Voronoi diagrams and Morse theory of the distance function

We consider the (minimal) distance function of a point in the plane to a set P of N points in the plane. The locus of non-dierentiability of this distance function consists (besides of the points of P) exactly of the Voronoi diagram of P. We show that the number of minima (m), maxima (M) and `saddle points' (s) of the distance function satisfy: m - s + M = 1. This is similar to the Morse type of statements for dierentiable functions. The saddle points occur exactly where a Delaunay edge cuts the corresponding Voronoi edge in its interior. The set of those edges form a subgraph of the Delaunay graph, which connects all minima and saddle points. This graph devides the plane into regions. In each of the compact regions, there is exactly one maximum, the non compact regions don't contain a local maximum. At the end we classify all those graphs if P contains of 3 or 4 points

Complements of hypersurfaces, variation maps and minimal models of arrangements

We prove the minimality of the CW-complex structure for complements of hyperplane arrangements in $\mathbb C^n$ by using the theory of Lefschetz pencils and results on the variation maps within a pencil of hyperplanes. This also provides a method to compute the Betti numbers of complements of arrangements via global polar invariants

On minimal round functions

We describe the structure of minimal round functions on closed surfaces and three-folds. The minimal possible number of critical loops is determined and typical non-equisingular round function germs are interpreted in the spirit of isolated line singularities. We also discuss a version of Lusternik-Schnirelmann theory suitable for round functions

Local embeddings of lines in singular hypersurfaces

Lines on hypersurfaces with isolated singularities are classied New normal forms of simple singularities with respect to lines are obtained Several invariants are introduce

Deformations of polynomials and their zeta functions

For an analytic family P_s of polynomials in n variables (depending on a complex number s, and defined in a neighborhood of s = 0), there is defined a monodromy transformation h of the zero level set V_s= {P_s=0} for s different from 0, small enough. The zeta function of this monodromy transformation is written as an integral with respect to the Euler characteristic of the corresponding local data. This leads to a study of deformations of holomorphic germs and their zeta functions. We show some examples of computations with the use of this technique.Comment: 9 page

Singularities at infinity and their vanishing cycles, II : monodromy

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 innity We construct a geometric monodromy with controlled behavior and dene global relative monodromy with respect to a general linear form We prove localization results for the relative monodromy and derive a zetafunction formula for the monodromy around an atypical value We compute the relative zeta function in several cases and emphasize the dierences to the classical local situatio

Vanishing cycles and singularities of meromorphic functions

We study vanishing cycles of meromorphic functions This gives a new and unitary point of view extending the study of the topology of holomorphic germs as initiated by Milnor in the sixties and of the global topology of polynomial functions which has been advanced more recently We dene singularities along the poles with respect to a certain weak stratication and prove local and global bouquet structure in case of isolated singularities In general splitting of vanishing homology at singular points and global PicardLefschetz phenomena occu

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 set

Some analogs of Zariski's Theorem on nodal line arrangements

For line arrangements in P^2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal basis of the abelianization. We consider higher dimensional analogs of the above situation. For these analogs, we give purely combinatorial complete descriptions of the following topological invariants (over an arbitrary field): the twisted homology of the complement, with arbitrary rank one coefficients; the homology of the associated Milnor fiber and Alexander cover, including monodromy actions; the coinvariants of the first higher non-trivial homotopy group of the Alexander cover, with the induced monodromy action.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-28.abs.htm