From the left to the right: B.A.Dubrovin, S.P.Tsarev, P.M.Akhmetiev,
V.M.Buchstaber, S.P.Novikov, A.P.Veselov, I.M.Krichever, O.I.Mokhov, I.A.Taimanov.
Edinburgh University, 1998.
The S.P.Novikov Seminar came into being in the mid-sixties. S.P.Novikov's
works on the topology of foliations and on the topological invariance of
rational Pontryagin classes were concluded in 1965. A new stage of research
was about to begin, devoted to extraordinary (generalized) cohomology theories
: complex cobodisms and K-theory. Novikov's older pupils (e.g. V.L.Golo)
had moved on to other fields.
By that time K-theory had become extremely popular in wide circles of
the mathematical community in connection with the work of M.Atiyah and
I.Singer on the index of elliptic operators; besides, J.F.Adams discovered
brilliant applications of K-theory in topology itself: the number of linearly
independent vector fields on spheres was found, important subgroups in
the stable homotopy groups of spheres were computed (the image of Whitehead's
J-homomorphism). The possibility of constructing regular methods, based
on K-theory, for the computation of homotopy groups (in particular those
of spheres), more effective than the classical methods of Cartan-Serre-Adams,
was being discussed (the Adams program).
Among the participants of the seminar during the first two-three years,
the names of V.Bukhshtaber, A.Mishchenko, I.Bernstein, I.Volodin, S.Smirnov,
S.Vishik, and F.Bogomolov should be singled out.
In 1966-67 S.P.Novikov  carried out his own program, creating methods
for the regular computation of stable homotopy groups on the basis of complex
cobodisms. It turned out that the Adams program for creating such methods
on the basis of K-theory could not be realized in principle. In contrast,
the new techniques of complex cobodisms were extremely rich in ideas and
applications to topology and algebra, namely:
formal groups, together with their applications in homotopy theory as
well as in the study of fixed points of finite and compact transformation
groups of smooth manifolds, including the remarquable properties of elliptic
genera (, , , , , , ) discovered much later;
the theory of multi-valued formal groups, together with its applications
to topology, algebra, analysis, including the relationship with generalized
shift operator theory (, , , );
the algebra of operations from complex cobordism theory with its numerous
topological applications and beautiful intrinsic algebraic structure that
later led to the "operator (Heisenberg) double" of Hopf algebras, which
is the quantum analog of the algebra of differential operators on a Lie
group (, , , ).
These ideas are developed in the article by V.Bukhshtaber which also
contains new results in this direction.
By the end of the sixties, the participants of the seminar, as well
as its subject matter, changed noticeably. The research interests of several
participants, e.g. I.Berstein and S.Vishik, shifted to other fields of
mathematics. A whole new generation of extremely talented young researchers
appeared in the seminar: G.Kasparov, O.Bogoyavlenskii, S.Gusein-Zade, S.Brakhman,
I.Krichever, V.Dubrovin, M.Bruk, V.Krasnov, M.Brodskii, S.Tankeev, F.Zak,
R.Nadiradze, A.Peresetskii, A.Shokurov, N.Panov, V.Vedenyapin, and others.
For about two years, B.G.Moishenzon became co-director of the seminar.
During this period its participants studied Kahler and agebraic geometry.
Later several members of the seminar began working on the problems of algebraic
Among the participants were several others, who, beginning in 1974,
greatly contributed to the application of methods coming from algebraic
geometry to modern mathematical physics (more specifically, to the periodicity
problems in soliton theory and in integrable systems). The work of these
participants, carried out in the seventies and eighties, became widely
known (, , , , , .
A striking example of feedback, namely, the application of the theory
of nonlinear equations to algebraic geometry, was Novikov's conjecture
on the characterization of the Jacobians of Riemann surfaces (the Riemann-Schottki
problem). The first notable advance in this problem was the work of B.Dubrovin
. A complete proof was given by T.Shiota .
During the second half of the sixties, several participants of the seminar
also studied the topology of nonsimplyconnected manifolds, developing the
ideas that had arisen in foliation theory, as well as those appearing in
the proof of the topological invariance of the rational Pontryagin classes.
At the time, the final formulation of S.P.Novikov's conjecture on higher
signatures  for manifolds with arbitrary fundamental group was put
forward, and was established for abelian fundamental groups. In this period
the seminar, besides its traditional contacts with the seminars of the
leading Moscow mathematical schools, was in constant very intense interaction
with the V.A.Rokhlin seminar in Leningrad.
This period also witnessed the appearance of ideas on the Hermitian
analogs of algebraic K-theory, based on the language and basic concepts
of the Hamiltonian formalism, an algebraic version of sorts of simplectic
The K-theory of infinite-dimensional complexes was then constructed,
leading up to complete answers in many important cases, including computations
for the classifying spaces of compact Lie groups and Eilenberg-MacLane
complexes . The correct higher analogs of algebraic K-theory (at the
same time as Quillen, but on the basis of a different idea) were obtained
Soon after that, the development of Fredholm representations was undertaken,
both for the construction of a topological K-homology theory and for the
higher signature problem (, ).
The S.P.Novikov conjecture on higher signatures eventually became widely
known in mathematics. A huge number of papers is devoted to this conjecture.
It gave the impetus for finding deep relationships between topology, algebra,
and functional analysis (, , , , ).
At the beginning of the seventies, the research interests of the participants
of the seminar diverged: new seminars were organized, where the branches
of topology and algebra mentioned above (cobordisms, formal groups, problems
of nonsimply-connected manifolds, including Hermitian K-theory, problems
of higher signatures and functional analysis techniques) were still studied,
e.g. at the V.Bukhshtaber seminar or the A.Mishchenko seminar.
In the second half of the seventies, only one pure topologist began
research at the S.P.Novikov seminar itself, on the then very new subjectmatter
related to Sullivan's ring approach to rational homotopy type: this was
I.Babenko (, ).
Around 1970-71, the seminar concentrated on the study of special and
generalized relativity (at the time one of the co-directors of the seminar
At the beginning of 1971, S.P.Novikov began working at the Landau Institute
of Theoretical Physics and the interests of the seminar shifted more and
more towards the mathematical problems of modern theoretical physics. Jointly
with several of his pupils (O.Bogoyavlenskii, B.Dubrovin, I.Krichever),
S.P.Novikov originated the qualitative theory of homogeneous cosmological
models, solved the periodic problems of soliton theory, developed the theory
of one-dimentional and two-dimensional Schrodinger operators in periodic
electric and magnetic fields (, , , ). Later these ideas
were developed and led to the creation of the theory of two-dimensional
periodic and rapidly decreasing operators (, , , , ).
In the late seventies and in the eighties, new participants of the seminar
joined in these directions of research, in particular A.Veselov, I.Taimanov,
P.Grinevich, O.Mokhov, S.Tsarev, A.Lyskova, P.Novikov .
In the process of solving problems of physical nature, S.P.Novikov returned
to topology: he found curious topological characteristics of the typical
two-dimensional Schrodinger operator in a magnetic field and in a periodic
lattice, that was later to play a crucial role in explaining the quantum
Hall effect, initiated the notion of multivalued variational calculus in
theoretical physics and mathematics, and constructed the analog of Morse
theory for multivalued functions and functionals (, , , ,
). A.Lyskova and I.Taimanov took part in these studies. The topic was
later developed by quite a few researchers.
These topics were those where a whole new generation of the seminar's
participants, working in topology, began their research: F.Voronov, A.Zorich,
A.Lazarev, D.Millionshchikov, A.Alaniya, Le Tu Thang, I.Dynnikov, and others
(, , , , , , , , ). In this volume these
ideas are developed in the joint article by P.Grinevich and S.P.Novikov
and the one by I.Dynnikov.
A curious cycle of new ideas in Euclidean geometry arose in the eighties
as the result of the interaction with young theoretical physicists from
the Landau Institute (Levitov, Kitaev, Kalugin): the beautiful concept
of quasi-crystallic subgroup of the isometry groups of Euclidean space
in the sense of Novikov-Veselov and other aspects of the geometry of quasi-crystals
were successfully exploited by S.Piunikhin, V.Sadov, and Le Tu Thang ;
the asymptotic problems of soliton theory led to the construction of a
Hamiltonian theory of systems of hydrodynamic type (i.e., quasi-linear
systems of the first order), which had not appeared in the entire hundred-year
history of this field. The Hamiltonian formalism of "hydrodynamic type"
was discovered by S.P.Novikov and B.Dubrovin in 1983, giving rise to a
new branch of Riemannian geometry (, , , ).
In the framework of this geometry, S.Tsarev found a method for integrating
such systems (, , ). As the result of a cycle of further studies
of several participants of the seminar (V.Avilov, S.P.Novikov, I.Krichever,
G.Potemin) the complete analytic solution for the dispersive analog of
the wave equation, undertaken in the early seventies by leading members
of the Landau school (A.B.Gurevich and L.P.Pitaevskii), was finally obtained.
The algebraic and geometrical realization of the Flaschke-MacLaughlin mean
of the (Wisem) soliton equations was developed by Krichever , .
These geometric ideas play an important role in the construction of
the now very popular two-dimensional topological quantum field theory ,
. In the present volume this cycle of ideas appears in the articles
by O.Mokhov, E.Feropontov, and in the subsequent one in those of B.Dubrovin,
In the eighties, several participants of the seminar made significant
contributions to the development of the geometry and topology of supermanifolds.
In particular, F.Voronov and A.Zorich constructed the correct superanalog
of the de Rham complex .
In connection with the operator quantization of boson strings and two-dimensional
conformal field theory, S.P.Novikov and I.Krichever defined the correct
analogs of Laurent and Fourier series for the expansion of meromorphic
tensor fields of any weight on Riemann surfaces. The operator quantization
program for strings was actively developed by physicists in 1970-74 (Veneciano,
Virasoro, Mandelstram, and others) for surfaces of genus 0. The absence
of analogs of Laurent-Fourier series on surfaces of positive genus stopped
further development of this topic in the mid-seventies.
Using a different approach (the continual integral), A.Polyakov succeeded
in quantizing the string for Riemann surfaces (diagrams) of any genus.
The implementation of the program of operator or algebraic quantization
of strings became possible as the result of the work of S.P.Novikov and
I.Krichever at the end of the eighties. The "Krichever-Novikov bases" and
the "Krichever-Novikov algebra" that they constructed are the Riemann analogs
of the Laurent-Fourier bases and of the Virasoro algebra .
 L.A.Alaniya, On manifolds of the Alexander type// Uspekhi Mat. Nauk,
V. 46 ,1991, P. 179-180.
 I.K.Babenko, On real homotopy properties of complete intersections//
Izv. Akad. Nauk SSSR, V.43, 1979, P. 104-240.
 I.K.Babenko, Growth and rationality problems in algebra and topology//
Uspekhi Mat. Nauk, V. 41, 1986, P. 95-142.
 O.I.Bogoyavlenskii, S.P.Novikov, Qualitative theory of homogeneous
cosmological models// Trudy sem. I.G.Petovskogo, V.1, 1975, P. 7-43.
 V.M.Bukhshtaber, The Chern-Dold character in cobordism// Mat.Sbornik,
V. 83, 1970, P. 575-595.
 V.M.Bukhshtaber, Two-valued formal groups. Algebraic theory and
application to cobordism I// Izv. Akad. Nauk SSSR, V. 39, 1975, P. 1044-1064.
 V.M.Bukhshtaber, Two-valued formal groups. Algebraic theory and
application to cobordism II// Izv. Akad. Nauk SSSR, V. 39, 1976, P. 289-325.
 V.M.Bukhshtaber, Topological applications of the theory of two-valued
formal groups// Izv. Akad. Nauk SSSR, V. 42, 1978, P. 130-184.
 V.M.Bukhshtaber, Operator doubles and semigroups of maps into groups//
Doklady Akad. Nauk, 1995, V. 341
 V.M.Bukhshtaber, A.S.Mishchenko, A K-theory on the category of
infinite cell complexes// Izv.Akad.Nauk SSSR, V.32, 1968, P. 460-604.
 V.M.Bukhshtaber, A.S.Mishchenko, S.P.Novikov, Formal groups and
their role in the apparatus of algebraic topology// Uspekhi Mat. Nauk,
V. 26, 1971, 575-595.
 V.M.Bukhshtaber, S.P.Novikov, Formal groups, power systems and
Adams operators// Mat.Sbornik, V. 84, 1971, P. 81-118.
 V.M.Bukhshtaber, A.B.Shokurov, The Landweber-Novikov algebra and
formal vector fields on the line// Funkt. Analiz i Prolozhen., V. 12, 1978,
 A.Connes, M.Gromov, H.Moscovici, Conjecture de Novikov et fibres
presque plats// C.R.A.S. Paris, V.310, 1990, P. 273-277.
 A.Connes, M.Gromov, H.Moscovici, Group cohomology with Lipschitz
control and Higher Signatures// Geometric and Funct. Anal., V. 3, 1993,
 A.Connes, H.Moscovici, Cyclic cohomology, the Novikov conjecture
and hyperbolic group// Topology, V. 29, 1990, P. 1345-1388.
 B.A.Dubrovin, Theta-functions and nonlinear equations// Uspekhi
Mat. Nauk, V. 36, 1981, P. 11-80.
 B.A.Dubrovin, // Izv. Akad. Nauk SSSR, V. 46, 1982, P. 285-296.
 B.A.Dubrovin, On the differetial geometry of strongly integrable
system of hydrodynamical type// Funkt. Analiz Prilozh. V. 24, 1990, P.
 B.A.Dubrovin, // Progr. Math., Boston, Birkhauser, V. 115, 1989,
 B.A.Dubrovin, Differential geometry of the space of orbits of a
Coxeter group// Preprint SISSA-89/94/FM, Trieste
 B.A.Dubrovin, I.M.Krichever, S.P.Novikov Schr"odinger equation
in a periodic field and Riemann surfaces Doklady Akad.Nauk SSSR V. 229,
1976, P. 15-18.
 B.A.Dubrovin, I.M.Krichever, S.P.Novikov, Integrable systems I
// Sovremennye problemy matematiki, Fund. napravleniya, VINITI V. 4, 1985,
 B.A.Dubrovin, V.B.Matveev, S.P.Novikov, Non-linear equations of
the KdV type, finite zone linear operators and Abelian varieties// Uspekhi
Mat. Nauk, V. 31, 1976, P. 55-136. English transl. in Russian Math.Surveys
V. 31, 1976.
 B.A.Dubrovin, S.P.Novikov, Hydrodynamics of weakly deformed soliton
lattices. Differential geometry and Hamiltonian theory// Uspekhi Mat. Nauk,
V. 44, 1989, P. 29-98. English transl. in Russian Math. Surveys V. 44,
1989, P. 35-124
 B.A.Dubrovin, S.P.Novikov, Hydrodynamics of soliton lattices//
Sov. Sci. C Math Phys., V. 9, 1993, P.1-136.
 B.A.Dubrovin, S.P.Novikov , Poisson brackets of hydrodynamic type//
Doklady Akad. Nauk SSSR, V.79, 1984, P. 294-297
 I.A.Dynnikov, Proof of the S.P.Novikov conjecture on the semiclassical
motion of the electron// Mat. Zametki, V. 53, 1993, P. 57-68.
 P.G.Grinevich, R.G.Novikov, Analogs of multisoliton potentials
for the two-dimensional Schr"odinger operator//Funkt. Analiz Prilozh.,
V. 19, 1985, P. 32-42.
 P.G.Grinevich, S.P.Novikov, Two-dimensional inverse scattering
problem for negative energies and generalized analytic functions I. Energies
below ground state // Funkt. Analiz Prilozh., V. 22, 1988, P. 23-33.
 S.M.Gusein-Zade, U-actions of the circle and fixed points// Izv.
Akad. Nauk SSSR, V. 35, 1971, P. 1120-1136.
 G.G.Kasparov Topological invariants of elliptic operators I: K-homology//Izv.Akad.Nauk
SSSR, V. 39, 1975, P. 796-839.
 G.G.Kasparov, Equivariant KK-theory and the Novikov conjecture//
Invent. math., V. 91, 1988, P. 147-201.
 I.M.Krichever, Formal groups and the Atiyah--Hirzebruch formula//1
Izv. Akad. Nauk SSSR, V. 38, 1974, P. 1289-1304.
 I.M.Krichever, // Uspekhi Mat. Nauk, V. 32, 1977, P. 55-136.
 I.M.Krichever, The method of averaging for two-dimensional "integrable"
equations// Funkt. Analiz Prilozh., V. 22, 1988, P. 37-52.
 I.M.Krichever, Spectral theory of two-dimensional periodic operators
and its applications// Uspekhi Mat. Nauk, V. 44, 1989, P. 121-184.
 I.M.Krichever, Generalized elliptic genus and Baker-Akhiezer functions//
Mat. Zametki, V. 47, 1990, P. 132-142.
 I.M.Krichever, Perturbation theory in periodic problems for two-dimensional
integrable systems// Sov. Sci. C Math Phys. part 2., V. 9, 1991,
 I.M.Krichever, // Comm. Pure Appl. Math., V. 47, 1994, P. 437-475.
 I.M.Krichever, S.P.Novikov, Holomorphic fiber bundles over algebraic
curves and nonlinear equations// Uspekhi Mat. Nauk, V.35, 1980,, P. 47-68.
 I.M.Krichever, S.P.Novikov, Virasoro-type algebra, Riemann surfaces
and soliton theory// Funkt. Analiz Prilozh., V. 21, 1987, P. 46-63.
 A.Yu.Lazarev, Novikov homology in knot theory// Mat. Zametki, V.
51, 1992, P. 53-57.
 T.T.Le, Structure of level surfaces of the Morse form// Mat. Zametki,
V. 44, 1988, P. 124-133.
 D.V.Millionshchikov, Embedding of the minimal model k-homotopy
type in the algebra of smooth forms $\Lambda^* (M)$// Uspekhi Mat. Nauk,
V. 43, 1988, P. 147-148.
 D.V.Millionshchikov, Some spectral sequences in analytical homotopy
theory// Mat. Zametki, V. 47, 1990, P. 52-61.
 A.S.Mishchenko, Hermitian K-theory. The theory of characteristic
classes and metods of functional analysis// Uspekhi Mat. Nauk, V. 31, 1976,
 A.S.Mishchenko, Infinite representations of discrete groups and
higher signatures// Izv. Akad. Nauk SSSR, V. 38, 1974, P. 81-106.
 S.P.Novikov, Analytic topology methods from the viewpoint of cobordism
theory// Izv. Akad. Nauk SSSR, V. 31, 1967, P. 885-951.
 S.P.Novikov, Adams operators and fixed points// Izv. Akad. Nauk
SSSR, V. 32, 1968, P. 1245-1263.
 S.P.Novikov, Analytic construction and properties of Hermitian
analogs of k-theory over rings with involution from the point of view of
Hamiltonian formalism. Some applications to differential topology and to
the theory of characteristic classes// Izv.Akad.Nauk SSSR, V. 34, 1970
I N2, P. 253-288; II: N3, P. 475-500. English transl. in Actes Congr. Intern.
Math., V. 2, 1970, P. 39-45.
 S.P.Novikov, The periodicity problem for the Corteveg de Fries
equation. I.// Funkt. Analiz Prilozh., V. 8, 1974, P. 54-66.
 S.P.Novikov, Magnetic Bloch functions and vector bundles. Typical
dispersion laws and their quantum numbers// Doklady Akad. Nauk SSSR, V.
257, 1981, P. 538-543.
 S.P.Novikov, The Hamiltonian formalism and a multivalued analog
of Morse theory// Uspekhi Mat. Nauk, V. 37, 1982, P. 3-49.
 S.P.Novikov, Two-dimensional Shr"odinger operators in periodic
fields// Itogi nauki i tech. Sovremennye problemy matematiki, VINITI, V.
23, 1983, P. 3-23.
 S.P.Novikov, I.A.Taimanov, Periodic extremals of multivalued and
not everywhere positive functionals// Doklady Akad.Nauk SSSR, V. 274, 1984,
 S.P.Novikov, A.P.Veselov, Two-dimensional Shr"odinger operator:
inverse scattering and evolutional equations// Physica, Amsterdam North
Holland, V. 18D, 1986, P. 267-273.
 S.P.Novikov, Various doubles of Hopf algebras. Operator algebras
on quantum groups, complex cobordisms// Uspekhi Akad. Nauk SSSR, V. 47,
1992, P. 189-190.
 S.Oshanine, Sur les genres multiplicatifs d'efinis par des int'egrales
elliptiques// Topology, V. 26, 1987, P. 143-151.
 S.A.Piunikhin, T.T.Le, V.A.Sadov, The geometry of quasicrystals//
Uspekhi Mat. Nauk, V. 48, 1993, P. 41-102.
 T.Shiota, Characterization of Jacobian varieties in terms of soliton
equations// Invent. Math., V. 83, 1986, P. 333-382.
 I.A.Taimanov, Closed extremals on two-dimensional manifolds// Uspekhi
Mat. Nauk, V. 47, 1992, P. 143-185.
 S.P.Tsarev, Liouville Poisson brakets and one-dimensional Hamiltonian
systems of the hydrodynamic type which arise in the Bogolyubov-Whitman
theory of averaging// Uspekhi Mat. Nauk, V. 39, 1984, P. 209-210.
 S.P.Tsarev, Poisson brackets and one-dimensional Hamiltonian systems
of the hydrodynamic type// Doklady Akad.Nauk SSSR, V. 282, 1987, P. 534-537.
 S.P.Tsarev, The geometry of Hamiltonian systems of hydrodynamic
type. The generalized hodograph method// Izv.Akad.Nauk SSSR, V. 54, 1990,
P. 1048-1068. English translation in: Math. USSR Izvestia, V. 37, 1991,
 I.A.Volodin, Algebraic K-theory as an extraordinary homology theory
on the category of associative rings with unit// Izv. Akad. Nauk SSSR,
V. 35, 1971, P. 884-873.
 T.Voronov, Geometric integration theory on supermanifolds// Sov.
Sci. C Math Phys., V. 9, 1992, P. 1-138.
 F.F.Voronov, A.V.Zorich, Complexes of forms on supermanifolds//
Funkt. Analiz Prilozh., V. 20, 1986, P. 58-59.
 A.V.Zorich, Quasiperiodic structure of level surfaces of a Morse
1-form, close to a rational one, an S.P.Novikov problem// Izv.Akad.Nauk
SSSR, V. 51, 1987, P. 1322-1344.
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