Dirk Siersma

Singularities at infinity and their vanishing cycles

Let f C n C be any polynomial function By using global polar methods we introduce models for the bers of f and we study the monodromy at atypical values of f including the value in nity We construct a geometric monodromy with controlled behavior and de ne global relative monodromy with respect to a general linear form We prove localization results for the relative monodromy and derive a zeta function formula for the monodromy around an atypical value We compute the relative zeta function in several cases and emphasize the di erences to the classical local situation key words topology of polynomial functions singularities at in nity relative monodromy

Singularities with critical locus a 1-dimensional complete intersection and transversal type A1

In this paper we study germs of holomorphic functions f: (Cn+1,0)→C with the following two properties: (i) the critical set Σ of f is a 1-dimensional isolated complete intersection singularity (icis); (ii) the transversal singularity of f in points of Σ−{0} is of type A1. We first compute the homology of the Milnor fibre F of f in terms of numbers of special points in certain deformations. Next we show that the homotopy type of the Milnor fibre F of f is a bouquet of spheres. There are two cases: (a) general case Sn v⋯v Sn (b) special case Sn-1 v sn v⋯v Sn.

The Vanishing Topology of Non Isolated Singularities

We consider holomorphic function germs f : (ℂ n+1,O)→ (ℂ,0) and allow arbitrary singularities (isolated or non-isolated). We are interested in the topology of this situation, especially the so called vanishing homology.

New developments in singularity theory

Preface. Part A: Singularities of real maps. Classifications in Singularity Theory and Their Applications J.W. Bruce. Applications of Flag Contact Singularities V. Zakalyukin. On Stokes Sets Y. Baryshnikov. Resolutions of discriminants and topology of their complements V. Vassiliev. Classifying Spaces of Singularities and Thom Polynomials M. Kazarian. Singularities and Noncommutative Geometry J.-P. Brasselet. Part B: Singular complex varieties. The Geometry of Families of Singular Curves G.-M. Greuel, C. Lossen. On the preparation theorem for subanalytic functions A. Parusinski. Computing Hodge-theoretic invariants of singularities M. Schulze, J. Steenbrink. Frobenius manifolds and variance of the spectral numbers C. Hertling. Monodromy and Hodge Theory of Regular Functions A. Dimca. Bifurcations and topology of meromorphic germs S. Gusein-Zade, et al. Unitary reflection groups and automorphisms of simple hypersurface singularities V.V. Goryunov. Simple Singularities and Complex Reflections P. Slodowy. Part C: Singularities of holomorphic maps. Discriminants, vector fields and singular hypersurfaces A.A. du Plessis, C.T.C. Wall. The theory of integral closure of ideals and modules: Applications and new developments T. Gaffney. Nonlinear Sections of Nonisolated Complete Intersections J. Damon. The Vanishing Topology of Non Isolated Singularities D. Siersma.

Properties of conflict sets in the plane

This paper studies the smoothness and the curvature of conict sets of the distance function in the plane Conict sets are also well known as bisectors We prove smoothness in the case of two convex sets and give a formula for the curvature We generalize moreover to weighted distance functions the socalled JohnsonMehl model

Critical configurations of planar robot arms

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.

Curvatures of conflict surfaces in Euclidean 3-space

This article extends to three dimensions results on the curvature of the conflict curve for pairs of convex sets of the plane established by Siersma In the present case a conflict surface arises equidistant from the given convex sets The Gaussian Mean Curvatures and the location of Umbilic Points on the conflict surface are determined here Initial results on the Darbouxian type of Umbilic Points on conict surfaces are also established The results are expressed in terms of the principal directions and on the curvatures of the borders of the given convex sets

Vanishing homology of projective hypersurfaces with 1-dimensional singularities

We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus

Topology of polynomial functions and monodromy dynamics

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