Geometric bases and topological equivalence

There are many algebraic and topological invariants associated to a singular point of a complex analytic function. The intent here is to discuss some of these invariants and the topological classification of singularities. Specifically, we establish that the topological type is determined by the Lefschetz vanishing cycles obtained by unfolding the singularity and certain local monodromy operators defined by Gabrielov. In Brieskorn's terminology singularities with the same geometric bases are topologically indistinguishable. Thus the higher invariants in the hierarchy of Brieskorn are necessary to understand the geometry of higher singularities. As a corollary to our main theorem, we obtain the result of Lê-Ramanujam which states that the topological type is constant in a oneparameter family of singularities with constant Milnor number.     

Cobordism of algebraic knots via Seifert forms

Summary We prove that, for anyn strictly greater than 2, there exist nonisotopic algebraic spherical knots of dimension 2n–1 which are cobordant. We first consider plane curve singularities. In that case we determine the Witt-class of the associated rational Seifert form and we attach to such a singularity a finite abelian group which is an invariant of the integral monodromy. This allows us to gather information about cobordism and isotopy classes of the higher dimensional algebraic knots obtained after suspension, by means of the dictionary relating knots and Seifert forms.A recent paper of Szczepanski [SZ] seemed to give partial results about the cobordism of algebraic knots. However, we shall show that these results cannot be true.Oblatum 28-VIII-1991 & 15-V-1992     

Fibration of classifying spaces in the cobordism theory of singular maps

In the cobordism theory of singular smooth maps there exist classifying spaces (analogues of Thom spectra) depending on the set of allowed singularity types. The so-called “key fibration” introduced by A. Szűcs connects these classifying spaces for different sets of allowed singularities. Here we prove the existence of such a fibration using a new, more simple and general argument than that of Szűcs. This makes it possible to extend the range of applications to some negative codimension maps.     

On corner singularities in Reissner-Mindlin's theory of elastic plates

In the framework of linear elasticity, singularities occur in domains with non-smooth boundaries. Particularly in Fracture Mechanics, the local stress field near stress concentrations is of interest. In this work, singularities at re-entrant corners or sharp notches in Reissner-Mindlin plates are studied. Therefore, an asymptotic solution of the governing system of partial differential equations is obtained by using a complex potential approach which allows for an efficient calculation of the singularity exponent λ. The effect of the notch opening angle and the boundary conditions on the singularity exponent is discussed. The results show, that it can be distinguished between singularities for symmetric and antisymmetric loading and between singularities of the bending moments and the transverse shear forces. Also, stronger singularities than the classical crack tip singularity with free crack faces are observed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simple Singularities for Hamilton-Jacobi Equations with Max-Concave Hamiltonians and Generalized Characteristics

We consider piecewise smooth viscosity solutions to a Hamilton-Jacobi equation, for which the Hamiltonian is the maximum of a finite number of smooth concave functions. We describe the possible types of “basic” singularities, namely jump discontinuities in either the derivative of the solution or the Hamiltonian, which occur across a smooth hypersurface. Each such type of singularity is described in terms of the properties of the Hamiltonian and the classical characteristics in the regions on either side of the singularity. Where appropriate, singular characteristic equations of the types developed by Melikyan are formulated which can be used in constructions.     

三维拟线性热弹性力学方程区域内部解的奇性传播规律

本文利用频率分析对角化的方法,研究了三维拟线性热弹性力学方程区域内部解的奇性传播规律. 首先从微局部观点出发,利用仿微分算子和拟微分算子将方程仿线性化和对角化.然后,利用穿梭法和经典的双曲方程和抛物方程理论,证明了区域内部解的奇性传播也是沿耦合方程组的双曲算子的零次特征带传播,并且当初值的奇性沿方程组的双曲算子的前向光锥传播时,时间t也具有很好的正则性.     

Geometry of singularities for the steady Boussinesq equations

Analysis and computations are presented for singularities in the solution of the steady Boussinesq equations for two-dimensional, stratified flow. The results show that for codimension 1 singularities, there are two generic singularity types for general solutions, and only one generic singularity type if there is a certain symmetry present. The analysis depends on a special choice of coordinates, which greatly simplifies the equations, showing that the type is exactly that of one dimensional Legendrian singularities, generalized so that the velocity can be infinite at the singularity. The solution is viewed as a surface in an appropriate compactified jet space. Smoothness of the solution surface is proved using the Cauchy-Kowalewski Theorem, which also shows that these singularity types are realizable. Numerical results from a special, highly accurate numerical method demonstrate the validity of this geometric analysis. A new analysis of general Legendrian singularities with blowup, i.e., at which the derivative may be infinite, is also presented, using projective coordinates.Research supported in part by the ARPA under URI grant number #N00014092-J-1890.Research supported in part by the NSF under grant number #DMS93-02013.Research supported in part by the NSF under grant #DMS-9306488.     

Algebraic cobordism revisited

We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative Donaldson–Thomas theory. We use double point cobordism to prove all the degree 0 conjectures in Donaldson–Thomas theory: absolute, relative, and equivariant.     

A theory of cobordism for non-spherical links

We define an equivalence relation, called algebraic cobordism, on the set of bilinear forms over the integers. When , we prove that two 2n - 1 dimensional, simple fibered links are cobordant if and only if they have algebraically cobordant Seifert forms. As an algebraic link is a simple fibered link, our criterion for cobordism allows us to study isolated singularities of complex hypersurfaces up to cobordism. Received: August 24, 1995     

Closed-form solutions for stress singularities at bi-material wedges in Reissner-Mindlin plates

In this work, stress singularities in isotropic bi-material junctions are investigated using Reissner-Mindlin plate theory by means of a complex potential formalism. The governing system of partial differential equations is solved employing methods of asymptotic analysis. The resulting asymptotic near-fields including the singularity exponent λ are obtained in a closed-form analytical manner as solutions of a corresponding eigenvalue problem. The singular solution character is discussed for different geometrical configurations. In particular, the present study investigates the influence of the material constants on the singularity exponent. It is shown, that the Reissner-Mindlin theory allows for distinguishing between singularities of the bending moments and the transverse shear forces. Further, stronger singularities than the classical crack-tip singularity are observed. The results allow for further application such as a combination with numerical methods. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of the attractive force of an ellipsoid

A numerical method for computing the attractive force of an ellipsoid is proposed that does not involve separating subdomains with singularities. The sought function is represented as a triple integral such as the inner integral of the kernel can be evaluated analytically with the kernel treated as a weight function. The inner integral is approximated by a quadrature for the product of functions, of which one has an integrable singularity. As a result, the integrand obtained before the second integration has only a weak logarithmic singularity. The subsequent change of variables yields an integrand without singularities. Based on this approach, at each stage of integral evaluation with respect to a single variable, quadrature formulas are derived that do not have singularities at integration nodes and do not take large values at these nodes. For numerical experiments, a rather complicated test function is constructed that is the exact attractive force of an ellipsoid of revolution with an elliptic density distribution.     

Resolution except for minimal singularities II. The case of four variables

In this sequel to Bierstone and Milman [4], we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities. This result develops the philosophy of [4], that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities.     

Simple singularities and topology of symplectically filling 4-manifold

Topological restrictions of symplectically filling 4-manifolds of links around simple singularities are studied by using the Seiberg-Witten monopole equations. In particular, the intersection form of minimal symplectically filling 4-manifolds of the singularity of type E ₈ is determined. Moreover, for the case of simply elliptic singularities, similar restrictions are obtained. In the proof, a vanishing theorem of the Seiberg-Witten invariant is discussed. Received: June 9, 1998.     

Elements of functional calculus andL 2 regularity for some classes of fourier integral operators

We employ the method of slices to develop a rudimentary calculus describing the nature of operators T*T (respectively, TT*), where T are Fourier integral operators with one-sided right (respectively, left) singularities; this idea has its roots in the work of Greenleaf and Seeger. Such a result allows us to reduce the L² regularity problem for operators in n dimensions with one-sided singularities to the L² regularity problem for operators with two-sided singularities in n − 1 dimensions. As a consequence we deduce almost sharp L²-Sobolev estimates for operators in three-dimensions; an interesting special case is provided by certain restricted X-ray transforms associated to line complexes which are not well curved. We also provide a proof of almost-sharpness by looking at a restricted X-ray transform associated to the line complex generated by the curve t → (t, t^k). Appropriate notions of singularity, strong singularity, and type are also developed.     

Equianalytic and equisingular families of curves on surfaces

Summary We consider flat families of reduced curves on a smooth surfaceS such that each memberC has the same number of singularities and each singularity has a fixed singularity type (up to analytic resp. topological equivalence). We show that these families are represented by a schemeH and give sufficient conditions for the smoothness ofH (atC). Our results improve previously known criteria for families with fixed analytic singularity type and seem to be quite sharp for curves in ℙ² of small degree. Moreover, for families with fixed topological type this paper seems to be the first in which arbitrary singularities are treated. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Resolution of Corank 1 Singularities of a Generic Front

We construct a resolution of singularities for wave fronts having only stable singularities of corank 1. It is based on a transformation that takes a given front to a new front with singularities of the same type in a space of smaller dimension. This transformation is defined by the class A_µ of Legendre singularities. The front and the ambient space obtained by the A_µ-transformation inherit topological information on the closure of the manifold of singularities A_µ of the original front. The resolution of every (reducible) singularity of a front is determined by a suitable iteration of A_µ-transformations. As a corollary, we obtain new conditions for the coexistence of singularities of generic fronts.     

On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups

In [Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In this work, by using Quinn's Transversality Theorem [Proc. Sympos. Pure. Math. 15 (1970) 213-222], it will be shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It will be shown that the Thom isomorphism in this theory will be satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps will be proved. After we discuss the relation between this theory and classical cobordism, we describe some applications to the complex cobordism of flag varieties of loop groups and we do some calculations.     

Variation homological diagrams,¶and corner singularities

We describe the general homological framework (the variation arrays and variation homological diagrams) in which can be studied hypersurface isolated singularities as well as boundary singularities and corner singularities from the point of view of duality. We then show that any corner singularity is extension, in a sense which is defined, of the corner singularities of less dimension on which it is built. This framework is also used to rewrite Thom–Sebastiani type properties for isolated singularities and to establish them for boundary singularities. Received: 27 June 2000 / Revised version: 18 October 2000     

A geometrical treatment of singular trajectories

The nonlinear field equations often arising in geometrodynamical theories of matter generally exhibit nonremovable singularities. Assuming that the field equations are either (1) analytic, or (2) structurally stable, we show that the Christoffel symbols of the second kind have certain properties (4), (9). The singularities are such that wedge-shaped sets can be found containing n-parameter families of trajectories emanating from a given point on a singularity. In particular cases where the singularity is an isolated point, entire neighborhoods have been found, composed of trajectories. The latter situation is especially convenient in that a generalized tangent space can be defined, in which various manipulations of other field equations can be done (separation of variables, potential theory) and for which an exponential map can be set up. We show that under (4) the geodesies (trajectories) vary continuously with respect to limit tangent vectors at the singularity. Under a slightly stronger condition (23), trajectories vary differentiably with respect to limit tangent vectors. The limit tangent vectors are the elements of the generalized tangent space.