Volume 3 (2003), Number 2. Abstracts O. Bogoyavlenskij. Infinite Families of Exact Periodic Solutions to the Navier—Stokes Equations [PDF] A complete classification of all periodic solutions to the 3-dimensional Navier—Stokes equations with pairwise non-interacting Fourier modes is obtained. The corresponding sets of the wave vectors k ∈ Z3 necessarily belong either to the straight lines, the planes, the circumferences or the spheres. The constructed exact periodic solutions are smooth and exist for all values of the time variable t > 0. Keywords. Navier—Stokes equations, periodic solutions, Fourier modes, Beltrami equation. 2000 Mathematics Subject Classification. 76X05, 35Q99. E. Brieskorn, A. Pratoussevitch, and F. Rothenhäusler. The Combinatorial Geometry of Singularities and Arnold's Series E, Z, Q [PDF] We consider discrete subgroups Γ of the simply connected Lie group SU&tilde(1,1) of finite level. This Lie group has the structure of a 3-dimensional Lorentz manifold coming from the Killing form. Γ acts on SU˜(1,1) by left translations. We want to describe the Lorentz space form Γ\SU˜(1,1) by constructing a fundamental domain F for Γ. We want F to be a polyhedron with totally geodesic faces. We construct such F for all Γ satisfying the following condition: The image ¯Γ of Γ in PSU(1,1) has a fixed point u in the unit disk of order larger than the level of Γ. The construction depends on Γ and Γu. For co-compact Γ the Lorentz space form Γ\SU˜(1,1) is the link of a quasi-homogeneous Gorenstein singularity. The quasi-homogeneous singularities of Arnold's series E, Z, Q are of this type. We compute the fundamental domains for the corresponding group. They are represented by polyhedra in Lorentz 3-space shown on Tables 1—13. Each series exhibits a regular characteristic pattern of its combinatorial geometry related to classical uniform polyhedra. Keywords. Lorentz space form, polyhedral fundamental domain, quasihomogeneous singularity, Arnold singularity series. 2000 Mathematics Subject Classification. Primary 53C50; Secondary 14J17, 20H10, 30F35, 30F60,32G15, 32S25, 51M20, 52B10. J. Bruce. On Families of Symmetric Matrices [PDF] In this paper we consider germs of smooth families of symmetric matrices. Using the natural notion of equivalence, that is smooth change of parameters and parameterised conjugation, we obtain a list of all simple germs and investigate their geometry. Keywords. Symmetric matrices, families of smooth mappings, singularities. 2000 Mathematics Subject Classification. 15A21, 58K50, 58K60. J. Damon. On the Legacy of Free Divisors II: Free* Divisors and Complete Intersections [PDF] We provide a criterion that for an equivalence group G on holomorphic germs, the discriminant of a G-versal unfolding is a free divisor. The criterion is in terms of the discriminant being Cohen—Macaulay and generically having Morse-type singularities. When either of these conditions fails, we provide a criterion that the discriminant have a weaker free* divisor structure. For nonlinear sections of a free* divisor V, we obtain a formula for the number of singular vanishing cycles by modifying an earlier formula obtained with David Mond and taking into account virtual singularities. Keywords. Discriminants, versal unfoldings, free divisors, free* divisors, liftable vector fields, Morse-type singularities, Cohen—Macaulay condition. 2000 Mathematics Subject Classification. Primary: 14B07, 14M12, 32S30; Secondary: 13C12, 14B10. Yu. Drozd, G.-M. Greuel, and I. Kashuba. On Cohen—Macaulay Modules on Surface Singularities [PDF] We study Cohen—Macaulay modules over normal surface singularities. Using the method of Kahn and extending it to families of modules, we classify Cohen—Macaulay modules over cusp singularities and prove that a minimally elliptic singularity is Cohen—Macaulay tame if and only if it is either simple elliptic or cusp. As a corollary, we obtain a classification of Cohen—Macaulay modules over log-canonical surface singularities and hypersurface singularities of type Tpqr especially they are Cohen—Macaulay tame. We also calculate the Auslander—Reiten quiver of the category of Cohen—Macaulay modules in the considered cases. Keywords. Cohen—Macaulay modules, Cohen—Macaulay tame and wild rings, normal surface singularities, minimally elliptic singularities, cusp singularities, log-canonical singularities, hypersurface singularities, Auslander—Reiten quiver. 2000 Mathematics Subject Classification. Primary 13C14, 13C05; Secondary: 16G50, 14J17. I. Dynnikov and S. Novikov. Geometry of the Triangle Equation on Two-Manifolds [PDF] 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 of triangles. In the present work, we develop a theory of these operators and equations showing their similarity to the complex derivatives ∂ and ¯∂. Keywords. Discrete connection, discrete analog of complex derivatives, triangle equation, first order difference operator. 2000 Mathematics Subject Classification. 39A12 (39A70). W. Ebeling and S. Gusein-Zade. Indices of 1-forms on an Isolated Complete Intersection Singularity [PDF] There are some generalizations of the classical Eisenbud—Levine—Khimshiashvili formula for the index of a singular point of an analytic vector field on Rn to vector fields on singular varieties. We offer an alternative approach based on the study of indices of 1-forms instead of vector fields. When the variety under consideration is a real isolated complete intersection singularity (icis), we define an index of a (real) 1-form on it. In the complex setting we define an index of a holomorphic 1-form on a complex icis and express it as the dimension of a certain algebra. In the real setting, for an \textbf{icis} V = f−1(0), f: (Cn, 0) → (Ck, 0), f is real, we define a complex analytic family of quadratic forms parameterized by the points \epsilon of the image (Ck, 0) of the map f, which become real for real ε and in this case their signatures defer from the real'' index by χ(Vε)−1, where χ(Vε) is the Euler characteristic of the corresponding smoothing Vε = f−1(ε) ∩ Bδ of the icis V. Keywords. Singular varieties, 1-forms, singular points, indices. 2000 Mathematics Subject Classification. 14B05, 32S99. G. Felder, L. Stevens, and A. Varchenko. Modular Transformations of the Elliptic Hypergeometric Functions, Macdonald Polynomials, and the Shift Operator [PDF] We consider the space of elliptic hypergeometric functions of the sl2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in terms of values at roots of unity of Macdonald polynomials of the sl2 type. Keywords. Elliptic hypergeometric functions, conformal\linebreak blocks, Macdonald polynomials. 2000 Mathematics Subject Classification. Primary 39Axx; Secondary 11Fxx, 20Gxx, 32G34, 33Dxx. A. Givental. An−1 Singularities and nKdV Hierarchies [PDF] According to a conjecture of E. Witten [21] proved by M. Kontsevich [12], a certain generating function for intersection indices on the Deligne—Mumford moduli spaces of Riemann surfaces coincides with a certain tau-function of the KdV hierarchy. The generating function is naturally generalized under the name the total descendent potential in the theory of Gromov—Witten invariants of symplectic manifolds. The papers [6], [4] contain two equivalent constructions, motivated by some results in Gromov—Witten theory, which associate a total descendent potential to any semisimple Frobenius structure. In this paper, we prove that in the case of K. Saito's Frobenius structure [17] on the miniversal deformation of the An−1-singularity, the total descendent potential is a tau-function of the nKdV hierarchy. We derive this result from a more general construction for solutions of the nKdV hierarchy from n−1 solutions of the KdV hierarchy. Keywords. Singularities, vanishing cycles, oscillating integrals, vertex operators, Hirota quadratic equations, Frobenius structures, the phase form. 2000 Mathematics Subject Classification. 14N35, 17B69, 32S30, 37K30. V. Goryunov and V. Zakalyukin. Simple Symmetric Matrix Singularities and the Subgroups of Weyl Groups Aμ, Dμ, Eμ [PDF] We analyse the classification of simple symmetric matrix singularities depending on two parameters which was obtained recently by Bruce and Tari. We show that these singularities are classified by certain reflection subgroups Y of the Weyl groups X = Aμ, Dμ, Eμ. The Dynkin diagram of such a subgroup is obtained from the affine diagram of X by deleting vertices of total marking 2: deletion of two 1-vertices corresponds to a 2 × 2 matrix singularity, and deletion of one 2-vertex gives rise to a 3 × 3 matrix. The correspondence is based on an isomorphism of the discriminants and on the description of a relevant monodromy group of the determinantal curve. Moreover, the base of a miniversal deformation of a simple matrix singularity turns out to be isomorphic to the quotient of the complex configuration space of the group X by the subgroup Y. We discuss lattice properties of symmetric matrix families in two variables which, in the case of simple singularities, define the choice of the subgroups. Keywords. Simple singularities of families of symmetric matrices, monodromy group, Weyl groups, sublattices in the vanishing homology, determinantial varieties. 2000 Mathematics Subject Classification. 58C27, 53A25, 65F15, 58K. Yu. Ilyashenko and V. Moldavskis. Morse—Smale Circle Diffeomorphisms and Moduli of Elliptic Curves [PDF] To any circle diffeomorphism there corresponds, by a classical construction of V. I. Arnold, a one-parameter family of elliptic curves. Arnold conjectured that, as the parameter approaches zero, the modulus of the corresponding elliptic curve tends to the (Diophantine) rotation number of the original diffeomorphism. In this paper, we disprove the generalization of this conjecture to the case when the diffeomorphism in question is Morse—Smale. The proof relies on the theory of quasiconformal mappings. Keywords. Circle diffeomorphism, rotation number, moduli of elliptic curves, quasiconformal mappings. 2000 Mathematics Subject Classification. 37E10, 37F30. S. Natanzon. Effectivisation of a String Solution of the 2D Toda Hierarchy and the Riemann Theorem About Complex Domains [PDF] Let 0 ∈ D+ be a connected domain with analytic boundary on the complex plane C. Then according to the Riemann theorem there exists a function w(z) = (1/r)z + ∑j=0&infin pj z-j, mapping biholomorphically D&minus = C\D+ to the exterior of the unit disk {w∈C: |w|>1}. From Wiegmann's and Zabrodin's rezults it follows that this function is described by the formula log w = log z − ∂t0 ((1/2)∂t0 + ∑k≥1 (z-k/k) ∂tk)v, where v = v(t0, t1, ¯t1, t2, ¯t2, ...) is a function of an infinite number of harmonic moments ti of the domain D−. This function is independent from the domain and satisfies the dispersionless Hirota equation for the 2D Toda lattice hierarchy. In the paper we find recursion relations for coefficients of the Taylor series of v. Keywords. Integrable systems, Toda lattice, Riemann theorem. 2000 Mathematics Subject Classification. 30C, 37K. D. Novikov and S. Yakovenko. Quasialgebraicity of Picard—Vessiot Fields [PDF] We prove that under certain spectral assumptions on the monodromy group, solutions of {Fuchsian systems} of linear equations on the Riemann sphere admit explicit global uniform bounds on the number of their isolated zeros in a way remotely resembling algebraic functions of one variable. Keywords. Fuchsian systems, complex zeros, monodromy. 2000 Mathematics Subject Classification. Primary 34C08, 34M10; Secondary 34M15, 14Q20, 32S40, 32S65. G. Paternain, L. Polterovich, and K. F. Siburg. Boundary Rigidity for Lagrangian Submanifolds, Non-Removable Intersections, and Aubry—Mather Theory [PDF] The paper establishes a link between symplectic topology and Aubry—Mather theory. We show that certain Lagrangian submanifolds lying in an optical hypersurface cannot be deformed into the domain bounded by the hypersurface. Even when this rigidity fails, we find that the intersection between the deformed Lagrangian submanifold and the hypersurface always contains a dynamically significant set related to Aubry—Mather theory. This phenomenon, although in a weaker form, still persists in the non-optical case. Keywords. Lagrangian submanifold, optical hypersurface, characteristic foliation, Liouville class, symplectic shape, generating function, Aubry set. 2000 Mathematics Subject Classification. 53D12, 37J50, 57R17, 57R30. I. Scherbak and A. Varchenko. Critical Points of Functions, sl2 Representations, and Fuchsian Differential Equations with only Univalued Solutions [PDF] Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at z1, ..., zn with exponents (ρ1,1,ρ2,1), ..., (ρ1,n,ρ2,n). Let the exponents at infinity be (ρ1,&infin,ρ2,&infin). Then for fixed generic z1, ..., zn, the number of such Fuchsian equations is equal to the multiplicity of the irreducible sl2 representation of dimension |ρ2,&infin − ρ1,&infin| in the tensor product of irreducible sl2 representations of dimensions |ρ2,1 − ρ1,1|, ..., |ρ2,n &minus ρ1,n|. To show this we count the number of critical points of a suitable function which plays the crucial role in constructions of the hypergeometric solutions of the sl2 KZ equation and of the Bethe vectors in the sl2 Gaudin model. As a byproduct of this study we conclude that the set of Bethe vectors is a basis in the space of states for the sl2 inhomogeneous Gaudin model. Keywords. Critical points, Bethe ansatz, polynomial solutions of differential equations. 2000 Mathematics Subject Classification. Primary 14Qxx; Secondary 32Sxx, 33Cxx, 34Mxx. B. Shapiro and A. Vainshtein. Counting Real Rational Functions with All Real Critical Values [PDF] We study the number #R of real rational degree n functions (considered up to a linear fractional transformation of the independent variable) with a given set of 2n−2 distinct real critical values. We present a combinatorial interpretation of these numbers and provide exact and asymptotic enumeration results for certain particular cases. Keywords. Real rational functions; real critical values; chord diagrams; enumeration. 2000 Mathematics Subject Classification. 14N10; 14Pxx; 26C15. D. Siersma and M. Tibãr. Deformations of Polynomials, Boundary Singularities and Monodromy [PDF] We study the topology of polynomial functions by deforming them generically. We explain how the non-conservation of the total quantity'' of singularity in the neighbourhood of infinity is related to the variation of topology in certain families of boundary singularities along the hyperplane at infinity. Keywords. Deformations of polynomials, singularities at infinity, monodromy, boundary singularities. 2000 Mathematics Subject Classification. 32S30, 14D05, 32S40, 58K60, 14B07, 58K10, 55R55. S. Tabachnikov. On Skew Loops, Skew Branes and Quadratic Hypersurfaces [PDF] A skew brane is an immersed codimension 2 submanifold in affine space, free from pairs of parallel tangent spaces. Using Morse theory, we prove that a skew brane cannot lie on a quadratic hypersurface. We also prove that there are no skew loops on embedded ruled developable discs in 3-space. Keywords. Skew loops and skew branes, quadratic hypersurfaces, double normals, Morse theory, developable surfaces. 2000 Mathematics Subject Classification. 53A05, 53C50, 58E05. V. Vladimirov and K. Ilin. Virial Functionals in Fluid Dynamics [PDF] The aim of this paper is to show that functionals similar to the virial' function of classical mechanics can be introduced for several dynamical systems of fluid mechanics provided that those dynamical systems can be described by Hamilton's principle of least action. The main requirement to virials' is their increasing by virtue of equations of motion. Applications of those functionals to hydrodynamic stability theory are reviewed and further perspectives are discussed. Keywords. Inviscid fluid, instability, virial. 2000 Mathematics Subject Classification. Primary 76B99, 76E99; Secondary 76M99. V. Yudovich. Eleven Great Problems of Mathematical Hydrodynamics [PDF] The key unsolved problems of mathematical fluid dynamics, their current state and outlook are discussed. These problems concern global existence and uniquness theorems for basic boundary and initial-boundary value problems in the theory of ideal and viscous incompressible fluids, the spectral problems in hydrodynamic stability theory for steady and time periodic flows, creation of secondary, tertiary, etc... flow regimes as a result of bifurcations and the asymptotics of vanishing viscosity. Several new problems are formulated. Keywords. Incompressible fluid, unsolved problems, existence, uniqueness, stability, asymptotics. 2000 Mathematics Subject Classification. 37Nxx, 35Qxx.
 Moscow Mathematical Journal is distributed by the American Mathematical Society for the Independent University of Moscow Online ISSN 1609-4514 © 2003, Independent University of Moscow Comments:mmj@mccme.ru