Real and Complex Singularities

PREFACE.

LIST OF PARTICIPANTS.

ORGANIZING COMMITTEE.

CONTENTS.

We study integral curves in 3-dimensional contact space and Engel space. Comparing the proofs of the classification of topologically trivial Legendrian knots in the standard contact 3-space and the classification of integral circles in the standard Engel space, we discuss the differences and similarities of them.

In this article we discuss some representation problem, related to the equisingularity problem for real algebraic singularities. Given a polynomial mapping f : ℝⁿ→ ℝ^p with f(0) = 0, we show that f is represented by a blowing-up and, in the case n ≥ p, is represented by a desingularisation map for some real algebraic variety. We further treat these properties also in the polynomially bounded o-minimal structure.

We suggest a reformulation of the problem of determining algebraically the Blanchfield pairing (Seifert form) of the link of an isolated singularity, in terms of seeking an hermitian isomorphism of the local Gauß-Manin connection with its dual in the category of such connections.

In this survey article we show several recent results on the local classification of irreducible plane curve singularities, plane branches, based on Oscar Zariski’s lectures [17], and that of Legendre curve singularities. We compare exact classifications, the classification of plane branches and the classifications of Legendre curves, in the cases of simple and uni-modal singularities, and, as a consequence, we observe remarkable difference and mysterious similarity of these classification results. Also the comparison is achieved for a bi-modal case of characteristic (6, 7) which Zariski treated.

The theory of multivariate polynomial interpolation has made a big progress at the beginning of the nineties. The interpolation of Hermite type reduces to the theories of holonomic systems of PDEs with constant coefficients and Gröbner basis. There is another scheme of interpolation: the theory of the least interpolation spaces. This method allow us to treat more general linear functionals such as the integrations (by compactly supported Radon measures).

This is a sketch of the theory of germs of analytic mappings of analytic spaces and related theories. There are two major streams in singularity theory. One consists of the research of commutative algebra and algebraic geometry. They treat mainly rings and spaces rather than mappings. Another consists of theories of singularities of smooth mappings of manifolds. This is originated by topologists.

In this note we are mainly concerned with local properties of singularities of general analytic mappings of singular spaces. This theory has a flavour of analysis and we may say that the theme of this note belongs to differential analysis or analytic singularity theory. We work mostly in the framework of the complex analytic geometry for the sake of simplicity.

We point out some relations of this theory to the problems in other areas of mathematics also and show a list of related literature (of course, not complete). This note has overlapping parts with the postscript of the Japanese book [64].

This is an elementary survey on the results of the singularity theory of smooth mapping and its applications.

In this paper we outline a program of connecting some old problems in algebraic and symplectic geometry with new methods coming from Homological Mirror Symmetry. We discuss several examples and applications.

Resolution of Singularities (Desingularization) and Equisingularities (Equivalence of Singularities) are neighboring research fronts in Algebraic/Analytic Geometry; they involve sophisticated mathematics and much jargons. However, a number of fundamental concepts can be fully explained to undergraduate students, using elementary examples in ℝ². This we try to achieve. Student readers are encouraged to do the exercises, most of which are easy.

Consider a self-similar space X. A typical situation is that X looks like several copies of itself glued to several copies of another space Y, and Y looks like several copies of itself glued to several copies of X—or the same kind of thing with more than two spaces. Thus, we have a system of simultaneous equations in which ‘higher-dimensional formulas’ specify how to glue the spaces together.

This idea is developed in detail elsewhere [1,2]. The present account explains the theory in outline.

By reformulating a result of Ernström [6] in terms of Chow groups, we obtain a class formula, i.e. a formula for the degrees of dual varieties, which is applicable also for projective varieties with positive dual defect. Various generalized Plücker-Teissier-Kleiman type formulas will be obtained. By our approach, topological invariants such as Milnor numbers along the singular locus of the original projective variety naturally appear in the class formulas.

We review the notion of derived Picard groups of algebras and the relations with automorphism groups of derived categories of coherent sheaves on some projective varieties and the non-commutative space which was introduced by Kontsevich-Rosenberg.

One of the the properties which characterizes the higher dimensional complex analysis is so-called the Hartogs’ phenomenon as well as the uniqueness of analytic continuation. This enables us to analyze complex singularities by means of smooth manifolds-technique on its regular part. In this paper, we will give an overview on the approach to deformation theory of isolated singularities under this guiding principle.

Homological mirror symmetry conjectures that the derived category of coherent sheaves on X should be equivalent to the Fukaya category of its mirror pair. In the light of recent work by Krause, it is natural to ask for an infinite version.

A topological invariant of a polynomial map p : X → B from a complex surface containing a curve C ⊂ X to a one-dimensional base is its rational second homology class in the compactification of the moduli space of genus g curves with n labeled points . Here the generic fibre of p has genus g and intersects C in n points. In this paper we give an efficient method to calculate this homology class. We apply this to any polynomial in two complex variables p : ℂ² → ℂ where the n points on a fibre are its points at infinity.

Let f : (Rⁿ, 0) → (R^p, 0) be a C^∞ map-germ. We are interested in whether the number modulo 2 of stable singular points of codimension n that appear near the origin in a generic perturbation of f is a topological invariant. In this paper we concentrate on investigating the problem when p is 2n – 1, where stable singular points of codimension n are only Whitney’s umbrellas.

This is a survey article on the existence problem of fold maps. Let Mⁿ be a closed n-manifold and f : Mⁿ → ℝ^p a generic smooth map into the p-dimensional Euclidean space with n ≥ p. If f has only fold singularities as its singularities, then f is called a fold map. When p = 1, a fold map is nothing but a Morse function. Then we will consider a problem belonging to global singularity theory: Find the necessary and/or sufficient condition(s) for the existence of fold maps. In the problem for the cases where p = 3 and p = 7, the existence of fold maps has a special position if we assume that the Euler characterisitc of the source manifold should be odd. The main purpose of the paper is to discuss the problem in the case that p = 3, 7 respectively by summarizing recent results in relation with the Thom polynomials, the Eliashberg-Ando h-principle theorem, etc. We will also give a new result on the necessary and sufficient condition for the existence problem of fold maps of odd dimensional manifolds into ℝ³.

A complete solution of a first order ordinary differential equation is a certain one-parameter family of geometric solutions on the equation surface. In this survey, we shall discuss the existence conditions, properties, duality, a generic classification of completely integrable (regular) first order ordinary differential equations and of bifurcations of them. Moreover, we consider a general Clairaut type equations which is a generalized notion of the classical first order Clairaut equations, but the equations are not necessarily regular first order ordinary differential equations. We also give a generic classification and bifurcations of general Clairaut type equations as an application of singularity theory.

No abstract received.

The author [24] have shown that under certain conditions, the Euler number of the source 4-manifold has the same parity as the total number of certain singular fibres of stable maps. In this paper, we show that there is no Euler number formula of the same type if we have no condition on the stable maps.

AUTHOR INDEX.