Singularities of Families of Chords

A chord is a straight line joining two points of a pair of hypersurfaces in an affine space such that the tangent hyperplanes at these points are parallel. We classify the singularities of envelopes of the families of chords determined by generic pairs of plane curves and surfaces in three-space. The list contains all bifurcation diagrams of simple boundary singularities (of the corresponding multiplicity).     

Vector fields and functions on discriminants of complete intersections and on bifurcation diagrams of projections

The paper studies vector fields that preserve the discriminants of isolated singularities of complete intersections and bifurcation diagrams of projections to the straight line. The results are applied to find stable functions on discriminants of simple complete intersections and normal forms of functions of general position on bifurcation diagrams of projections of low codimension.Translated from Itogi Nauki i Tekhniki, Seriya Sovermennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 33, pp. 31–54, 1988.     

A method of desingularization for analytic two-dimensional vector field families

It is well known that isolated singularities of two dimensional analytic vector fields can be desingularized: after a finite number of blowing up operations we obtain a vector field that exhibits only elementary singularities. In the present paper we introduce a similar method to simplify the periodic limit sets of analytic families of vector fields. Although the method is applied here only to reduce to families in which the zero set has codimension at least two, we conjecture that it can be used in general. This is related to the famouss Hibert's problem about planar vector fields.     

Hypersurface singularities,codimension two complete intersections and tangency sets

Some codimension 2 projective complete intersections are related to local hypersurface singularities. Necessary and sufficient conditions are given such that the singularities obtained are isolated. Incidentally we prove the finitude of the tangency set of two equidimensional smooth complete intersections with distinct multidegrees.     

On codimension one foliations with prescribed cuspidal separatrix

In this paper, we will construct a pre-normal form for germs of codimension one holomorphic foliation having a particular type of separatrix, of cuspidal type. As an application, we will explain how this form could be used in order to study the analytic classification of the singularities via the projective holonomy, in the generalized surface case. We will also give an application to the analytic classification of singularities, and a sufficient condition, in the quasi-homogeneous, three-dimensional case, for these foliations to be of generalized surface type.     

Note on the unfolding of degenerate Hopf bifurcation germs

In an article published in this journal (J. Differential Equations41 (1981), 375–415) M. Golubitsky and W. F. Langford provide a classification by codimension of local Hopf bifurcation problems which do not satisfy the classical nondegeneracy conditions. They describe all the stable perturbations of a oneparameter family of codimension 3 problems. The stable bifurcation diagrams for this family are presented in Fig. (4.6) of the above article, where one of the cases is indicated as N.A.—not applicable. It is shown here that this case is applicable, and the missing bifurcation diagram is provided.     

Quelques problèmes en géométrie feuilletée pour les 60 années de l’IMPA

We propound several problems concerning codimension one holomorphic foliations either in a local context (theory of singularities) or in a global situation (algebraic foliations). Each problem is illustrated by suitable examples.     

Nowhere minimal CR submanifolds and Levi-flat hypersurfaces

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is studied and characterized. This question is completely resolved for algebraic submanifolds of codimension 2 and a sufficient condition for noncontainment is given for non algebraic submanifolds. As a consequence, an example of a submanifold of codimension 2, not biholomorphically equivalent to an algebraic one, is given. We also investigate the structure of singularities of Levi-flat hypersurfaces.     

Classification and unfoldings of degenerate Hopf bifurcations

This paper initiates the classification, up to symmetry-covariant contact equivalence, of perturbations of local Hopf bifurcation problems which do not satisfy the classical non-degeneracy conditions. The only remaining hypothesis is that ±i should be simple eigenvalues of the linearized right-hand side at criticality. Then the Lyapunov-Schmidt method allows a reduction to a scalar equation G(x, λ) = 0, where G(?x, λ) = ?G(x, λ). A definition is given of the codimension of G, and a complete classification is obtained for all problems with codimension ?3, together with the corresponding universal unfoldings. The perturbed bifurcation diagrams are given for the cases with codimension ?2, and for one case with codimension 3; for this last case one of the unfolding parameters is a “modal” parameter, such that the topological codimension equals in fact 2. Formulas are given for the calculation of the Taylor coefficients needed for the application of the results, and finally the results are applied to two simple problems: a model of glycolytic oscillations and the Fitzhugh nerve equations.     

Generic bifurcations of low codimension of planar Filippov Systems

In this article some qualitative and geometric aspects of non-smooth dynamical systems theory are discussed. The main aim of this article is to develop a systematic method for studying local (and global) bifurcations in non-smooth dynamical systems. Our results deal with the classification and characterization of generic codimension-2 singularities of planar Filippov Systems as well as the presentation of the bifurcation diagrams and some dynamical consequences.     

Resolution except for minimal singularities I

The philosophy of this article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of resolution of singularities of a variety or a divisor, except for simple normal crossings (i.e., which avoids blowing up simple normal crossings, and ends up with a variety or a divisor having only simple normal crossings singularities). (2) For more general normal crossings (in a local analytic or formal sense), such a result does not hold. We find the smallest class of singularities (in low dimension or low codimension) with which we necessarily end up if we avoid blowing up normal crossings singularities. Several of the questions studied were raised by Kollár.     

Puiseux parametric equations of analytic sets

We prove the existence of local Puiseux-type parameterizations of complex analytic sets via Laurent series convergent on wedges. We describe the wedges in terms of the Newton polyhedron of a function vanishing on the discriminant locus of a projection. The existence of a local parameterization of quasi-ordinary singularities of complex analytic sets of any codimension will come as a consequence of our main result.

     

On the geometry of conjugacy classes in classical groups

Summary We study closures of conjugacy classes in the Lie algebras of the orthogonal and symplectic groups and determine which ones are normal varieties. Furthermore we give a complete classification of the minimal singularities which arise in this context, i.e. the singularities which occur in the open classes in the boundary of a given conjugacy class. In contrast to the results for the general linear group ([KP1], [KP2]) there are classes with non normal closure; they are branched in a class of codimension two and give rise to normal minimal singularities. The methods used are (classical) invariant theory and algebraic geometry. Supported in part by the SFB Theoretische Mathematik, University of Bonn, and by the University of Hamburg     

Phase Transformation and Singularity Theory

Phase transformation plays an important role in thermodynamics and materials science. Based on the theory of singularities, a new method to construct phase diagrams is presented. Analysing singularities on base of root sequences, see Tamaschke [16], will help to develop singularity graphs, where workings by H. Whitney, R. Thom, and V. I. Arnold provide fundamentals. The generated singularity graphs build the starting point for singularity phase diagrams. A powerful characteristic of such singularity graphs is that higher-dimensional surfaces can be transformed to a two-dimensional diagram. The attained singularity diagram can be used in materials science for analytical models of temperature-concentrated diagrams. As tools from algebra and analysis build a sound basis for singularity diagrams, it is possible to evolve computer software generating these phase diagrams. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rigid and Complete Intersection Lagrangian Singularities

In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in [SvS03]. The result can be applied to show the rigidity of all open swallowtails of dimension 2. In the case of lagrangian complete intersection singularities the lagrangian de Rham complex turns out to be perverse. We also show that lagrangian complete intersections in dimension greater than two cannot be regular in codimension one.     

Generic singularities of implicit systems of first order differential equations on the plane

For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well-known singularities of generic vector fields on the plane and the singularities described by a generic first order implicit differential equations, there exists only one generic singularity described by the implicit first order equation supplied by Whitney umbrella surface generically embedded to the space of directions on the plane.     

Smoothing of quiver varieties

We show that Gorenstein singularities that are cones over singular Fano varieties provided by so-called flag quivers are smoothable in codimension three. Moreover, we give a precise characterization about the smoothability in codimension three of the Fano variety itself.     

Nikolai Nikolaevich Krasovskii

We consider the classification of germs of functions up to a nonstandard equivalence relation similar to the quasi boundary equivalence and quasi equivalence of projections recently introduced by the second author. In fact, it is more rough than the classification of functions with respect to the group of diffeomorphisms preserving a corner (that is, a union of a pair of transversal hypersurfaces). We present the list of all simple classes and discuss its relation to the singularities of Lagrangian projections with corners. Also, we describe the bifurcation diagrams and caustics of simple quasi corner singularities.     

Simple Cohen–Macaulay Codimension 2 Singularities

In this article, we provide a complete list of simple isolated Cohen–Macaulay codimension 2 singularities together with a list of adjacencies which is complete in the case of fat point and space curve singularities.     

A Characterization of Quasi-Homogeneous Purely Elliptic Complete Intersection Singularities

It is well-known that quasi-homogeneity is characterized by equality of the Milnor and Tjurina numbers for isolated complex analytic hypersurface singularities and for certain low-dimensional singularities. In this paper we prove that this characterization extends to isolated purely elliptic complete intersection singularities, with bounds on neither the embedding codimension nor the dimension of the singularity.