Related papers: A formula for Euler characteristic of line singula…

We obtain a recursive formula for the characteristic number of degree $d$ curves in $\mathbb{P}^2$ with prescribed singularities (of type $A_k$) that are tangent to a given line. The formula is in terms of the characteristic number of…

We prove a unified and general criterion for the uniqueness of critical points of a functional in the presence of constraints such as positivity, boundedness, or fixed mass. Our method relies on convexity properties along suitable paths and…

The Aluffi algebra is algebraic definition of characteristic cycles of a hypersurface in intersection theory. In this paper we focus on the Aluffi algebra of quasi-homogeneous and locally Eulerian hypersurface with isolated singularities.…

A theorem of Mumford states that, on complex surfaces, any normal isolated singularity whose link is diffeomorphic to a sphere is actually a smooth point. While this property fails in higher dimensions, McLean asks whether the contact…

Using the semicontinuity of the Milnor number and the {\L}ojasiewicz exponent, we give a simple proof of Yau's characterization of isolated homogenous hypersurface singularities.

We find an algorithm to compute the quadratic Euler characteristic of a smooth projective complete intersection of hypersurfaces of the same degree. As an example, we compute the quadratic Euler characteristic of a smooth projective…

We study one parameter deformations of a pair consisting of an analytic singular space $X_0$ and a function $f_0$ on it, in case this defines an isolated singularity. We prove, under general conditions, a bouquet decomposition of the Milnor…

Convenient mixed functions with strongly non-degenerate Newton boundaries have Milnor fibrations, as the isolatedness of the singularity follows from the convenience. In this paper, we consider the Milnor fibration for non-convenient mixed…

We compute the Euler characteristics of the recently discovered series of Gothic Teichm\"{u}ller curves. The main tool is the construction of 'Gothic' Hilbert modular forms vanishing at the images of these Teichm\"{u}ller curves. Contrary…

For any connected component $H_0$ of the space of real meromorphic functions we build a compactification $N(H_0)$ of the space $H_0$. Then we express the Euler characteristics of the spaces $H_0$ and $N(H_0)$ in terms of topological…

For any plane curve singularity defined by an analytic function germ $f$, we construct a spine on each Milnor fiber simultaneously, that realizes the vanishing topology. In order to do so, we study the separatrices at the origin of the…

Let $L$ be a cyclic $L_\infty$-algebra of dimension $3$ with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic $L_\infty$-algebra structure on the cohomology $H^*(L)$. The inner product…

The purpose of this note is to prove an Euler-type formula for partitions of the M\"obius strip. This formula was introduced in our joint paper with R.~Kiwan, "Courant-sharp property for Dirichlet eigenfunctions on the M\"obius strip"…

We consider the problem of computing the Euler characteristic of an abstract simplicial complex given by its vertices and facets. We show that this problem is #P-complete and present two new practical algorithms for computing Euler…

We study functions on isolated singularities and prove some results of type Milnor number = Tjurina number. We use them to endow the base space of their miniversal deformation with the structure of F-manifold.

We study the boundary of the Milnor fibre of real analytic singularities $f: (\bR^m,0) \to (\bR^k,0)$, $m\geq k$, with an isolated critical value and the Thom $a_f$-property. We define the vanishing zone for $f$ and we give necessary and…

We give in this work an explicit combinatorial algorithm for the description of the Milnor fiber of a Newton non degenerate surface singularity as a graph manifold. This is based on a previous work by the author describing a general method…

Mechanical fields over thin elastic surfaces can develop singularities at isolated points and curves in response to constrained deformations (e.g., crumpling and folding of paper), singular body forces and couples, distributions of isolated…

In this work we address the realizability of a Lyapunov graph labeled with GS singularities, namely regular, cone, Whitney, double crossing and triple crossing singularities, as continuous flow on a singular closed $2$-manifold…

Assume that the coefficients of a polynomial in a complex variable are Laurent polynomials in some complex parameters. The parameter space (a complex torus) splits into strata corresponding to different combinations of coincidence of the…