We develop two methods for expressing the global index of the gradient of a 2 variable polynomial function $f$: in terms of the atypical fibres of $f$, and in terms of the clusters of Milnor arcs at infinity. These allow us to derive upper bounds for the global index, in particular refining the one that was found by Durfee in terms of the degree of $f$.

We develop a Thom–Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather–Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with isolated instability which are not necessarily versal. We prove a frontal version of Mond's conjecture in dimension 1. Finally, we classify stable frontal multigerms and give a complete classification of corank 1 stable frontals from $\mathbb {C}^3$ to $\mathbb {C}^4$.

We obtain a complete topological classification of $k$-folding map-germs on generic surfaces in $\mathbb {R}^3$, discover new robust features of surfaces and recover, in a unified way, many of the robust features that were obtained previously by considering the contact of a surface with lines, planes or spheres.

In this paper, we give the generic classification of the singularities of 3-parameter line congruences in $\mathbb {R}^{4}$. We also classify the generic singularities of normal and Blaschke (affine) normal congruences.

In this paper, we prove the Fukui–Kurdyka–Paunescu conjecture, which says that sub-analytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (respectively, degree) of real and complex analytic (respectively, algebraic) sets. For instance, still in the real case, we prove a global version of the Fukui–Kurdyka–Paunescu conjecture. In the complex case, one of the results that we prove is the following: if $(X,0)\subset (\mathbb {C}^{n},0)$, $(Y,0)\subset (\mathbb {C}^{m},0)$ are germs of analytic sets and $h\colon (X,0)\to (Y,0)$ is a semi-bi-Lipschitz homeomorphism whose graph is a complex analytic set, then the germs $(X,0)$ and $(Y,0)$ have the same multiplicity. One of the results that we prove in the global case is the following: if $X\subset \mathbb {C}^{n}$, $Y\subset \mathbb {C}^{m}$ are algebraic sets and $\phi \colon X\to Y$ is a semi-algebraic semi-bi-Lipschitz homeomorphism such that the closure of its graph in $\mathbb {P}^{n+m}(\mathbb {C})$ is an orientable homological cycle, then ${\rm deg}(X)={\rm deg}(Y)$.

Let $f\colon (\mathbb {R}^{3},0)\to (\mathbb {R}^{4},0)$ be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of $f$ with a small enough sphere $S^{3}_\epsilon$ centred at the origin in $\mathbb {R}^{4}$. If $f$ is of fold type, we define a tree, that we call dual tree, that contains all the topological information of the link and we prove that in this case it is a complete topological invariant. As an application we give a procedure to obtain normal forms for any topological class of fold type.

In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations

\begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*}

$0<\alpha <1$

$\Omega = \mathbb {R}^{N}$

$\Omega$

$\Gamma$

$\Omega$

$f(t)$

$t\in [0,\,\infty )$

$f(t)/t^{({N+2\alpha })/({N-2\alpha })}$

$t$

$t$

$t>0$

\begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*}

$\Gamma =\{0\}$

We consider the relative Bruce–Roberts number $\mu _{\textrm {BR}}^{-}(f,\,X)$ of a function on an isolated hypersurface singularity $(X,\,0)$. We show that $\mu _{\textrm {BR}}^{-}(f,\,X)$ is equal to the sum of the Milnor number of the fibre $\mu (f^{-1}(0)\cap X,\,0)$ plus the difference $\mu (X,\,0)-\tau (X,\,0)$ between the Milnor and the Tjurina numbers of $(X,\,0)$. As an application, we show that the usual Bruce–Roberts number $\mu _{\textrm {BR}}(f,\,X)$ is equal to $\mu (f)+\mu _{\textrm {BR}}^{-}(f,\,X)$. We also deduce that the relative logarithmic characteristic variety $LC(X)^{-}$, obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the component corresponding to the complement of $X$ in the ambient space, is Cohen–Macaulay.

Let $f\,{:}\,(\mathbb R^n,0)\to (\mathbb R,0)$ be an analytic function germ with non-isolated singularities and let $F\,{:}\, (\mathbb{R}^{1+n},0) \to (\mathbb{R},0)$ be a 1-parameter deformation of f. Let $ f_t ^{-1}(0) \cap B_\epsilon^n$ , $0 < \vert t \vert \ll \epsilon$ , be the “generalized” Milnor fiber of the deformation F. Under some conditions on F, we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui.

Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications.

In order to investigate envelopes for singular surfaces, we introduce one- and two-parameter families of framed surfaces and the basic invariants, respectively. By using the basic invariants, the existence and uniqueness theorems of one- and two-parameter families of framed surfaces are given. Then we define envelopes of one- and two-parameter families of framed surfaces and give the existence conditions of envelopes which are called envelope theorems. As an application of the envelope theorems, we show that the projections of singular solutions of completely integrable first-order partial differential equations are envelopes.

We obtain a new theorem for the non-properness set $S_f$ of a non-singular polynomial mapping $f:\mathbb C^n \to \mathbb C^n$. In particular, our result shows that if f is a counterexample to the Jacobian conjecture, then $S_f\cap Z \neq \emptyset $, for every hypersurface Z dominated by $\mathbb C^{n-1}$ on which some non-singular polynomial $h: \mathbb C^{n}\to \mathbb C$ is constant. Also, we present topological approaches to the Jacobian conjecture in $\mathbb C^n$. As applications, we extend bidimensional results of Rabier, Lê and Weber to higher dimensions.

We consider the possible disentanglements of holomorphic map germs f: (ℂn, 0) → (ℂN, 0), 0 < n < N, with nonisolated locus of instability Inst (f). The aim is to achieve lower bounds for their (homological) connectivity in terms of dim Inst (f). Our methods apply in the case of corank 1.

Let X ⊂ ℂn be an equidimensional complex algebraic set and let f: X → ℂ be a polynomial function. For each c ∈ ℂ, we define the global Brasselet number of f at c, a global counterpart of the Brasselet number defined by the authors in a previous work, and the Brasselet number at infinity of f at c. Then we establish several formulas relating these numbers to the topology of X and the critical points of f.

We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations.

We consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair is an immersion. We define an evolute and a focal surface of a framed immersion in the Euclidean space. The evolutes and focal surfaces of framed immersions are generalizations of each object of regular space curves. We give relationships between singularities of the evolutes and of the focal surfaces. Moreover, we consider properties of the evolutes, focal surfaces and repeated evolutes.

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

The detection of the bifurcation set of polynomial mapping ℝn → ℝp, n ⩾ p, in more than two variables remains an unsolved problem. In this note we provide a solution for n = p + 1 ⩾ 3.

We consider finitely determined map germs f : (ℝ3, 0) → (ℝ2, 0) with f–1(0) = {0} and we look at the classification of this kind of germ with respect to topological equivalence. By Fukuda's cone structure theorem, the topological type of f can be determined by the topological type of its associated link, which is a stable map from S2 to S1. We define a generalized version of the Reeb graph for stable maps γ : S2→ S1, which turns out to be a complete topological invariant. If f has corank 1, then f can be seen as a stabilization of a function h0: (ℝ2, 0) → (ℝ, 0), and we show that the Reeb graph is the sum of the partial trees of the positive and negative stabilizations of h0. Finally, we apply this to give a complete topological description of all map germs with Boardman symbol Σ2, 1.

We show that there exist an infinite number of topological orbits in classes of complex map germs from the plane to the plane that have a representative of type (xy, xa + yb ), with (a, b) ≠ = (2, 3) or (2, 5). Our key tool to prove this existence is the existence (or not) of stems in the class; these germs are not -finitely determined and allow the determination of a non-finite number of topological orbits. We also show that the class (xy, x 2 + y 5) has two topological orbits.