On the coexistence of corank 1 multisingularities of a stable smooth mapping of equidimensional manifolds

We study the topology of stable smooth mappings of smooth closed real manifolds of the same dimension. We calculate all universal linear relations between the Euler characteristics of the manifolds of multisingularities of mappings that have only corank 1 singularities. Published in Russian in Trudy Matematicheskogo Instituta imeni V. A. Steklova, 2007, vol. 258, pp. 201–226. Dedicated to Vladimir Igorevich Arnold on the occasion of his 70th birthday     

The topology of corank 1 multi-singularities of stable smooth mappings of equidimensional manifolds

We study conditions for the coexistence of singularities of a stable smooth mapping of a closed manifold into a manifold of the same dimension n. Assuming that this mapping has only singularities of corank 1, we find universal linear relations between the Euler characteristics of the manifolds of multi-singularities in the image of the considered mapping. To cite this article: V.D. Sedykh, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Divergence in 3-manifold groups

The divergence of the fundamental group of compact irreducible 3-manifolds satisfying Thurston's geometrization conjecture is calculated. For every closed Haken 3-manifold group, the divergence is either linear, quadratic or exponential, where quadratic divergence occurs precisely for graph manifolds and exponential divergence occurs when a geometric piece has hyperbolic geometry. An example is given of a closed 3-manifoldN with a Riemannian metric of nonpositive curvature such that the divergence is quadratic and such that there are two geodesic rays in the universal coverN whose divergence is precisely quadratic, settling in the negative a question of Gromov's.Partially supported by NSF grant DMS-9200433.     

On Euler characteristics of manifolds of singularities of wave fronts

All linear relations with real coefficients between the Euler characteristics of manifolds of singularities of wave fronts are found for all generic compact fronts in all spaces of dimension at most 6.     

Actions of discrete groups on spheres and real projective spaces

In this paper, we first define discrete, smooth actions on , whose limit sets are Cantor sets wildly embedded in (Antoine's necklaces). Secondly, we define Schottky groups on real projective spaces of odd dimensions, . We lift these actions to (locally) projective actions on the sphere and consider the quotient space of the domain of discontinuity by the group to obtain new examples of manifolds with real projective structures. *This work was partially supported by PROMEP (SEP). **This work was partially supported by CONACyT (México), grant U1 55084, PAPIIT (UNAM) grant IN102108.     

On stability of projections of Lagrangian varieties

We show that Lagrangian and Legendre varieties associated with matrix singularities and singularities of composite functions are stable in the sense of a natural modification of Giventals notion of stability of Lagrangian projections.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 13–21, 2004Original Russian Text Copyright © by V. V. Goryunov and V. M. ZakalyukinThe second authors research was partially supported by RFBR grant 02-01-00099 and Universities of Russia grant 04-01-016.     

Colouring Cubic Graphs by Small Steiner Triple Systems

Given a Steiner triple system , we say that a cubic graph G is -colourable if its edges can be coloured by points of in such way that the colours of any three edges meeting at a vertex form a triple of . We prove that there is Steiner triple system of order 21 which is universal in the sense that every simple cubic graph is -colourable. This improves the result of Grannell et al. [J. Graph Theory 46 (2004), 15–24] who found a similar system of order 381. On the other hand, it is known that any universal Steiner triple system must have order at least 13, and it has been conjectured that this bound is sharp (Holroyd and Škoviera [J. Combin. Theory Ser. B 91 (2004), 57–66]). Research partially supported by APVT, project 51-027604. Research partially supported by VEGA, grant 1/3022/06.     

The Abstract Titchmarsh-Weyl M-function for Adjoint Operator Pairs and its Relation to the Spectrum

In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction A_B of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces and such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples – one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line – which together indicate that the abstract results are probably best possible. James Hinchcliffe and Serguei Naboko wish to thank the British EPSRC for financial support under grant EP/C008324/1 “Spectral Problems on Families of Domains and Operator M-functions”. Serguei Naboko wishes to thank the Russian RFBR for grant 06-01-00219. All authors wish to thank INTAS for financial support under INTAS Project No. 051000008-7883. The authors wish to thank the referee for many helpful comments.     

Sojourn times,manifolds with infinite cylindrical ends,and an inverse problem for planar waveguides

We prove that two particular entries in the scattering matrix for the Dirichlet Laplacian on ℝ × (−γ, γ) determine an analytic strictly convex obstacle . With an additional symmetry assumption, one entry suffices. Part of the proof is an integral identity involving an entry in the scattering matrix and a distribution related to the fundamental solution of the wave equation. This identity holds for general manifolds with infinite cylindrical ends. A consequence of this is a relationship between the singularities of the Fourier transform of an entry in the scattering matrix and the sojourn times of certain geodesics. Partially supported by NSF grant DMS 0500267.     

Conditions of topological conjugacy of gradient-like diffeomorphisms on irreducible 3-manifolds

In this paper, we obtain the topological classification of gradient-like diffeomorphisms and the conditions of topological conjugacy of Morse-Smale diffeomorphisms with finite sets of heteroclinic trajectories on three-dimensional manifolds.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 73–80, January, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-01-407, by the International Science Foundation under grant R99000, and by the Foundation Cultural Initiative.     

An improved Hardy-Sobolev inequality in W 1,p and its application to Schrödinger operators

In this paper we prove new Hardy-like inequalities with optimal potential singularities for functions in W^1,p(Ω), where Ω is either bounded or the whole space and also some extensions to arbitrary Riemannian manifolds. We also study the spectrum of perturbed Schr?dinger operators for perturbations which are just below the optimality threshold for the corresponding Hardy inequality.     

Singular measures with smalH(p, q)-projections

We construct a singular probability measure μ on the complex sphere such that the Poisson integral of μ is a pluriharmonic function in the ball and the Fourier transform of μ is asp→∞. Supported by the Centre de Recerca Matemàtica (Institut d’Estudis Catalans, Barcelona) under a grant from DGICYT (Spain); partially supported by the RFFI grant 96-01-00693.     

Embeddability of smooth Cauchy-Riemann manifolds

Summary The main purpose of this paper is to give a sufficient condition for global embeddability of smooth Cauchy-Riemann manifolds (CR-manifolds) into complex manifolds with boundary. Namely, let M be a smooth CR-manifold of real dimension 2n – 1 and CR-dimension n – 1, where n 2, which is locally CR-embeddable into a complex manifold. Assume further that the Levi form of M is non-vanishing at each point. The main result of this paper is that such a CR-manifold is globally CR-embeddable into an n-dimensional complex manifold with boundary. Moreover if the Levi form has at each point of M eigenvalues of opposite signs, then M embeds into a complex manifold without boundary.This research is supported by a grant from Consiglio Nazionale delle Ricerche in Italy.     

On the Classification of Cartan Actions

We study higher-rank Cartan actions on compact manifolds preserving an ergodic measure with full support. In particular, we classify actions by with k ≥ 3 whose one-parameter groups act transitively as well as nondegenerate totally nonsymplectic -actions for k ≥ 3. The first author is supported in part by NSF grants DMS-0140513. The second author is supported in part by NSF grant DMS-0203735. Received: July 2005 Revision: March 2006 Accepted: March 2006     

Generalized symmetrization in enveloping algebras

Let g be a Lie algebra,S(g) the symmetric algebra,U(g) the universal enveloping algebra, andZ(g) the center ofU(g). The aim of this paper is to discuss a construction of a class of linear isomorphisms :S(g)U(g) which commute with the adjoint representation. Applications to constructing a basis inZ(g) for classical g are also sketched.Supported by the Russian Foundation for Basic Research under grant 95-01-00814     

Liouville theorems for harmonic maps

Summary We prove several Liouville theorems for harmonic maps between certain classes of Riemannian manifolds. In particular, the results can be applied to harmonic maps from the Euclidean space (R ^m ,g ₀) to a large class of Riemannian manifolds. Our assumptions on the harmonic maps concern the asymptotic behavior of the maps at .Oblatum 28-XII-1990 & 11-II-1991Supported by NSF grant DMS-8610730     

Multishifts in Hilbert spaces

We introduce and study a multishift structure in a Hilbert space. This structure is a noncommutative analog of the (simple one-sided) shift operator, well known in function theory and functional analysis. Subspaces invariant under the multishift are described. A theorem on the factorization into an inner and an outer factor is established for operators commuting with the multishift.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 69–81, 2005Original Russian Text Copyright © by P. A. TerekhinSupported by the RFBR under grant No. 03-01-00390, the program Leading Scientific Schools of the Russian Federation under grant No. NSh-1295.2003.1, and the INTAS under grant No. 99-00089.Translated by V. E. Nazaikinskii     

Chern–Weil map for principal bundles over groupoids

The theory of principal G-bundles over a Lie groupoid is an important one unifying various types of principal G-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal G-bundles. In this paper, we study differential geometry of these objects, including connections and holonomy maps. We also introduce a Chern–Weil map for these principal bundles and prove that the characteristic classes obtained coincide with the universal characteristic classes. As an application, we recover the equivariant Chern–Weil map of Bott–Tu. We also obtain an explicit chain map between the Weil model and the simplicial model of equivariant cohomology which reduces to the Bott–Shulman map when the manifold is a point. P. Xu Research partially supported by NSF grant DMS-03-06665.     

Singularities of linear systems and the Waring problem

The Waring problem for homogeneous forms asks for additive decomposition of a form into powers of linear forms. A classical problem is to determine when such a decomposition is unique. In this paper we answer this question when the degree of is greater than the number of variables. To do this we translate the algebraic statement into a geometric one concerning the singularities of linear systems of with assigned singularities.