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\head {\smc The S.P.Novikov Seminar } \endhead
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Our 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 [49]
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:
$\bullet$ 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
([50], [11], [5], [31], [34], [59], [38])
discovered much later;
$\bullet$ the theory of multi-valued formal groups, together with its
applications to topology, algebra, analysis, including the relationship
with generalized shift operator theory ([12], [6], [7], [8]);
$\bullet$ 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 ([49], [13], [58], [9]).
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 geometry itself.
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 ([24], [52], [35], [41], [17], [23].
In this volume these ideas are developed in the papers by I.~Krichever,
S.~Natanzon, A.~Veselov. 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 [18].
A complete proof was given by T.~Shiota [61].
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 [51]
for manifolds with arbitrary fundamental group was put forward, and was
established for abelian fundamental groups $\pi_1$. 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 geometry [51].
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 [10].
The correct higher analogs of algebraic $K$-theory $K^0$, $K^1$ (at the
same time as Quillen, but on the basis of a different idea) were
obtained [66].
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 ([32], [47]).
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 ([48], [33],
[16], [14], [15]).
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 ([2], [3]).
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 was V.~P.~Myasnikov).
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 Schr\"odinger operators in periodic
electric and magnetic fields ([4], [24], [22], [53]).
Later these ideas were developed and led to the creation of the theory
of two-dimensional periodic and rapidly decreasing operators
([57], [29], [30], [37], [39]).
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 [26].
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 Schr\"odinger 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 ([53], [54], [55], [56], [62]).
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 ([68], [67], [69], [43], [45], [46], [1], [44],
[28]).
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 [60];
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 ([27], [25], [26], [19]).
In the framework of this geometry, S.~Tsarev found a method for
integrating such systems ([63], [64], [65]).
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 [36], [40].
These geometric ideas play an important role in the construction of the
now very popular two-dimensional topological quantum field theory [20],
[21].
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, M.~Pavlov, V.~Alekseev.
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 [67].
The ideas of supersymmetery are developed in this volume in the article
by F.~Voronov.
In connection with the operator quantization of boson strings and
two-dimen\-sio\-nal 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 [42].
This topic is that of numerous papers by many authors. In this volume
it is represented by O.~Sheinman's article.
\Refs
\ref \no1
\by L.~A.~Alaniya,
\paper On manifolds of the Alexander type
\jour Uspekhi Mat. Nauk
\vol 46 \yr 1991 \pages 179--180
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no2
\by I.~K.~Babenko
\paper On real homotopy properties of complete intersections
\jour Izv. Akad. Nauk SSSR
\vol 43 \yr 1979 \pages 104--240
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no3
\by I.~K.~Babenko
\paper Growth and rationality problems in algebra and topology
\jour Uspekhi Mat. Nauk
\vol 41 \yr 1986 \pages 95--142
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no4
\by O.~I.~Bogoyavlenskii, S.~P.~Novikov
\paper Qualitative theory of homogeneous cosmological models
\jour Trudy sem. I.~G.~Petovskogo
\vol 1 \yr 1975 \pages 7--43
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no5
\by V.~M.~Bukhshtaber
\paper The Chern-Dold character in cobordism
\jour Mat.Sbornik
\vol 83 \yr 1970 \pages 575--595
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no6
\by V.~M.~Bukhshtaber
\paper Two-valued formal groups. Algebraic theory and application to
cobordism I
\jour Izv. Akad. Nauk SSSR
\vol 39 \yr 1975 \pages 1044--1064
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no7
\by V.~M.~Bukhshtaber
\paper Two-valued formal groups. Algebraic theory and application to
cobordism II
\jour Izv. Akad. Nauk SSSR
\vol 40 \yr 1976 \pages 289--325
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no8
\by V.~M.~Bukhshtaber
\paper Topological applications of the theory of two-valued formal
groups
\jour Izv. Akad. Nauk SSSR
\vol 42 \yr 1978 \pages 130--184
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no9
\by V.~M.~Bukhshtaber
\paper Operator doubles and semigroups of maps into groups
\jour Doklady Akad. Nauk
\yr 1995
\vol 341
\pages
\transl\nofrills English transl. in
\endref
\ref \no10
\by V.~M.~Bukhshtaber, A.~S.~Mishchenko
\paper A $K$-theory on the category of infinite cell complexes
\jour Izv.Akad.Nauk SSSR
\vol 32 \yr 1968 \pages 460--604
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no11
\by V.~M.~Bukhshtaber, A.~S.~Mishchenko, S.~P.~Novikov
\paper Formal groups and their role in the apparatus of algebraic
topology
\jour Uspekhi Mat. Nauk \vol 26 \yr 1971 \pages 575--595
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no12
\by V.~M.~Bukhshtaber, S.~P.~Novikov
\paper Formal groups, power systems and Adams operators
\jour Mat.Sbornik
\vol 84 \yr 1971 \pages 81--118
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no13
\by V.~M.~Bukhshtaber, A.~B.~Shokurov
\paper The Landweber--Novikov algebra and formal vector fields on the
line
\jour Funkt. Analiz Prolozhen.
\vol 12
\yr 1978
\pages 1-11
\transl\nofrills English transl. in
\endref
\ref \no14
\by A.~Connes, M.~Gromov, H.~Moscovici
\paper Conjecture de Novikov et fibr\'es presque plats
\jour C.R.A.S. Paris
\vol 310 \yr 1990 \pages 273--277
\endref
\ref \no15
\by A.~Connes, M.~Gromov, H.~Moscovici
\paper Group cohomology with Lipschitz control and Higher Signatures
\jour Geometric and Funct. Anal.
\vol 3 \yr 1993 \pages 1--78
\endref
\ref \no16
\by A.~Connes, H.~Moscovici
\paper Cyclic cohomology, the Novikov conjecture and hyperbolic group
\jour Topology
\vol 29 \yr 1990 \pages 1345--1388
\endref
\ref \no17
\by B.~A.~Dubrovin
\paper Theta-functions and nonlinear equations
\jour Uspekhi Mat. Nauk
\jour Uspekhi Mat. Nauk \vol 36 \yr 1981
\pages 11-80
\transl\nofrills English transl. in
\jour Russian Math. Surveys
\endref
\ref \no18
\by B.~A.~Dubrovin
\paper
\jour Izv. Akad. Nauk SSSR
\vol 46 \yr 1982 \pages 285--296
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no19
\by B.~A.~Dubrovin
\paper On the differetial geometry of strongly integrable system
of hydrodynamical type
\jour Funkt. Analiz Prilozh.
\vol 24 \yr 1990 \pages 25--30
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no20
\by B.~A.~Dubrovin
\paper
\jour Progr. Math., Boston, Birkhauser,
\vol 115 \yr 1989 \pages 313--361
\endref
\ref \no21
\by B.~A.~Dubrovin
\paper Differential geometry of the space of orbits of a Coxeter group
\jour Preprint SISSA-89/94/FM, Trieste
\endref
\ref \no22
\by B.~A.~Dubrovin, I.~M.~Krichever, S.~P.~Novikov
\paper Schr\"odinger equation in a periodic field and Riemann surfaces
\jour Doklady Akad.Nauk SSSR
\vol 229 \yr 1976 \pages 15-18
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no23
\by B.~A.~Dubrovin, I.~M.~Krichever, S.~P.~Novikov
\paper Integrable systems I
\jour Sovremennye problemy matematiki, Fund. napravleniya, VINITI
\vol 4 \yr 1985
\pages 179-284
\transl\nofrills English transl. in
\endref
\ref \no24
\by B.~A.~Dubrovin, V.~B.~Matveev, S.~P.~Novikov
\paper Non-linear equations of the $K\,dV$ type, finite zone linear
operators and Abelian varieties
\jour Uspekhi Mat. Nauk
\vol 31 \yr 1976 \pages 55--136
\transl\nofrills English transl. in
\jour Russian Math.Surveys \vol 31 \yr 1976
\endref
\ref \no25
\by B.~A.~Dubrovin, S.~P.~Novikov
\paper Hydrodynamics of weakly deformed soliton lattices.
Differential geometry and Hamiltonian theory
\jour Uspekhi Mat. Nauk \vol 44 \yr 1989 \pages 29--98
\transl\nofrills English transl. in
\jour Russian Math. Surveys \vol 44 \yr 1989 \pages 35--124
\endref
\ref \no26
\by B.~A.~Dubrovin, S.~P.~Novikov
\paper Hydrodynamics of soliton lattices
\jour Sov. Sci. C Math Phys.
\vol 9 \yr 1993 \pages 1--136
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no27
\by B.~A.~Dubrovin, S.~P.~Novikov
\paper n Poisson brackets of hydrodynamic type
\jour Doklady Akad. Nauk SSSR
\vol 279 \yr 1984 \pages 294--297
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no28
\by I.~A.~Dynnikov
\paper Proof of the S.~P.~Novikov conjecture on the semiclassical
motion of the electron
\jour Mat. Zametki \vol 53 \yr 1993 \pages 57--68
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no29
\by P.~G.~Grinevich, R.~G.~Novikov
\paper Analogs of multisoliton potentials for the two-dimensional
Schr\"odinger operator
\jour Funkt. Analiz Prilozh. \vol 19 \yr 1985 \pages 32--42
\transl\nofrills English transl. in \jour \vol \yr \endref
\ref \no30
\by P.~G.~Grinevich, S.~P.~Novikov
\paper Two-dimensional inverse scattering problem for negative energies
and generalized analytic functions I. Energies below ground state
\jour Funkt. Analiz Prilozh.
\vol 22 \yr 1988 \pages 23--33
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no31
\by S.~M.~Gusein-Zade
\paper $U$-actions of the circle and fixed points
\jour Izv. Akad. Nauk SSSR
\vol 35 \yr 1971 \pages 1120--1136
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no32
\by G.~G.~Kasparov
\paper Topological invariants of elliptic operators I: $K$-homology
\jour Izv.Akad.Nauk SSSR
{\bf 39} (1975), 796-839,
\vol 39 \yr 1975 \pages 796--839
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no33
\by G.~G.~Kasparov
\paper Equivariant $KK$-theory and the Novikov conjecture
\jour Invent. math.
\vol 91 \yr 1988 \pages 147--201
\endref
\ref \no34
\by I.~M.~Krichever
\paper Formal groups and the Atiyah--Hirzebruch formula
\jour Izv. Akad. Nauk SSSR
\vol 38 \yr 1974 \pages 1289--1304
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no35
\by I.~M.~Krichever
\paper
\jour Uspekhi Mat. Nauk \vol 32 \yr 1977
\pages 55-136
\endref
\ref \no36
\by I.~M.~Krichever
\paper The method of averaging for two-dimensional `integrable'
equations
\jour Funkt. Analiz Prilozh.
\vol 22 \yr 1988 \pages 37--52
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no37
\by I.~M.~Krichever
\paper Spectral theory of two-dimensional periodic operators and
its applications
\jour Uspekhi Mat. Nauk
\vol 44 \yr 1989 \pages 121--184
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no38
\by I.~M.~Krichever
\paper generalized elliptic genus and Baker-Akhiezer functions
\jour Mat. Zametki
\vol 47 \yr 1990 \pages 132--142
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no39
\by I.~M.~Krichever
\paper Perturbation theory in periodic problems for two-dimensional
integrable systems
\jour Sov. Sci. C Math Phys. part 2.
\vol 9 \yr 1991 \pages
\endref
\ref \no40
\by I.~M.~Krichever
\paper
\jour Comm. Pure Appl. Math.
\vol 47 \yr 1994 \pages 437--475
\endref
\ref \no41
\by I.~M.~Krichever, S.~P.~Novikov
\paper Holomorphic fiber bundles over algebraic curves and nonlinear
equations
\jour Uspekhi Mat. Nauk \vol 35 \yr 1980
\pages 47-68
\endref
\ref \no42
\by I.~M.~Krichever, S.~P.~Novikov
\paper Virasoro-type algebra, Riemann surfaces and soliton theory
\jour Funkt. Analiz Prilozh.
\vol 21 \yr 1987 \pages 46--63
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no43
\by A.~Yu.~Lazarev
\paper Novikov homology in knot theory
\jour Mat. Zametki
\vol 51 \yr 1992 \pages 53--57
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no44
\by T.~T.~Le
\paper Structure of level surfaces of the Morse form
\jour Mat. Zametki
\vol 44 \yr 1988 \pages 124--133
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no45
\by D.~V.~Millionshchikov
\paper Embedding of the minimal model $k$-homotopy type in the algebra
of smooth forms $\Lambda^* (M)$
\jour Uspekhi Mat. Nauk \vol 43
\yr 1988 \pages 147--148 \transl\nofrills English transl. in \jour
\vol \yr \endref
\ref \no46
\by D.~V.~Millionshchikov
\paper Some spectral sequences in analytical homotopy theory
\jour Mat. Zametki
\vol 47 \yr 1990 \pages 52--61
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no47
\by A.~S.~Mishchenko
\paper Hermitian $K$-theory. The theory of characteristic classes and
metods of functional analysis
\jour Uspekhi Mat. Nauk
\vol 31 \yr 1976 \pages 71--138
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no48
\by A.~S.~Mishchenko
\paper Infinite representations of discrete groups and higher
signatures
\jour Izv. Akad. Nauk SSSR
\vol 38 \yr 1974 \pages 81--106
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no49
\by S.~P.~Novikov
\paper Analytic topology methods from the viewpoint of cobordism theory
\jour Izv. Akad. Nauk SSSR
\vol 31 \yr 1967 \pages 885--951
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no50
\by S.~P.~Novikov
\paper Adams operators and fixed points
\jour Izv. Akad. Nauk SSSR
\vol 32 \yr 1968 \pages 1245--1263
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no51
\by S.~P.~Novikov
\paper 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
\jour Izv.Akad.Nauk SSSR \vol 34 \yr 1970 \nofrills {I: N2,}
\pages 253-288 \moreref
\nofrills {II: N3,} \pages 475--500
\transl\nofrills English transl. in
\jour Actes Congr. Intern. Math. \vol 2 \yr 1970 \pages 39--45 \endref
\ref \no52
\by S.~P.~Novikov
\paper The periodicity problem for the Corteveg de Fries equation. I.
\jour Funkt. Analiz Prilozh. \vol 8 \yr 1974
\pages 54-66
\endref
\ref \no53
\by S.~P.~Novikov
\paper Magnetic Bloch functions and vector bundles. Typical dispersion
laws and their quantum numbers
\jour Doklady Akad. Nauk SSSR
\vol 257 \yr 1981 \pages 538--543 \transl\nofrills
English transl. in \jour \vol \yr \endref
\ref \no54
\by S.~P.~Novikov
\paper The Hamiltonian formalism and a multivalued analog of Morse
theory
\jour Uspekhi Mat. Nauk \vol 37 \yr 1982 \pages 3--49
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no55
\by S.~P.~Novikov
\paper Two--dimensional Shr\"odinger operators in periodic fields
\jour Itogi nauki i tech. Sovremennye problemy matematiki, VINITI,
\vol 23 \yr 1983 \pages 3--23
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no56
\by S.~P.~Novikov, I.~A.~Taimanov
\paper Periodic extremals of multivalued and not everywhere positive
functionals
\jour Doklady Akad.Nauk SSSR, \vol 274 \yr 1984
\pages26--28 \transl\nofrills English transl. in \jour \vol \yr
\endref
\ref \no57
\by S.~P.~Novikov, A.~P.~Veselov
\paper Two-dimensional Shr\"odinger operator: inverse scattering
and evolutional equations
\jour Physica, Amsterdam North Holland,
\vol 18D \yr 1986 \pages 267--273
\endref
\ref \no58
\by S.~P.~Novikov
\paper Various doubles of Hopf algebras. Operator algebras on quantum
groups, complex cobordisms
\jour Uspekhi Akad. Nauk SSSR
\vol 47 \yr 1992
\pages 189-190
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no59
\by S.~Oshanine
\paper Sur les genres multiplicatifs d\'efinis par des int\'egrales
elliptiques
\jour Topology
\vol 26 \yr 1987 \pages 143--151
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no60
\by S.~A.~Piunikhin, T.~T.~Le, V.~A.~Sadov
\paper The geometry of quasicrystals
\jour Uspekhi Mat. Nauk
\vol 48 \yr 1993 \pages 41--102
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no61
\by T.~Shiota
\paper Characterization of Jacobian varieties in terms of soliton
equations
\jour Invent. Math.
\vol 83 \yr 1986 \pages 333--382
\endref
\ref \no62
\by I.~A.~Taimanov
\paper Closed extremals on two-dimensional manifolds
\jour Uspekhi Mat. Nauk
\vol 47 \yr 1992 \pages 143--185
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no63
\by S.~P.~Tsarev
\paper Liouville Poisson brakets and one-dimensional Hamiltonian
systems of the hydrodynamic type which arise in the Bogolyubov-Whitman
theory of averaging
\jour Uspekhi Mat. Nauk
\vol 39 \yr 1984 \pages 209--210
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no64
\by S.~P.~Tsarev
\paper Poisson brackets and one-dimensional Hamiltonian systems of
the hydrodynamic type
\jour Doklady Akad.Nauk SSSR
\vol 282 \yr 1987 \pages 534--537
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no65
\by S.~P.~Tsarev
\paper The geometry of Hamiltonian systems of hydrodynamic type. The
generalized hodograph method
\jour Izv.Akad.Nauk SSSR \vol 54 \yr 1990 \pages
1048--1068 \transl\nofrills English transl. in
\jour Math. USSR Izvestia \vol 37 \yr 1991 \page 397--419
\endref
\ref \no66
\by I.~A.~Volodin
\paper Algebraic $K$-theory as an extraordinary homology theory on the
category of associative rings with unit
\jour Izv. Akad. Nauk SSSR \vol 35 \yr 1971 \pages 884--873
\transl\nofrills English transl. in \jour \vol \yr \endref
\ref \no67
\by T.~Voronov
\paper Geometric integration theory on supermanifolds
\jour Sov. Sci. C Math Phys.
\vol 9 \yr 1992 \pages 1--138
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no68
\by F.~F.~Voronov, A.~V.~Zorich
\paper Complexes of forms on supermanifolds
\jour Funkt. Analiz Prilozh.
\vol 20 \yr 1986 \pages 58-59
\transl\nofrills English transl. in
\jour \vol \yr
\endref
\ref \no69
\by A.~V.~Zorich
\paper Quasiperiodic structure of level surfaces of a Morse $1$-form,
close to a rational one, an S.~P.~Novikov problem
\jour Izv.Akad.Nauk SSSR \vol 51 \yr 1987 \pages 1322--1344
\transl\nofrills English transl. in \jour \vol \yr
\endref
\endRefs
\enddocument