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Serguei Novikov

Distinguished University Professor, Joint/Mathematics,
Fields Medal,
Foreign Associate of the National Academy of Sciences
Mathematics

Professor Novikov was born March 20, 1938 in Gorki, into a family of outstanding mathematicians. His father, Petr Sergeevich Novikov (1901-1975), was an academician, an outstanding expert in mathematical logic, algebra, set theory, and function theory; his mother, Lyudmila Vsevolodovna Keldysh (1904-1976), was a professor, a well-known expert in geometric topology and set theory. Novikov received his mathematical education in the Faculty of Mathematics and Mechanics of Moscow University (1955-1960), and he has worked there since 1964 in the Department of Differential Geometry; since 1983 he has been head of the Department of Higher Geometry and Topology of Moscow University. Beginning in 1992 he regularly worked at the University of Maryland at College Park as a Visiting Professor. In September 1996 he became a full-time professor at the University of Maryland at College Park in the Department of Mathematics and the Institute for Physical Science and Technology.

In 1960 Novikov enrolled as a research student at the Steklov Institute of Mathematics, where his supervisor was M. M. Postnikov; since 1963 he has been on the staff there. He was awarded the degree of Ph.D. there in 1964, and that of Doctor of Science in 1965. In 1966 he was elected Corresponding Member of the Academy of Sciences of the USSR, and in 1981 a full member. Since 1984 he has been head of the Department of Geometry and Topology of the Mathematical Institute of the Academy of Sciences, and in charge of the problem committee of Geometriya i topologiya (Geometry and Topology) at the Mathematics Division of the Academy of Sciences of the USSR. He has been head of the Mathematics Division at the L. D. Landau Institute for Theoretical Physics of the Academy of Sciences since 1971, where he works closely with the physicists. During the period 1985-1996 Novikov served as President of the Moscow Mathematical Society, and during 1986-1990 he was also a Vice-President of the International Association in Mathematical Physics.

Since 1971 his scientific work has played an important part in building a "bridge" between modern mathematics and theoretical physics. Some of Novikov's papers can be divided as follows:

Papers before 1971

• Methods of calculating stable homotopy groups, Complex cobordism theory.
• The classification of smooth simply-connected manifolds of dimension n greater than or equal to 5 with respect to diffeomorphisms. Topological Invariance of rational Pontryagyin classes, higher signatures.
• The qualitative theory of foliations of codimension 1 on three-dimensional manifolds.

Papers after 1971

• Methods of qualitative theory of dynamical systems in the theory of homogeneous cosmological models (of spatially homogeneous solutions to Einstein equations).
• Periodic problems in the theory of solitons (non-linear waves) and in the spectral theory of linear operators, Riemann surfaces and theta-functions in mathematical physics.
• The Hamiltonian formalism of completely integrable systems, Hamiltonian hydrodynamic type systems, and applications of Riemannian geometry.
• Ground states for a two-dimensional non-relativistic particle with spin ½ in a doubly-periodic topologically non-trivial magnetic field. Topolgical invariants for generic operators; Laplace transformations and exactly solvable two-dimensional Schrödinger operators in magnetic fields, and a discrete analogue of this theory.
• Multi-valued functional in mechanics and quantum field theory. Analogue of the Morse theory for the closed 1-forms; foliations given by the closed 1-forms; the special case of 3-torus; and applications in the quantum theory of normal metal, observable topological numbers.
• Analogues of the Fourier-Laurent series on Riemann surfaces, Virasoro algebras, operator construction of string theory.

Novikov's main area of current scientific interests: Geometry, Topology and Mathematical Physics.

Awards and Honors

• 1966-1981 Corresponding member of the Academy of Sciences of the USSR
• 1967 Lenin Prize
• 1970 Fields Medal of the International Mathematical Union
• 1981 Lobachevskii International Prize of the Academy of Sciences of the USSR
• 1981 Full Member of the Academy of Sciences of the USSR
• 1987 An Honorary Member of the London Math. Society
• 1988 Honorary Member of the Serbian Academy of Art and Sciences
• 1988 Honorary Doctor of the University of Athens
• 1991 Foreign Member of the "Academia de Lincei", Italy
• 1992 Member of Academia Europea
• 1994 Foreign Member of the National Academy of Sciences of US
• 1996 Member of Pontifical Academy of Sciences (Vatican)

Students of Sergei Novikov

More than 30 of Novikov's students have been awarded the Candidate Degree (equivalent to Ph.D.), and of these V. M. Buchstaber, A. S. Mishchenko, O. I. Bogoyavlenskii, I. M. Krichever, B. A. Dubrovin, G. G. Kasparov, F. A. Bogomolov, S. P.Tsarev, I. A.Tai manov, A. P.Veselov M. A. Brodskii, V. V. Vedenyapin, R. Nadiradze, V. L. Golo, S. M. Gusein-Zade have been awarded the degree of Doctor of Science (Scientific Degree, equivalent to the level of full professor in former USSR and in Russia).

In addition to those mentioned above, other Novikov pupils with the Candidate Degree (correspondent to Ph.D. level in the West) include I. A. Volodin, N. V. Panov, A. L. Brakhman, P. G. Grinevich, O. I. Mokhov, A. V. Zorich, F. A. Voronov, G. S. D. Grigoryan, A. S. Lyskova, E. Potemin, M. Pavlov, L. Alania, D. Millionshikov, V. Peresetski, I. Dynnikov, A. Maltsev, V. Sadov, Le Tu Thang, S. Piunikhin, A. Lazarev. Three subjects were developed during the last 3 years: (a) A nonstandard theory of the spectral symmetries for the low-dimensional Schrodinger Operators. In particular, a broad family of the exactly solvable two-dimensional Schrodinger Operators with two very special highly degenerate Landau-type energy levels were found for an electron in a magnetic field on lattice. The theory was extended to the discrete (difference) operators on the lattices and to the very general non regular configurations see [1] and [2]. (b) Topological properties of an electron in the quantum mechanics on graphs were found. Scattering theory on graphs with tails was constructed on the basis of topology and symplectic geometry see [3] and [4]. 3. New observable topological phenomena were found for the conductivity of a single crystal of a normal metal with complicated Fermi surface (such as a noble metal) in a strong magnetic field see [5]. Selected publications:

Selected publications:

[1] S. Novikov, A.Veselov. Exactly Solvable two-dimensional Schrodinger Operators and Laplace Transformations. AMS Translations, Series 2, vol 179, Advances in Math Sciences: Solitons, Geometry and Topology On the Crossroads. (edited by V. Bukstaber and S. Novikov), pp 109- 132.

[2] S. Novikov, I. Dynnikov. Spectral Symmetries of the low-dimensional Schrodinger operators and Laplace Transformations. Russia Math Surveys (1997), vol 52, n 5 pp 175-234.

[3] S. Novikov. Schrodinger Operators on Graphs and Topology. Russia Math Surveys (1997), vol 52 n 6 pp 177-178.

[4] S. Novikov. Schrodinger operators on Graphs and Symplectic Geometry, to appear in the Arnoldfest, vol 2, Fields Institute, Toronto (dedicated to the 60th birthday of V. Arnold).

[5] S. Novikov, A. Maltsev. Topological Phenomena in Normal Metals. Reviews of Topical Problems, Physics-Uspekhi 41 (3), pp 231-239 (1998).