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:
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;
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]);
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].
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. 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 (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 Schrodinger 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 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 ([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].
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 [42].
[1] L.A.Alaniya, On manifolds of the Alexander type// Uspekhi Mat. Nauk, V. 46 ,1991, P. 179-180.
[2] I.K.Babenko, On real homotopy properties of complete intersections// Izv. Akad. Nauk SSSR, V.43, 1979, P. 104-240.
[3] I.K.Babenko, Growth and rationality problems in algebra and topology// Uspekhi Mat. Nauk, V. 41, 1986, P. 95-142.
[4] O.I.Bogoyavlenskii, S.P.Novikov, Qualitative theory of homogeneous cosmological models// Trudy sem. I.G.Petovskogo, V.1, 1975, P. 7-43.
[5] V.M.Bukhshtaber, The Chern-Dold character in cobordism// Mat.Sbornik, V. 83, 1970, P. 575-595.
[6] V.M.Bukhshtaber, Two-valued formal groups. Algebraic theory and application to cobordism I// Izv. Akad. Nauk SSSR, V. 39, 1975, P. 1044-1064.
[7] V.M.Bukhshtaber, Two-valued formal groups. Algebraic theory and application to cobordism II// Izv. Akad. Nauk SSSR, V. 39, 1976, P. 289-325.
[8] V.M.Bukhshtaber, Topological applications of the theory of two-valued formal groups// Izv. Akad. Nauk SSSR, V. 42, 1978, P. 130-184.
[9] V.M.Bukhshtaber, Operator doubles and semigroups of maps into groups// Doklady Akad. Nauk, 1995, V. 341
[10] 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.
[11] 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.
[12] V.M.Bukhshtaber, S.P.Novikov, Formal groups, power systems and Adams operators// Mat.Sbornik, V. 84, 1971, P. 81-118.
[13] V.M.Bukhshtaber, A.B.Shokurov, The Landweber-Novikov algebra and formal vector fields on the line// Funkt. Analiz i Prolozhen., V. 12, 1978, P. 1-11.
[14] A.Connes, M.Gromov, H.Moscovici, Conjecture de Novikov et fibres presque plats// C.R.A.S. Paris, V.310, 1990, P. 273-277.
[15] A.Connes, M.Gromov, H.Moscovici, Group cohomology with Lipschitz control and Higher Signatures// Geometric and Funct. Anal., V. 3, 1993, P. 1-78.
[16] A.Connes, H.Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic group// Topology, V. 29, 1990, P. 1345-1388.
[17] B.A.Dubrovin, Theta-functions and nonlinear equations// Uspekhi Mat. Nauk, V. 36, 1981, P. 11-80.
[18] B.A.Dubrovin, // Izv. Akad. Nauk SSSR, V. 46, 1982, P. 285-296.
[19] B.A.Dubrovin, On the differetial geometry of strongly integrable system of hydrodynamical type// Funkt. Analiz Prilozh. V. 24, 1990, P. 25-30.
[20] B.A.Dubrovin, // Progr. Math., Boston, Birkhauser, V. 115, 1989, P. 313-361.
[21] B.A.Dubrovin, Differential geometry of the space of orbits of a Coxeter group// Preprint SISSA-89/94/FM, Trieste
[22] 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.
[23] B.A.Dubrovin, I.M.Krichever, S.P.Novikov, Integrable systems I // Sovremennye problemy matematiki, Fund. napravleniya, VINITI V. 4, 1985, P. 179-284.
[24] 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.
[25] 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
[26] B.A.Dubrovin, S.P.Novikov, Hydrodynamics of soliton lattices// Sov. Sci. C Math Phys., V. 9, 1993, P.1-136.
[27] B.A.Dubrovin, S.P.Novikov , Poisson brackets of hydrodynamic type// Doklady Akad. Nauk SSSR, V.79, 1984, P. 294-297
[28] 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.
[29] 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.
[30] 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.
[31] S.M.Gusein-Zade, U-actions of the circle and fixed points// Izv. Akad. Nauk SSSR, V. 35, 1971, P. 1120-1136.
[32] G.G.Kasparov Topological invariants of elliptic operators I: K-homology//Izv.Akad.Nauk SSSR, V. 39, 1975, P. 796-839.
[33] G.G.Kasparov, Equivariant KK-theory and the Novikov conjecture// Invent. math., V. 91, 1988, P. 147-201.
[34] I.M.Krichever, Formal groups and the Atiyah--Hirzebruch formula//1 Izv. Akad. Nauk SSSR, V. 38, 1974, P. 1289-1304.
[35] I.M.Krichever, // Uspekhi Mat. Nauk, V. 32, 1977, P. 55-136.
[36] I.M.Krichever, The method of averaging for two-dimensional "integrable" equations// Funkt. Analiz Prilozh., V. 22, 1988, P. 37-52.
[37] I.M.Krichever, Spectral theory of two-dimensional periodic operators and its applications// Uspekhi Mat. Nauk, V. 44, 1989, P. 121-184.
[38] I.M.Krichever, Generalized elliptic genus and Baker-Akhiezer functions// Mat. Zametki, V. 47, 1990, P. 132-142.
[39] I.M.Krichever, Perturbation theory in periodic problems for two-dimensional integrable systems// Sov. Sci. C Math Phys. part 2., V. 9, 1991,
[40] I.M.Krichever, // Comm. Pure Appl. Math., V. 47, 1994, P. 437-475.
[41] I.M.Krichever, S.P.Novikov, Holomorphic fiber bundles over algebraic curves and nonlinear equations// Uspekhi Mat. Nauk, V.35, 1980,, P. 47-68.
[42] I.M.Krichever, S.P.Novikov, Virasoro-type algebra, Riemann surfaces and soliton theory// Funkt. Analiz Prilozh., V. 21, 1987, P. 46-63.
[43] A.Yu.Lazarev, Novikov homology in knot theory// Mat. Zametki, V. 51, 1992, P. 53-57.
[44] T.T.Le, Structure of level surfaces of the Morse form// Mat. Zametki, V. 44, 1988, P. 124-133.
[45] 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.
[46] D.V.Millionshchikov, Some spectral sequences in analytical homotopy theory// Mat. Zametki, V. 47, 1990, P. 52-61.
[47] A.S.Mishchenko, Hermitian K-theory. The theory of characteristic classes and metods of functional analysis// Uspekhi Mat. Nauk, V. 31, 1976, P. 71-138.
[48] A.S.Mishchenko, Infinite representations of discrete groups and higher signatures// Izv. Akad. Nauk SSSR, V. 38, 1974, P. 81-106.
[49] S.P.Novikov, Analytic topology methods from the viewpoint of cobordism theory// Izv. Akad. Nauk SSSR, V. 31, 1967, P. 885-951.
[50] S.P.Novikov, Adams operators and fixed points// Izv. Akad. Nauk SSSR, V. 32, 1968, P. 1245-1263.
[51] 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.
[52] S.P.Novikov, The periodicity problem for the Corteveg de Fries equation. I.// Funkt. Analiz Prilozh., V. 8, 1974, P. 54-66.
[53] 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.
[54] S.P.Novikov, The Hamiltonian formalism and a multivalued analog of Morse theory// Uspekhi Mat. Nauk, V. 37, 1982, P. 3-49.
[55] 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.
[56] S.P.Novikov, I.A.Taimanov, Periodic extremals of multivalued and not everywhere positive functionals// Doklady Akad.Nauk SSSR, V. 274, 1984, P. 26-28.
[57] 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.
[58] 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.
[59] S.Oshanine, Sur les genres multiplicatifs d'efinis par des int'egrales elliptiques// Topology, V. 26, 1987, P. 143-151.
[60] S.A.Piunikhin, T.T.Le, V.A.Sadov, The geometry of quasicrystals// Uspekhi Mat. Nauk, V. 48, 1993, P. 41-102.
[61] T.Shiota, Characterization of Jacobian varieties in terms of soliton equations// Invent. Math., V. 83, 1986, P. 333-382.
[62] I.A.Taimanov, Closed extremals on two-dimensional manifolds// Uspekhi Mat. Nauk, V. 47, 1992, P. 143-185.
[63] 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.
[64] S.P.Tsarev, Poisson brackets and one-dimensional Hamiltonian systems of the hydrodynamic type// Doklady Akad.Nauk SSSR, V. 282, 1987, P. 534-537.
[65] 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, P. 397-419.
[66] 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.
[67] T.Voronov, Geometric integration theory on supermanifolds// Sov. Sci. C Math Phys., V. 9, 1992, P. 1-138.
[68] F.F.Voronov, A.V.Zorich, Complexes of forms on supermanifolds// Funkt. Analiz Prilozh., V. 20, 1986, P. 58-59.
[69] 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.