.
• with V. I. Arnold as editor and co-author: Dynamical systems, 1994, Encyclopedia of mathematical sciences, Springer
• Topology I: general survey, V. 12 of Topology Series of Encyclopedia of mathematical sciences, Springer 1996; 2013 edition
• Solitons and geometry, Cambridge 1994
• as editor, with Buchstaber: Solitons, geometry and topology: on the crossroads, AMS, 1997
• with Dubrovin and Krichever: Topological and Algebraic Geometry Methods in contemporary mathematical physics V.2, Cambridge ISBN 9780415299190, OL 10188321M
• My generation in mathematics, Russian Mathematical Surveys V.49, 1994, p. 1 doi:10.1070/RM1994v049n06ABEH002446