Serguei Novikov

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 (see S.P.Novikov Seminar). 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 Persident 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. Calculation of cobordisms; homotopy groups of spheres and the Adams-Novikov spectral sequence, the technique of homological algebra, unitary cobordisms as a theory of homology, characteristic classes and the Landweber-Novikov algebra, formal groups (papers 1, 2, 6, 30-32, 38, 39).
• The classification of smooth simply-connected manifolds of dimension n greater than or equal to 5 with respect to diffeomorphisms. A theorem on the topological invariance of integrals of Pontryagin classes over cycles with respect to continuous homeomorphisms, the finiteness of a number of smooth structures. Non-simply connected manifolds, Pontryagin classes, Hermitian K-theory and algebraic aspects of Hamiltonian formalism, Novikov's conjecture about higher signatures. The homotopy properties of the group of diffeomorphisms of a sphere (papers 3-5, 7, 8, 10, 18-21, 24, 34-37).
• The qualitative theory of foliations of codimension 1. Three-dimensional manifolds. Theorem on a closed leave (papers 11-13, 23).

Papers after 1971

• Methods of qualitative theory of dynamical systems in the theory of homogeneous cosmological models (of spatially homogeneous solutions to Einstein equations) (papers 40, 42, 46, 50, 61).
• 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. Finite-zone linear operators and finite-zone (algebraic-geometric) solutions of non-linear systems. A solution of rank greater than or equal to one, commuting operators and non-linear systems. Inverse problems for a one-dimensional and two-dimensional Schrödinger operator with periodic and quasi-periodic potential in the class of algebraic-geometric operators, Prym theta-functions, and non-linear systems. Rapidly decreasing two-dimensional potentials with energy less than the ground state (papers 43-45, 48, 49, 53, 55-57, 59, 62, 63, 66, 67, 73, 75-79, 81, 90-92, 98, 99, 105, 112, 113).
• The Hamiltonian formalism of completely integrable systems. Action variables and Riemann surfaces. The Hamiltonian formalism of field-theoretical systems of hydrodynamics type and differential geometry, an averaging method of Whitham for field-theoretical systems. The problem of the involution of many-valued functions, numerical and analytic research (papers 47, 48, 74-76, 80, 85, 88, 92-95, 98, 100, 107, 109).
• Ground states of a two-dimensional non-relativistic particle with spin 1/2 in a doubly-periodic topologically non-trivial magnetic field (with non-zero flux). The case of integer and rational flux. The influence of electric potential. Magnetic Bloch functions and vector bundles. Topological invariants of typical dispersion relations and their Chern classes (papers 64, 65, 71, 81).
• Multi-valued functional in mechanics and field theory. An analogue of the Morse theory for many-valued functions (closed 1-forms). Morse's theorem and the fundamental group. Analytic theory of homotopy groups (papers 68-70, 74, 82-84, 87, 96-98, 102-104, 115).
• Analogues of the Fourier-Laurent series on Riemann surfaces, Virasoro algebras, operator construction of string theory (papers 110, 111, 114).

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.Taimanov, 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 (corre-spondent 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.
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List of Novikov's papers

2. Some problems in the topology of manifolds connected with the theory of Thom spaces, Dokl. Akad. Nauk SSSR 132 (1960) 1031-1034. = Soviet Math. Dokl. 1 (1960) 717-719.

3. On embedding simply-connected manifolds in Euclidean space, Dokl. Akad. Nauk SSSR 138 (1961) 775-778. = Soviet Math. Dokl. 2 (1961) 718-721.

4. Smooth manifolds of general homotopy type, Internat. Cong. Math., Stockholm 1962, section 4, 139.

5. On the diffeomorphism of simply-connected manifolds, Dokl. Akad. Nauk SSSR 143 (1962) 1046-1049. = Soviet Math. Dokl. 3 (1962) 540-543.

6. Homotopy properties of Thom complexes, Mat. Sb. 57 (1962) 407-422.

7. Homotopy properties of the group of diffeomorphisms of the sphere, Dokl. Akad. Nauk SSSR 148 (1963) 32-35. = Soviet Math. Dokl. 4 (1963) 32-35.

97. Analytical homotopy theory. Rigidity of homotopic integrals, Dokl. Akad. Nauk SSSR, 1985, v. 283, N 5, 1088-1091.

98. Integrable systems, I. Current problem in mathematics. Fundamental directions. VINITI, 1985, v. 4, 179-284 (with B. A. Dubrovin and I. M. Krichever).

99. Two-diinensioirlal periodic Schrödinger operators and Prym's theta-fiinctions, in Geometry Today, Internat. Conf. Rome, June 1984, Boston, 1985, 106-118 (with A. P.Veselov and I. M. Krichever).

100. Differential geometly and the averaging method for field-theoretic systems, Proc. III Internat. Symp. on Selected Problem in Statistical Mechanics (Dubna, 1984), Joint. Institute of Nuclear Research, Dubna, 1985, vol. 2, 106-118.

101. Modern geometry. Methods and applications, 2nd revised edition, Nauka, Moscow 1986 (with B. A. Dubrovin and A. T. Fomenko).

102. Bloch homology. Critical points of functions and closed I-forms, Dokl. Akad. Nauk SSSR, 1986, v. 287, N 6, 1321-1324.

103. Morse inequalities and von Neumann factors, Dokl. Akad. Nauk SSSR, 1986, v. 289, N 2, 289-292.

104. Topology I. Current problem in mathematics. Fundamental directions, VINITI, 1986, v. 12, 5-251.

105. Two-dimensional Schrödinger operators: Inverse scattering transform and evolutional equations, Phys. D18, 1986, 267-273 (with A. P. Veselov).

106. Vladimir Abramovich Rokhiin (obituary), Uspekhi Mat, Nauk, 1986, v. 41, N 3, 159-163 (with V. I. Arnol'd, D. B. Fuks, A. N. Kolmogorov, Ya. G. Sinai, A. M. Vershik, and 0. Ya. Viro).

107. Evolution of the Whitham zone in the Korteweg-de Vries theory, Dokl. Akad. Nauk SSSR, 1987, v. 294, N 2, 325-229 (with V. V. Avilov).

108. Elements of differential geometry and topology, Nauk, Moscow, 1987 (with A. T. Fomenko).

109, Evolution of the Whitham zone in the Korteweg-de Vries theory, Dokl. Akad. Nauk SSSR, 1987, v. 295, N 2, 345-349 (with V. V. Avilov and I. M. Krichever).

110. Algebras of Virasoro type, Riemann surfaces and structures in the theory of solitons, Funktsional. Anal. i Prilozhen., 1987, v. 21, N 2, 46-63 (with I. M. Krichever).

111. Algebras of Virasoro type, Riemann surfaces, and strings in Minkowski space, Funktsional. Anal. i Prilozhen., 1937, v. 21, N 4, 47-61 (with I. M. Krichever).

112. Two-dimensiottal Schrödinger operator and solitons 3 -dimensional integrable systems VIII Internat. Congr. on Math. Physics, Marseille 1986, Wered Scientific Publ. 1987, 226-241.

Papers of the 1988-1997 period

114. Analytical theory of homotopy groups, in Topology and Geometry, Rochlin Seminar, Lecture Notes in Math., 1988, vol. 1346, 99-112, Springer-Verlag.

115. Virasoro-type algebras, pseudo-tensor of energy-momentum and operator expansions on the Riemann surfaces, Funktsional. Anal. i Prilozhen. 1989, v. 23, N 1 (with I. M. Krichever).

116. Riemanti surfaces, operator fields, strings. Analogues of the Fourier-Laurent bases, in Memorial Volume for Vadim Knizhnik, "Physics and Mathematics of Strings", eds. L. Brink, D. Friedan, A. M. Polyakov, World Scientific, Singapore, 1989, 356-388 (with I. M. Krichever).

117. Hydrodynamics of the soliton lattices. Differential geometry and Hamiltonian formalism. Uspekhi Mat. Nauk, 1989, v. 44, N 6, 29-98 (with B. A. Dubrovin).

118. Qn the quantization of finite-zoned potentials in connection with string theory, Funktsional. Anal. i Prilozhen., 1990. v 24, N 4.

119. On the equation [L,A] = e.1, RIMS publications, 1991 (Transactions of the Kyoto workshop in the string theory. May 5-21, 1990).

120. Hydrodynamics of the soliton lattices and differential geometry. (Collection of the survey articles. Potsdam. 1992, edited by A. Fokas.)

121. Quasiperiodic structures in topology, in the Proceeding of the Conference: Topological Methods in Modern Mathematics", June 15-22, 1991, dedicated to the 60-th birthday of T. Milnor. Stonybrook University, 1993.

122. Different doubles of the Hopf algebras. Operator algebras on the quantum groups and complex cobordisms, Uspekhi Math. Mauk, 1992, v. 47 iss. 5, pp. 189-190.

123. Integrability in mathematics and theoretical physics: Solitons. The Mathematical Intelligencer, 1992, Vol. 14, N 4. Springer-Veriag, New York.

124. On the Liouville form of the Poisson bracket of hydrodynamic type and nonlinear WKB. Uspechi Math. Nauk, 1993, v. 48, iss. 1 (with A. Maltzev.)

125. Solitons and geometry. Fermi lectures 1992. Scuola Norm. Sup. di Pisa, Cambridge University Press, 1994.

126. String Equation -2. Physical Solution, Algebra and Analysis, 1994, v. 6 iss. 3, pp. 118-140 (with P. G. Grinevich).

127. The semiclassical electron in a magnetic field and lattice. Some problems of the Low Dimensional "Periodic" Topology, Geometric and Functional Analysis, 1995, v. 5, N 2.

128. Topology- 1. Enclyclopedia of Mathematical Sciences, v. 12, Springer Verlag, 1995.

129. Nonselfinersecting magnetic orbits on the plane. Proof of Principle of the Overthrowing of the Cycles, Topics in Topology and Mathematical Physics, 1995, AMS Translations (2), v. 170 (with P. G. Orinevich).

130. Exactly solvable periodic 2-D Schrödinger operators, Russia Math. Surveys, 1995, N 6 (in Russiah) (with A. P. Veselov).

131. Topological quantum numbers observable in the conductivity of normal metals, 1996, Letters of JETF, v. 63, iss. 10, pp. 809-813 (in Russian) (with A. Maltzev).

132. Laplace transformations and exactly solvable 2-dimensional Schrödinger operators, 1997, AMS Translations, ser. 2, v. 179, pp. 109-132 (with A. P. Veselov).

133. Algebraic properties of 2D discrete Schrödinger operators, Russia Math. Surveys, 1997, v. 32, N 1, pp. 223-224 (in Russian).

134. Discrete spectral symmetries of low dimensional differential and difference Schrödinger operators on the Euclidean lattices and 2-manifolds, Russia Math Surveys, 1997, v. 32 N 5 (in Russian) (with I. A. Dynnikov).