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Discriminants and local invariants of planar fronts

The aim of this study is the description of all the local additive invariants of the plane wave fronts. A generic wave front is a curve whose only singularities are the transversal self-intersections on the semicubical cusps (see fig. 1a, b). The invariants that we shall find are “dual” to different strata of the discriminant formed by the nongeneric wave fronts.
Francesca Aicardi

Crofton Densities, Symplectic Geometry and Hilbert’s Fourth Problem

We study the relation between the integral geometric and the symplectic construction of Desarguesian metrics on R n and show that these constructions characterize all Desarguesian Finsler metrics.
J. C. Alvarez, I. M. Gelfand, M. Smirnov

Projective convex curves

We deal with convex curves and surfaces in real projective space. Questions concerning the affinity of convex curves, the flattening points and the loss of convexity are regarded. Finally, we prove that there are no convex surfaces except curves and hypersurfaces.
S. Anisov

Topological classification of real trigonometric polynomials and cyclic serpents polyhedron

The goal of this paper is the study of the manifold of the real trigonometric polynomials of degree n having the maximal possible number (2n) of real critical points (M-polynomials).
V. Arnold

Singularities of short linear waves on the plane

The subject of the paper is the geometrical optics of short linear waves on plane. We describe perestroikas of momentary fronts and scattering of rays when the light hypersurface has generic conical singularities. They appear if the waves propagate in a nonhomogeneous anisotropic medium and are controlled by a hyperbolic variational principle.
Ilia A. Bogaevski

New generalizations of Poincaré’s geometric theorem

Poincaré’s geometric theorem claims that an area-preserving diffeomorphism of an annulus which shifts the boundary circles at opposite directions has at least two fixed points. The present paper consists of two parts. In the first one, we show that such a diffeomorphism has more than just two fixed points provided the shift of the boundaries is large enough. In the second part, we prove symplectic fixed point theorems which can be viewed as generalizations of Poincaré’s geometric theorem to higher dimensions.
Yu V. Chekanov

Explicit formulas for Arnold’s generic curve invariants

We review the explicit formulas for Arnold’s generic curve invariants due to Viro, Shumakovich and Polyak and add some remarks concerning the invariants of spherical curves and curves immersed in arbitrary orientable surfaces.
S. Chmutov, S. Duzhin

Nonlinear integrable equations and nonlinear Fourier transform

In this paper we study nonlocal functionals whose kernels are homogeneous generalized functions. We also use such functionals to solve the Korteweg-de Vries (KdV), the nonlinear Schrödinger (NLS) and the Davey-Stewartson (DS) equations.
A. S. Fokas, I. M. Gelfand, M. V. Zyskin

Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions

Various results in algebra, analysis, and geometry can be generalized by replacing the ordinary numbers (integer, real or complex) by their trigonometric analogues. For x ∈ ℂ, the trigonometric number [x] h ∈ ℂ is defined by
$$[x]_h = \frac{{\sin (\pi hx)}}{{\sin (\pi h)}}$$
where h ∈ ℂ\ℤ is a fixed parameter. It is clear that \(\mathop {\lim }\limits_{h \to 0} [x]_h = x\) thus, [x] h may be viewed as a one-parameter deformation of x. The trigonometric numbers are not additive: generally speaking [x+y] h ≠ [x] h +[y] h . However, they satisfy a kind of additivity of “second order”: for any x, y, z ∈ ℂ,
$$\left[ {x + z} \right]h\left[ {x - z} \right]h = \left[ {x + y} \right]h\left[ {x - y} \right]h\left[ {y + z} \right]h\left[ {y - z} \right]h.$$
Many identities between ordinary numbers can be proved using only the additivity of second order and therefore allow a trigonometric deformation.
Igor B. Frenkel, Vladimir G. Turaev

Combinatorics of hypergeometric functions associated with positive roots

In this paper we study the hypergeometric system on unipotent matrices. This system gives a holonomic D-module. We find the number of independent solutions of this system at a generic point. This number is equal to the famous Catalan number. An explicit basis of Γ-series in solution space of this system is constructed in the paper. We also consider restriction of this system to certain strata. We introduce several combinatorial constructions with trees, polyhedra, and triangulations related to this subject.
Israel M. Gelfand, Mark I. Graev, Alexander Postnikov

Local invariants of mappings of surfaces into three-space

Following Arnol’s and Viro’s approach to order 1 invariants of curves on surfaces [1, 2, 3, 20], we study invariants of mappings of oriented surfaces into Euclidean 3-space. We show that, besides the numbers of pinch and triple points, there is exactly one integer invariant of such mappings that depends only on local bifurcations of the image. We express this invariant as an integral similar to the integral in Rokhlin’s complex orientation formula for real algebraic curves. As for Arnold’s J + invariant [1, 2, 3], this invariant also appears in the linking number of two legendrian lifts of the image. We discuss a generalization of this linking number to higher dimensions.
Our study of local invariants provides new restrictions on the numbers of different bifurcations during sphere eversions.
Victor V. Goryunov

Theorem on six vertices of a plane curve via Sturm theory

We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the “projective version” of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of some general Sturm-type theorems.
L. Guieu, E. Mourre, V. Yu Ovsienko

The Arf—invariant and the Arnold invariants of plane curves

In [A] V.I.Arnold considered closed generic plane curves, i.e. immersions S 1R 2 , the images of which have no singularities except simple (double) self-intersections. In a generic one-parameter family of immersions three types of modifications (“perestroykas”) of generic curves can be met. They correspond to three natural strata in the set of non-generic immersions (the discriminant). These strata consist of immersions with a direct self-tangency (J +), with an inverse self-tangency (J ), and with a triple crossing (St) respectively. All three of them are coorientable. An invariant of generic plane curves is an invariant of the first order (in the sense of Vassiliev) if its change under a modification of crossing a stratum (in the positive direction) depends only on the stratum, but not on a (non-singular) point of its crossing. For closed plane curves V.I.Arnold defined basic invariants of the first order J+, J, and St corresponding to the described strata. The invariants J+, J, and St can be defined for so-called “long” curves (that is for curves going from the infinity to the infinity) as well.
S. M. Gusein-Zade, S. M. Natanzon

Produit cyclique d’espaces et opérations de Steenrod

Dans cet article nous poursuivons notre étude des opérations de Steenrod commencée dans [3], en utilisant le produit cyclique des espaces et, indirectement, leur produit symétrique infini par Dold et Thom [1]. Afin de faciliter sa lecture, nous commençerons par rappeler quelques définitions fondamentales.
Max Karoubi

Characteristic Classes of Singularity Theory

Theorems of the global singularity theory express topological invariants in terms of singularities of some geometrical objects: vector bundle mappings, the Thom-Boardman singularities of smooth mappings, Lagrangian and Legendrian singularities, ℝℂ-singularities ect. (cf. [P], [R], [L], [HL2], [AGLV], [V]). A classical example is Hopf’s theorem relating the singularities of vector fields to the Euler characteristic.
M. É. Kazarian

Value of Generalized Hypergeometric Function at Unity

Value of generalized hypergeometric function at a special point is calculated. More precisely, the value of certain multiple integral over vanishing cycle (all arguments collapse to unity) is calculated. The answer is expressed in terms of Γ-functions. The constant is relevant to the part of ρ in the Gindikin-Karpelevich formula for the c-function of Harish-Chandra. Calculation is an adaptation of classical calculations of Gelfand and Naimark (1950) to the Heckman-Opdam hypergeometric functions in the case of root system of type A n−1.
A. Kazarnovski-Krol

Harish-Chandra decomposition for zonal spherical function of type A n

Heckman-Opdam system of differential equations is holonomic, with regular singularities and has locally |W|-dimensional space of solutions (cf. corollary 3.9 of [12]), where |W| is the cardinality of the Weyl group W. The system is a generalization of radial parts of Laplace-Casimir operators on symmetric Riemannian spaces of nonpositive curvature and is isomorphic to Calogero-Sutherland model in the integrable systems.
A. Kazarnovski-Krol

Positive paths in the linear symplectic group

A positive path in the linear symplectic group Sp(2n) is a smooth path which is everywhere tangent to the positive cone. These paths are generated by negative definite (time-dependent) quadratic Hamiltonian functions on Euclidean space. A special case are autonomous positive paths, which are generated by time-independent Hamiltonians, and which all lie in the set u of diagonalizable matrices with eigenvalues on the unit circle. However, as was shown by Krein, the eigenvalues of a general positive path can move off the unit circle. In this paper, we extend Krein’s theory: we investigate the general behavior of positive paths which do not encounter the eigenvalue 1, showing, for example, that any such path can be extended to have endpoint with all eigenvalues on the circle. We also show that in the case 2n = 4 there is a close relation between the index of a positive path and the regions of the symplectic group that such a path can cross. Our motivation for studying these paths came from a geometric squeezing problem [16] in symplectic topology. However, they are also of interest in relation to the stability of periodic Hamiltonian systems [9] and in the theory of geodesies in Riemannian geometry [4].
François Lalonde, Dusa McDuff

Invariants of submanifolds in Euclidean space

The topology of the set of singular support hyperplanes and hyperspheres to a smooth submanifold in Euclidean space is studied. As a corollary, some relations between differential-geometric characteristics of a manifold are obtained. In particular, if a simple closed embedded generic curve in a plane has C global vertices (where the curvature circles are support circles to the curve) and T support circles touching the curve at three points, then CT = 4. Similar invariants are also obtained for submanifolds in higher-dimensional spaces.
V. D. Sedykh

On Combinatorics and Topology of Pairwise Intersections of Schubert Cells in SL n /B

Topological properties of intersections of pairs and, more generally, of k-tuples of Schubert cells belonging to distinct Schubert cell decompositions of a flag space are of particular importance in representation theory and have been intensively studied during the last 15 years, see e.g. [BB, KL1, KL2, Del, GS]. Intersections of certain special arrangements of Schubert cells are related directly to the representability problem for matroids, see [GS]. Most likely, for a somewhat general class of arrangements of Schubert cells their intersections are too complicated to analyze. Even the nonemptyness problem for such intersections in complex flag varieties is very hard. However, in the case of pairs of Schubert cells in the space of complete flags one can obtain a special decomposition of such intersections, and of the whole space of complete flags, into products of algebraic tori and linear subspaces. This decomposition generalizes the standard Schubert cell decomposition. The above strata can be also obtained as intersections of more than two Schubert cells originating from the initial pair. The decomposition considered is used to calculate (algorithmically) natural additive topological characteristics of the intersections in question, namely, their Euler E p,q -characteristics (see [DK]). Generally speaking, this decomposition of the space of complete flags does not stratify all pairwise intersections of Schubert cells, i.e. the closure of a stratum is not necessary a union of strata of lower dimensions. Still there exists a natural analog of adjacency, and its combinatorial description is available, see Theorem D. We discuss combinatorics of this special decomposition and some rather simple consequences for the cohomology and the mixed Hodge structure of intersections of Schubert cells in SL n /B.
Boris Shapiro, Michael Shapiro, Alek Vainshtein
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