Volume 6 (2006), Number 1. Abstracts

S. Anisov. Lower Bounds for Transversal Complexity of Torus Bundles over the Circle [PDF]

For a 3-dimensional manifold M³, its complexity c(M³), introduced by S. Matveev, is the minimal number of vertices of an almost simple spine of M³; in many cases it is equal to the minimal number of tetrahedra in a singular triangulation of M³. Usually it is straightforward to give an upper bound for c(M), but obtaining lower bounds remains very difficult. We consider manifolds fibered by tori over the circle, introduce transversal complexity tc(M) for such manifolds, and give a lower bound for tc(M) in terms of the monodromy of the fiber bundle; this estimate involves a very geometric study of the modular group action on the Farey tesselation of hyperbolic plane. As a byproduct, we construct pseudominimal spines of the manifolds fibered by tori over S¹. Finally, we discuss some potential applications of these ideas to other 3-manifolds.

Keywords. Complexity of 3-manifolds, T²-bundles over S¹, Farey tesselation

2000 Mathematics Subject Classification. 57M99 (primary); 57M20, 57M50, 57R05, 57R15, 57R22 (secondary)

V. Arnold. Statistics of Young Diagrams of Cycles of Dynamical Systems for Finite Tori Automorphisms [PDF]

A permutation of a set of N elements is decomposing this set into y cycles of lengths x_s, defining a partition N = x₁+…+x_y. The length X₁, the height y and the fullness λ=N/xy of the Young diagram x₁ ≥ x₂ ≥ … ≥ x_y behave for the large random permutation like x ∼ an, y ∼ b ln N, λ ∼ c/ln N.

The finite 2-torus M is the product Z_m×Z_m, and its Fibonacci automorphism sends (u,v) to (2u+v,u+v) (mod m). This permutation of N = m² points of the finite torus M defines a peculiar Young diagram, whose behavior (for large m) is very different from that of a random permutation of N points.

Keywords. Fibonacci numbers, permutations, symmetric group, projective line, chaos, cat mapping, modular group, randomness generating, Galois field, finite Lobachevsky plane, relativistic de Sitter world

2000 Mathematics Subject Classification. 05E10

V. Buchstaber. n-valued Groups: Theory and Applications [PDF]

We give a survey of the most important results in the theory of n-valued groups and some of their applications. Main directions of advanced research will be discussed.

We start with basic definitions. Further exposition follows a sequence of instructive examples originating from various branches of mathematics: algebra, analysis, representation theory, topology, and dynamical systems.

The contents is accessible to a broad audience.

Keywords. Multivalued multiplication and action, hypergroups, deformation of a group, algebraic representations, generalized shift, integrable multivalued dynamical systems

2000 Mathematics Subject Classification. 20N20, 16S34, 05E30

S. Duzhin and J. Mostovoy. A Toy Theory of Vassiliev Invariants [PDF]

We set up a theory of finite type invariants for smooth hypersurfaces in Rⁿ. For n=1, 2, 3 these invariants admit a complete description: they form a polynomial algebra on one generator.

Keywords. Vassiliev invariants, singularities, discriminant, embedded hypersurfaces

2000 Mathematics Subject Classification. 58K65, 57M27

T. Ekedahl, B. Shapiro, and M. Shapiro. First Steps towards Total Reality of Meromorphic Functions [PDF]

It was earlier conjectured by the second and the third authors that any rational curve γ: CP¹ → CPⁿ such that the inverse images of all its flattening points lie on the real line RP¹ ⊂ CP¹ is real algebraic up to a Möbius transformation of the image CPⁿ. (By a flattening point p on γ we mean a point at which the Frenet n-frame (γ′,γ″,…,γ⁽ⁿ⁾) is degenerate.) Below we extend this conjecture to the case of meromorphic functions on real algebraic curves of higher genera and settle it for meromorphic functions of degrees 2, 3 and several other cases.

Keywords. Total reality, meromorphic functions, flattening points

2000 Mathematics Subject Classification. 14P05, 14P25

V. Goryunov. Logarithmic Vector Fields for the Discriminants of Composite Functions [PDF]

The K_f-equivalence is a natural equivalence between map-germs φ: C^m → Cⁿ which ensures that their compositions f⋅φ with a fixed function-germ f on Cⁿ are the same up to biholomorphisms of C^m. We show that the discriminant Σ in the base of a K_f-versal deformation of a germ φ is Saito's free divisor provided the critical locus of f is Cohen–Macaulay of codimension m+1 and all the transversal types of f are A_k singularities. We give an algorithm to construct basic vector fields tangent to Σ. This is a generalisation of classical Zakalyukin's algorithm to write out basic fields tangent to the discriminant of an isolated function singularity. The case of symmetric matrix families in two variables is done in detail. For simple singularities, it is directly related to Arnold's convolution of invariants of Weyl groups.

Keywords. Logarithmic vector field, discriminant, composite function, free divisor, matrix singularities

2000 Mathematics Subject Classification. Primary 32S05; Secondary 58K20

D. Grantcharov and V. Serganova. Category of sp(2n)-modules with Bounded Weight Multiplicities [PDF]

Let g be a finite dimensional simple Lie algebra. Denote by B the category of all bounded weight g-modules, i.e. those which are direct sum of their weight spaces and have uniformly bounded weight multiplicities. A result of Fernando shows that infinite-dimensional bounded weight modules exist only for g = sl(n) and g = sp(2n). If g = sp(2n) we show that B has enough projectives if and only if n > 1. In addition, the indecomposable projective modules can be parameterized and described explicitly. All indecomposable objects are described in terms of indecomposable representations of a certain quiver with relations. This quiver is wild for n > 2. For n = 2 we describe all indecomposables by relating the blocks of B to the representations of the affine quiver A₃⁽¹⁾.

Keywords. Lie algebra, indecomposable representations, quiver, weight modules

2000 Mathematics Subject Classification. 17B10

A. Khovanskii and D. Novikov. On Affine Hypersurfaces with Everywhere Nondegenerate Second Quadratic Form [PDF]

An Arnold conjecture claims that a real projective hypersurface with second quadratic form of constant signature (k,l) should separate two projective subspaces of dimension k and l correspondingly. We consider affine versions of the conjecture dealing with hypersurfaces approaching at infinity two shifted halves of a standard cone. We prove that if the halves intersect, then the hypersurface does separate two affine subspaces. In the case of non-intersecting half-cones we construct an example of a surface of negative curvature in R³ bounding a domain without a line inside.

Keywords. Arnold conjecture, (k,l)-hyperbolic hypersurface, convex-concave set

2000 Mathematics Subject Classification. 52A30, 53A15

E. Soprunova. Zeros of Systems of Exponential Sums and Trigonometric Polynomials [PDF]

Gelfond and Khovanskii found a formula for the sum of the values of a Laurent polynomial at the zeros of a system of n Laurent polynomials in (C^×)ⁿ whose Newton polytopes have generic mutual positions. An exponential change of variables gives a similar formula for exponential sums with rational frequencies. We conjecture that this formula holds for exponential sums with real frequencies. We give an integral formula which proves the existence-part of the conjectured formula not only in the complex situation but also in a very general real setting. We also prove the conjectured formula when it gives answer zero, which happens in most cases.

Keywords. Exponential sums, trigonometric polynomials, quasiperiodic functions, mean value

2000 Mathematics Subject Classification. 14P15, 33B10

V. Tourtchine. What Is One-Term Relation for Higher Homology of Long Knots [PDF]

Vassiliev's spectral sequence for long knots is discussed. Briefly speaking we study what happens if the strata of non-immersions are ignored.

Various algebraic structures on the spectral sequence are introduced. General theorems about these structures imply, for example, that the bialgebra of chord diagrams is polynomial for any field of coefficients.

Keywords. Knot spaces, discriminant, bialgebra of chord diagrams, sphere, Hopf algebra with divided powers, simplicial algebra

2000 Mathematics Subject Classification. Primary: 57Q45; Secondary: 57Q35

A. Varchenko. Bethe Ansatz for Arrangements of Hyperplanes and the Gaudin Model [PDF]

We show that the Shapovalov norm of a Bethe vector in the Gaudin model is equal to the Hessian of the logarithm of the corresponding master function at the corresponding isolated critical point. We show that different Bethe vectors are orthogonal. These facts are corollaries of a general Bethe ansatz type construction, suggested in this paper and associated with an arbitrary arrangement of hyperplanes.

Keywords. Gaudin model, Bethe vectors, arrangements of hyperplanes, Orlik–Solomon algebra, flag complex

2000 Mathematics Subject Classification. Primary: 82C20; Secondary: 17B, 81R12, 82C23

O. Viro. Whitney Number of Closed Real Algebraic Affine Curve of Type I [PDF]

For a closed real algebraic plane affine curve dividing its complexification and equipped with a complex orientation, the Whitney number is expressed in terms of behavior of its complexification at infinity.

Keywords. Whitney number, real algebraic curve, curve of type I, complex orientation, blow up

2000 Mathematics Subject Classification. 14P25, 53A03, 57R42

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