相关论文: Verdier stratifications and [wf]-stratification in…

In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of Euclidean space is the restriction of a function that is continuously differentiable to order p. A necessary and sufficient…

We address the question of whether geometric conditions on the given data can be preserved by a solution in (1) the Whitney extension problem, and (2) the Brenner-Fefferman-Hochster-Koll\'ar problem, both for $\mathcal C^m$ functions. Our…

Let X be a definable sub-set of some o-minimal structure. We study the spectrum of X, in relation with the definability of types.

We show that separability and second-countability are first-order properties among topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$. We do so by introducing first-order characterizations -- definable separability and…

The paper deals with Henselian valued field with analytic structure. Actually, we are focused on separated analytic structures, but the results remain valid for strictly convergent analytic ones as well. A classical example of the latter is…

We develop the idea of self-indexing and the technology of gradient-like vector fields in the setting of Morse theory on a complex algebraic stratification. Our main result is the local existence, near a Morse critical point, of…

We investigate a model-theoretic property that generalizes the classical notion of "preservation under substructures". We call this property \emph{preservation under substructures modulo bounded cores}, and present a syntactic…

The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the…

We initiate a study of varieties of minimal degree in weighted projective spaces. We call a weighted projective space $\mathbf{P}(w_0,\dots,w_n)$ divisible if $w_i \mid w_{i+1}$ for all $i$. We provide sharp bounds for when a non-degenerate…

Given an o-minimal structure ${\mathcal M}$ with a group operation, we show that for a properly convex subset $U$, the theory of the expanded structure ${\mathcal M}'=({\mathcal M},U)$ has definable Skolem functions precisely when…

We give an abstract criterion for pasting pseudofunctors on two subcategories of a category into a pseudofunctor on the whole category. As an application we extend the variance theory of the twisted inverse image $(-)^!$ over schemes to…

Let $M$ be a stratum of a compact stratified space $A$. It is equipped with a general adapted metric $g$, which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, $g$ has a general…

Inspired by the Taubes-Wu construction of $\mathcal{C}^{1,\alpha}$ two-valued harmonic functions by the use of symmetry, we construct minimal surfaces with stratified branching sets as graphs of $\mathcal{C}^{1,\alpha}$ two-valued…

Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable…

We study $\mathbb{R}_{\textrm{an},\exp}$-definable functions $f:\mathbb{R}\to \mathbb{R}$ that take integer values at all sufficiently large positive integers. If $|f(x)|= O\big(2^{(1+10^{-5})x}\big)$, then we find polynomials $P_1, P_2$…

This paper addresses to Nichtnegativstellens\"atze for definable functions in o-minimal structures on $(\mathbb{R}, +, \cdot).$ Namely, let $f, g_1, \ldots, g_l \colon \mathbb{R}^n \to \mathbb{R}$ be definable $C^p$-functions ($p \ge 2$)…

New splitting theorems in a semi-Riemannian manifold which admits an irrotational vector field (not necessarily a gradient) with some suitable properties are obtained. According to the extras hypothesis assumed on the vector field, we can…

We show that the restriction to a smooth transversal section commutes to the computation of multiplier ideals and V-filtrations. As an application we prove the constancy of the spectrum along any stratum of a Whitney regular stratification.

We describe a class of sharply o-minimal structures, called analytically generated structures, whose definable sets and their complexity filtration are determined by the collection of definable complex cells. We prove a polynomially…

We show that if $\mathcal{F}$ is any "well-behaved" subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on $\pow(\mathbb{R})$ induced by $\mathcal{F}$ turns out to look like the Wadge hierarchy…