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We extend Wolff's "local smoothing" inequality to a wider class of not necessarily conical hypersurfaces of codimension 1. This class includes surfaces with nonvanishing curvature, as well as certain surfaces with more than one flat…

Let $f(\mathbb{z},\bar{\mathbb{z}})$ be a convenient Newton non-degenerate mixed polynomial with strongly polar non-negative mixed weighted homogeneous face functions. We consider a convenient regular simplicial cone subdivision $\Sigma^*$…

The Polyak-{\L}ojasiewicz (P{\L}) inequality extends the favorable optimization properties of strongly convex functions to a broader class of functions. In this paper, we prove a theorem (also obtained by Criscitiello, Rebjock and Boumal in…

We study a Lorentzian version of the well-known Calder\'{o}n problem that is concerned with determination of lower order coefficients in a wave equation on a smooth Lorentzian manifold, given the associated Dirichlet-to-Neumann map. In the…

Let $(f,g): (S,s) \to (\mathbb{C}^2, 0)$ be a finite morphism from a germ of normal complex analytic surface to the germ of $\mathbb{C}^2$ at the origin. We show that the affine algebraic curve in $\mathbb{C}^2$ defined by the initial…

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's…

Let $(M,g^{TM})$ be an odd dimensional ($\dim M\geq 3$) connected oriented noncompact complete spin Riemannian manifold. Let $k^{TM}$ be the associated scalar curvature. Let $f:M\to S^{\dim M}(1)$ be a smooth area decreasing map which is…

We prove that if $\{f_t\}$ is a family of line singularities with constant L\^e numbers and such that $f_0$ is a homogeneous polynomial, then $\{f_t\}$ is equimultiple. This extends to line singularities a well known theorem of A. M.…

In this paper, we consider deformations of singular complex curves on complex surfaces. Despite the fundamental nature of the problem, little seems to be known for curves on general surfaces. Let $C\subset S$ be a complete integral curve on…

We study local, analytic solutions for a class of initial value problems for singular ODEs. We prove existence and uniqueness of such solutions under a certain non-resonance condition. Our proof translates the singular initial value problem…

This paper contributes to the theory of singularities of meromorphic linear ODEs in traceless $2\times2$ cases, focusing on their deformations and confluences. It is divided into two parts: The first part addresses individual singularities…

Topological properties of the jacobian curve ${\mathcal J}_{\mathcal{F},\mathcal{G}}$ of two foliations $\mathcal{F}$ and $\mathcal{G}$ are described in terms of invariants associated to the foliations. The main result gives a decomposition…

We prove that for two germs of analytic mappings $f,g\colon (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^p,0)$ with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated…

The article starts with some introductory material about resolution graphs of normal surface singularities (definitions, topological/homological properties, etc). We then discuss the case when the normal surface singularity is an N-fold…

We obtain subelliptic estimates for the $\bar{\partial}$-problem on complex algebraic surfaces embedded in $\mathbb{C}^n$ with isolated singularities. $W^{\epsilon}$ Sobolev norms of a form, $f$, for $0< \epsilon < 1$ are estimated in terms…

We devise calculus rules for the Kurdyka-\L{}ojasiewicz exponent using the rank theorem and Lie group actions. They apply to a wide class of composite and invariant functions, and are particularly suitable for handling nonisolated local…

The jump of the Milnor number of an isolated singularity $f_0$ is the minimal non-zero difference between the Milnor numbers of $f_0$ and one of its deformations $(f_s)$. We give a formula for the jump in some class of surface singularities…

Let (X,O) be a real analytic isolated surface singularity at the origin o of a real analytic manifold M equipped with a real analytic metric g. Given a real analytic function f:(M,O) --> (R,0) singular at O, we prove that the gradient…

Superisolated surface singularities in $(\mathbb{C}^3,0)$ were introduced by I. Luengo to prove that the $\mu$-constant stratum may be singular. The main feature of this family is that it can bring information from the projective plane…

Let U be a neighbourhood of 0 \in C^n.We show that for a holomorphic mapping F = (f_1,...,f_m) : U -> C^m, F(0) = 0, the Lojasiewicz exponent of F at 0 is attained on the set {z \in U : f_1(z)...f_m(z) = 0}.