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This is a survey article dedicated to the study of topological quantities in theory of normal metals discovered in the works of the authors during the last years. Our results are based on the theory of dynamical systems on Fermi surfaces. The physical foundations of this theory (the so-called "Geometric Strong Magnetic Field Limit") were found by the school of I.M.Lifshitz many years ago. Here the new aspects in the topology of quasiperiodic functions are developed., Comment: 45 pages, 14 Encapsulated Postscript figures

A non-traditional approach to the discretization of differential-geometrical connections was suggested by the authors in 1997. At the same time we started studying first order difference ``black and white triangle operators (equations)'' on triangulated surfaces with a black and white coloring or triangles. In this work, we develop the theory of these operators and equations, showing their similarity with complex derivatives., Comment: latex, 23 pages, 10 figures

This work is dedicated to some new exotic homological constructions associated with the different Morse-type inequalities for differential forms and vector fields. It contains also survey of ideas developed by the present author in 1986 for this goal., Comment: 18 Pages, Latex, Theorem 1 is corrected, two more references added

An effective description of the inverse spectral data corresponding to the real periodic and quasiperiodic solutions for the sine-gordon equation is obtained. In particular, the explicit formula for the so-called topological charge of the solutions is found and proved. As it was understood already 20 years ago, it is very hard to extract any formula for this quantity from the Theta-functional expressions. A new method was developed by the authors for this goal. In the appendix 3 an analog of the Fourier-Laurent integral transform on the Riemann surfaces is defined, based on the constructions of the Soliton Theory. It is a natural continuous analog of the discrete Krichever-Novikov bases developed in the late 80s for the needs of the Operator Quantization of String Theory., Comment: 30 Pages, 4 Fig, Latex

We address the general classification problem of all stable associative product structures in the complex cobordism theory. We show how to reduce this problem to the algebraic one in terms of the Hopf algebra $S$ (the Landweber-Novikov algebra) acting on its dual Hopf algebra $S^*$ with a distinguished ``topologically integral'' part $\Lambda$ that coincids with the coefficient ring of the complex cobordism. We describe the formal group and its logarithm in terms of representations of $S$. We introduce one-dimensional representations of a Hopf algebra. We give series of examples of such representations motivated by well-known topological and algebraic results. We define and study the divided difference operators on an integral domain. We discuss certain important examples of such operators arising from analysis, representation theory, and noncommutative algebra. We give a special attention to the division operators by a noninvertible element of a ring. We give new general constructions of associative product structures (not necessarily commutative) using the divided difference operators. As application, we describe new classes of associative products in the complex cobordism theory., Comment: 21 pages

We study in this work the important class of nonlocal Poisson Brackets (PB) which we call weakly nonlocal. They appeared recently in some investigations in the Soliton Theory. However there was no theory of such brackets except very special first order case. Even in this case the theory was not developed enough. In particular, we introduce the Physical forms and find Casimirs, Momentum and Canonical forms for the most important Hydrodynamic type PB of that kind and their dependence on the boundary conditions., Comment: 45 pages, latex

Field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar field, advected by the Gaussian strongly anisotropic velocity field. Inertial-range anomalous scaling behavior is established, and explicit asymptotic expressions for the n-th order structure functions of scalar field are obtained; they are represented by superpositions of power laws with nonuniversal (dependent on the anisotropy parameters) anomalous exponents. In the limit of vanishing anisotropy, the exponents are associated with tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. For the finite anisotropy, the exponents cannot be associated with individual operators (which are essentially ``mixed'' in renormalization), but the aforementioned hierarchy survives for all the cases studied. The second-order structure function is studied in more detail using the renormalization group and zero-mode techniques., Comment: REVTEX file with EPS figures

Family of doublings of Hopf algeras based on the product of algebra and its dual are constructed and studied. Special cases of these construction may be considered as natural quantum analogs of rings of differential operators on groups. Such constructions appeared in 1966-67 in the authors works in the complex cobordism theory. These constructions do not leed to the Hopf algebras (except of the special case of the Drinfeld's quantum double which is not the same as the natural quantum analog of the ring of operators on groups). However, they have important ''almost Hopf'' properties important in particular in the topological and analytical applications., Comment: LATEX, 5 pages

Symplectic vector-valued scalar product is constructed on the spaces of solutions of the real discrete Shrodinger equation with fixed value of the spectral parameter on graphs. It takes values in the first homology group of the graph. This scalar product plays fundamental role in the Scattering theory for the graphs with tails. In particular, all unitarity properties of scattering follow from elementary symplectic geometry, Comment: LATEX2e, 3 pages

Discrete Lagrangian Systems on graphs are considered. Vector-valued closed differential 2-form on the space of solutions is constructed. This form takes values in the first homology group of the graph. This construction generalizes the Symplectic Wronskian for the linear self-adjoint operators on graphs found in 1997 by the first author and used for the needs of the Scattering Theory for graphs with tails, Comment: Latex, 4 pages

This work is the direct continuation of the author's note published in Russian Math Surveys 52 (1997), no 6. Discrete Schrodinger operators on graphs and higher dimensional simplicial complexes are considered. A vector-valued symplectic form on the space of solutions is consructed. This form, "Symplectic Wronskian", takes value in the group of 1-dimensional cycles. This construction has important applications for the Scattering Theory on graphs with tails. Effective diagonalization of the real Fermionic quadratic form is presented in the Appendix. This construction appeared first time in 1987 in the author's paper dedicated to an analogue of Morse theory for vector fields (it was published as an Appendix to the author's joint paper with M.Shubin, Soviet Math Dokl 34 (1987) no 1)., Comment: Latex, 25 pages. Published in The Arnoldfest: Proceedings of a Conference in Honour of V.I.Arnold for His Sixtieth Birthday, Fields Institute Comm, V 24

Commutative rings of one-dimensional difference operators of rank l>1 and their deformations are effectively constructed. Our analytical constructions are based on the so-called ''Tyurin parameters'' for the stable framed holomorphic vector bundles over algebraic curves of the genus equal to g and Chern number equal to lg. These parameters were heavily used by the present authors already in 1978-80 for the differential operators. Their deformations in the discrete case are governed by the 2D Toda Lattice hierarhy instead of KP. New integrable systems appear here in the case l=2,g=1. The theory of higher rank difference operators is much more rich than the rank one case where only 2-point constructions on the spectral curve were used in the previous literature (i.e. number of 'infinite points'' is equal to 2). One-point constructions appear in this problem for every even rank l=2k. Only in this case commutative rings depend on the functional parameters. Two-point constructions will be studied in the next work: even for higher rank l>1 this case can be solved in Theta-functions. It is not so for one-point constructions with rank l>1., Comment: Latex2e file, 4 pages

It is shown that the fourth order real self-adjoint difference operator on the Tivalent Tree admits nontrivial deformations preserving one energy level and therefore defines a nontrinial hierarhy of the completely integrable nonlinear systems representible through the ''L-A-B-triple''. The Laplace transformations for these operators are also constructed. Nothing like that exists for the second order difference operators on this tree., Comment: LATEX file, 3 pages

Different cases of sequences of the Laplace Transformations for the 2D Schrodinger operator in the periodic magnetic field and electric potential are considered. They lead to the exactly solvable operators with nonstandard spectral properties including the double-periodic operators with algebraic Fermi surface known from the periodic soliton theory. Two appendices are added. In the Appendix I (the author - S. Novikov) two discretizations of the 2D operators and Laplace transformations are constructed. In the Appendix II (the authors - S. Novikov and I. Taimanov) some nonstandard 1D discrete operators are constructed with very interesting spectral properties., Comment: AMSTeX, 25 pages

Euler-Darboux-Backlund and Laplace transformations are considered for the one- and two-dimensional Schrodinger operators. Their discrete analogs are constructed and generalized for the multidimensional lattices and two-manifolds with special "black-white" triangulations. Nonstandard generalizations of the connections and curvature are constructed for the simplicial complexes. Exactly solvable 2D Schrodinger operators with nonstandard spectral properties are constructed in the continuous and discrete cases using Laplace chains with different restrictions., Comment: LaTeX, 66 pages

We review basic ideas and basic examples of the theory of the inverse spectral problems., Comment: 36 pages, 5 figures, LaTeX

Topology of levels of the quasiperiodic functions with m=n+2 periods on the plane is studied. For the case of functions with m=4 periods full description is obtained for the open everywhere dense family of functions. This problem is equivalent to the study of Hamiltonian systems on the (n+2)-torus with constant rank 2 Poisson bracket. In the cases under investigation we proved that this system is topologically completely integrable in some natural sence where interesting integer-valued locally stable topological characteristics appear. The case of 3 periods has been extensively studied last years by the present author, Zorich, Dynnikov and Maltsev for the needs of solid state physics (''Galvanomagnetic Phenomena in Normal Metals''); The case of 4 periods might be useful for Quasicrystals., Comment: LATEX2e, 4 pages

This work is a continuation and extension of the note published in the Russian Math Surveys 1997 n 6. For any pair of solutions of the spectral problem for the second order selfadjoint real Schrodinger Operator on the graph their Symplectic Wronskian is a 1-cycle in the graph. This is a vector-valued symplectic 2-form on the space of solutions. This construction was applied to the Scattering Theory on the graphs with finite number of tails. The asymptotic values of solutions is a Lagrangian Plane of half dimension. This property determines all unitary properties of Scattering Matrix, which is also symmetric. All higher order discrete operators and operators on higher dimensional simplicial complexes are included in this scheme. Nonlinear analog of that was invented by the present author in collaboration with A.S.Schwarz., Comment: 20 pages, LaTeX, to appear in Asian Math. J. in the volume dedicated to the 70th birthday of Mikio Sato

This paper is devoted to topological phenomena in normal metals with rather complicated Fermi surface. The results of the article are based on the deep topological theorems concerning the geometry of non-compact plane sections of level surfaces of periodic function in 3-dimensional Euclidean space which are the quasi-classical electron orbits in the presence of homogeneous magnetic field. The main result is that the observation of electrical conductivity in strong magnetic fields can reveal such nontrivial topological characteristics of Fermi surface as integral planes, connected with conductivity tensor and locally stable under small rotations of magnetic field. This planes are connected with generic non-closed orbits on the Fermi surface. Some non-generic situations are also discussed., Comment: 21 pages, 9 Encapsulated Postscript figures

We present a brief survey of the results of the Theory of Solitons from the viewpoint of the periodic theory including some new results in the theory of 2-dimensional periodic Schrodinger Operators. The main subjects are: Periodic Solitons and Algebraic Geometry, The Theory of Solitons and the String equation, Topologically trivial and nontrivial periodic two-dimensional Schrodinger operators and Riemann surfaces and Cyclic and semicyclic chains of Laplace transformations (new results of the present author and A.Veselov)., Comment: 9 pages, Based on lectures delivered at the Erice Chalonge School, `String Gravity and Physics at the Planck Scale', 8-19 September 1995

Beginning from 1981 one of the present authors (S.Novikov) published a series of papers, (some of them in collaboration with I.Schmelzer and I.Taimanov) dedicated to the development of the analog of Morse theory for the closed 1-forms -- multivalued functions and functionals -- on the finite - and infinite-dimensional manifolds ({\bf Morse-Novikov Theory}). The notion of ``Multivalued action'' was understood and ``Topological quantization of the coupling constant'' for them was formulated by Novikov in 1981 as a Corollary from the requirement, that the Feinmann Amplitude should be one-valued on the space of fields-maps. Very beautiful analog of this theory appeared also in the late 80-ies in the Symplectic Geometry and Topology, when the so-called Floer Homology Theory was discovered. A very first topological idea of this theory, formulated in early 80-ies, was the so-called ``Principle of the Overthrowing of the Cycles''. It led to the results which were not proved rigorously until now. Our goal is to prove some of them. We study the motion of a classical charged particle on the Euclidean plane in a magnetic field orthogonal to this field. The trajectories of this motion can be characterized as extremals of the ``Maupertui--Fermat'' functional. We show that for any smooth everyvhere positive double periodic magnetic field for any fixed energy there exist at least two different periodic convex extremals, such that the value of the Maupertui-Fermat functional is positive for them. If all such extremals are nondegenerate in the sense of Morse in the space of nonparameterized curves then for any energy there exist at least 4 periodic convex extremals with the Morse indices (1,2,2,3)., Comment: 33 pages, LaTeX.

This paper is a continuation of the paper by S.P.Novikov in Funct.Anal.Appl., v.24(1990), No 4, pp 196-206. String equation is by definition the equation $[L,A]=1$ for the coefficients of two linear ordinary differential operators $L$ and $A$. For the ``double scaling limit'' of the matrix model we always have $L=-\partial_x^2+u(x)$, $A$ is some differential operator of the odd order $2k+1$. In the first nontrivial case $k=1$ we have the Painelev\'e-1 (P-1) equation. Only special real ``separatrix'' solutions of P-1 are important in the quantum field theory. By the conjecture of Novikov these ``physical'' solutions, which are analytically exceptional probably have much stronger symmetry then the other solutions but it is not proved until now. Two asymptotic methods were developed in the previous paper -- nonlinear semiclassics (or the Bogolubov-Whitham averaging method) and the linear semiclassics for the ``Isomonodromic'' method. The nonlinear semiclassics gives a good approximation for the general (``non-physical'') solutions of P-1 but fails in the ``physical'' case. In our paper the linear semiclasics for the ``physical'' solutions of the P-1 equations is studied. In particular connection between the semiclassics on Riemann surfaces and Hamiltonian foliations on these surfaces is established., Comment: 32 pages, LaTex, 4 pictures in separate files. Subj-class and Journal-ref added