Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but nonisomorphic fundamental groups. To do so, the Alexander Invariant and certain invariants of combinatorial line arrangements are presented and developed for combinatorics with only double and triple points. This is part of a more general project to better understand the relationship between topology and combinatorics of line arrangements.

We analyze the maximum averaged profit for one parameter families of dynamic inequalities and profit densities on a circle. Generic singularities of the profit for stationary strategies are classified. They are shown to be stable.

Thom polynomials compute the cohomology classes of degeneracy loci. In this paper we use a simple example to review the core ideas in different—mostly recently found—methods of computing Thom polynomials. Our goal is to show the underlying topology / geometry / algebra without involving combinatorics.

We show that finite order symmetries of the function singularity $J_{10}$ give rise to some of complex crystallographic groups listed in [12]. The groups are extracted from the equivariant monodromy of the function. This is the first appearance of affine reflection groups in a singularity context.

By (generalized) Thom polynomials we mean universal cohomology characteristic classes that express Poincaré duals to the singularity loci appearing in various context: singularities of maps, hypersurface singularities, complete intersection singularities, Lagrange and Legendre singularities, multisingularities, etc. In these notes we give a short review of the whole theory with a special account of discoveries of last years. We discuss existence of Thom polynomials, methods of their computations, relation between Thom polynomials for different classifications. Some of the theorems announced here are new and their proofs are not published yet. Some of known results acquire a new interpretation.

We show that every subset of $\mathbb{R}^n$ definable in an o-minimal structure can be decomposed into a finite number of definable sets that are quasi-convex i.e. have comparable, up to a constant, the intrinsic distance and the distance induced from the embedding. We apply this result to study the limits of secants of the trajectories of gradient vector field $\nabla f$ of a $C^1$ definable function $f$ defined in an open subset of $\mathbb{R}^n$. We show that if the o-minimal structure is polynomially bounded then the limit of such secants exists, that is an analog of the gradient conjecture of R. Thom holds. Moreover we prove that for $n = 2$ the result is true in any o-minimal structure.

Homotopy groups of the complements to divisors with ample components on non-singular projective varieties are considered as the modules over the fundamental group. We prove a vanishing theorem and consider the calculation of supports of these modules by relating them to the cohomology of local systems. We review previous work on the local study of isolated non-normal crossings. As an application, we obtain information about the support loci of homotopy groups of arrangements of hyperplanes.

It was shown by Kohno that all higher Massey products in the rational cohomology of a complex hypersurface complement vanish. We show that in general there exist non-vanishing triple Massey products in the cohomology with finite field coefficients.

The degree of mobility of a Riemannian metric $g$ is the dimension of the space of Riemannian metrics sharing the same geodesics with $g$. We prove that the degree of mobility of an irreducible Riemannian metric on a closed manifold is at most two, unless the sectional curvature is positive constant.

Goursat distributions are subbundles in the tangent bundles to manifolds having the flag of consecutive Lie squares of ranks not depending on a point and growing always by 1. It is known that moduli of the local classification of these objects (distributions determine their flags, and vice versa) are not functional, only continuous numeric, and appear in codimensions two and higher; singularities of codimension one are all simple. In the present work we show that most of the codimension-two singularities of Goursat flags is not simple. As to the precise modalities of those singularities, we give them at paper's end in the conjectural mode.

For controlled stratified maps $f : \mathcal{X} \to \mathcal{X}'$ between two stratified spaces, we define what it means for $f$ to be semi-differentiable, horizontally-$C^1$ and $\mathcal{F}$-semi-differentiable (where $\mathcal{F}$ is a foliation).

When $\mathcal{X}'$ is a smooth manifold, $f$ is always semi-differentiable.

In general, semi-differentiability is equivalent to $f$ being horizontally-$C^1$ with bounded differential.

Horizontally-$C^1$ regularity depends on the existence of $(a)$-regular horizontal stratified foliations of $\mathcal{X}$ and $\mathcal{X}'$, which gives a smooth version of the stratified fibration whose existence was conjectured by Whitney for analytic varieties in 1965, and implies a horizontally-$C^1$ version of Thom's first isotopy theorem.

We obtain finally the corresponding theorems for the finer property of $\mathcal{F}$-semi-differentiability.

We consider the class of submanifolds $M$ in an euclidean space $\mathbb{R}^n$ which admit a non-degenerate parallel normal vector field $\nu$. The image of the associated Gauss map $G_{\nu} : M \to S^{n-1}$ defines an immersed hyperspherical submanifold $M^{\nu}$ which has the following property: if $M$ has a contact of Boardman type $\Sigma^{i_1, \dots, i_k}$ with a hyperplane, then $M^{\nu}$ has the same contact type with the translated hyperplane. In particular, for a space curve $\alpha$ in $\mathbb{R}^3$, the spherical curve $\alpha^{\nu}$ has the same fiattenings and we deduce an extension of the Four Vertex Theorem. For an immersed surface $M$ in $\mathbb{R}^4$, it admits a local non-degenerate parallel normal vector field if and only if it is totally semi-umbilic and has non zero gaussian curvature $K$. Moreover, $G_{\nu}$ preserves the inflections and the asymptotic lines between $M$ and $M^{\nu}$. As a consequence, we deduce an extension for this class of surfaces of the classical Loewner and Carathéodory conjectures for umbilic points of analytic immersed surfaces in $\mathbb{R}^3$.

Based on the T. Fukui invariant and the recent motivic invariants proposed by S. Koike and A. Parusiński we give a simple classification of two variable quasihomogeneous polynomials by the blow-analytic equivalence.

This is a short and concise survey on recent results on the Milnor classes of global complete intersections. By definition the Milnor class of $X$ equals the difference between the Chern-Schwartz-MacPherson and the Fulton-Johnson classes of $X$ and we describe the results that express it in terms of the local and global invariants of the singular locus of $X$. In this survey we underline the characteristic cycle approach and its realtion to the vanishing Euler characteristic, as for instance to the Euler characteristic of the Milnor fibre in the hypersurface case.

A survey of recent results in the problem of the topological classification of 7th degree decomposable curves in the real projective plane is given.

We apply the method of constructing controlled vector fields to give sufficient conditions for the $\mathcal{A}$-topological triviality of deformations of map germs $f_t : (\mathbb{C}^n, 0) \to (\mathbb{C}^p, 0)$ of type $f_t(x) = f(x) + th(x)$, with $n \ge p$ or $n \le 2p$. These conditions are given in terms of an appropriate choice of Newton filtrations for $\mathcal{O}_n$ and $\mathcal{O}_p$ and are for the $\mathcal{A}$-tangent space of the germ $f$.

For the case $n \ge p$, we follow the technique used by M. A. S. Ruas in her Ph.D. Thesis [7] and construct control functions in the target and in the source to obtain, via a partition of the unit, a unique control function. We use the control function of the target to give an estimate for the case $p \ge 2n$. Moreover, in this case we show that if the coordinates of the map germ satisfy a Newton non-degeneracy condition, deformations by terms of higher filtration are topologically trivial.

As an application we obtain for both cases, $n \ge p$ and $p \ge 2n$, the results of Damon in [3] for deformations of weighted homogeneous map germs.

We study topology of symmetry sets, conflict sets and medial axes in the case when they have only stable singularities of corank 1. Singularities of these sets satisfy various conditions of coexistence. For example, isolated singularities and singularities forming smooth non-closed curves define a graph. If this graph is finite, then there is the following incidence relation: the sum of the local degrees of vertices of the graph is equal to the doubled number of its edges (the local degree of a vertex is the number of edges that are incident to this vertex; loops are counted twice). We give many-dimensional generalizations of this relation for sets mentioned above. These generalizations follow from some general facts on coexistence of wave front singularities found recently by the author.

We present sufficient conditions for the topological triviality of families of germs of functions defined on an analytic variety $V$. The main result is an infinitesimal criterion using the integral closure of a convenient ideal as the tangent space to a subset of the set of topologically trivial deformations of a given germ. Applications to the problem of equisingularity of families of sections of $V$ are also discussed.

In this paper are studied the nets of principal curvature lines on surfaces embedded in Euclidean 3–space near their end points, at which the surfaces tend to infinity.

This is a natural complement and extension to smooth surfaces of the work of Garcia and Sotomayor (1996), devoted to the study of principal curvature nets which are structurally stable –do not change topologically– under small perturbations on the coefficients of the equations defining algebraic surfaces.

This paper goes one step further and classifies the patterns of the most common and stable behaviors at the ends, present also in generic families of surfaces depending on one-parameter.

It is known that if two subanalytic strata satisfy Kuo's ratio test, then the normal cone of the larger stratum $Y$ along the smaller $X$ satisfies two nice properties: the fiber of the normal cone at any point of $X$ is the tangent cone to the fiber of $Y$ over that point; the projection of the normal cone to $X$ is open ("normal pseudo-flatness"). We present an example with $X$ a line and $Y$ a surface which is definable in any non polynomially bounded o-minimal structure such that the pair satisfies Kuo's ratio test, but neither of the above properties hold for the normal cone.

We give the principal notions in valuation theory, the value group and the residue field of a valuation, its rank, the compositions of valuations, and we give some classical examples. Then we introduce the Riemann-Zariski variety of a field, with the topology defined by Zariski. In the last part we recall the result of Zariski on local uniformization and give a sketch of the proof in the case of an algebraic surface.

The concept of motivic integration was invented by Kontsevich to show that birationally equivalent Calabi-Yau manifolds have the same Hodge numbers. He constructed a certain measure on the arc space of an algebraic variety, the motivic measure, with the subtle and crucial property that it takes values not in $\mathbb{R}$, but in the Grothendieck ring of algebraic varieties. A whole theory on this subject was then developed by Denef and Loeser in various papers, with several applications.

Batyrev introduced with motivic integration techniques new singularity invariants, the stringy invariants, for algebraic varieties with mild singularities, more precisely log terminal singularities. He used them for instance to formulate a topological Mirror Symmetry test for pairs of singular Calabi-Yau varieties. We generalized these invariants to almost arbitrary singular varieties, assuming Mori's Minimal Model Program.

The aim of these notes is to provide a gentle introduction to these concepts. There exist already good surveys by Denef-Loeser [DL8] and Looijenga [Loo], and a nice elementary introduction by Craw [Cr]. Here we merely want to explain the basic concepts and first results, including the $p$-adic number theoretic pre-history of the theory, and to provide concrete examples.

The text is a slightly adapted version of the 'extended abstract' of the author's talks at the 12th MSJ-IRI "Singularity Theory and Its Applications" (2003) in Sapporo. At the end we included a list of various recent results.

We give two geometric methods of constructing plane curves giving cable knots of torus knots via A'Campo's divide knot theory, related to both singularity theory and knot theory. We point out a relationship between "area" of the plane curves and the coefficients of finite Dehn surgery, which is Dehn surgery yielding three-dimensional manifolds with finite fundamental group.