Le spectre réel et la topologie des variétés algébriques sur un corps réel clos
Le topos étale réel d'un anneau
Lefschetz Fibrations and real Lefschetz fibrations
This note is based on the lectures that I have given during the winter school Winter Braids IV, School on algebraic and topological aspects of braid groups held in Dijon on 10 - 13 February 2014. The aim of series of three lectures was to give an overview of geometrical and topological properties of 4-dimensional Lefschetz fibrations. Meanwhile, I could briefly introduce real Lefschetz fibrations, fibrations which have certain symmetry, and could present some interesting features of them.This note...
Levels in algebra and topology.
Lie ideals and Jordan triple derivations in rings
Lipschitz stratification of real analytic sets
Lipschitz stratifications and Lipschitz isotopies
Local approximation of semialgebraic sets
Let be a closed semialgebraic subset of euclidean space of codimension at least one, and containing the origin as a non-isolated point. We prove that, for every real , there exists an algebraic set which approximates to order at . The special case generalizes the result of the authors that every semialgebraic cone of codimension at least one is the tangent cone of an algebraic set.
Local semianalytic geometry.
Łojasiewicz Exponent of Overdetermined Mappings
A mapping is called overdetermined if m > n. We prove that the calculations of both the local and global Łojasiewicz exponent of a real overdetermined polynomial mapping can be reduced to the case m = n.
Łojasiewicz inequalities for sets definable in the structure
We consider some variants of Łojasiewicz inequalities for the class of subsets of Euclidean spaces definable from addition, multiplication and exponentiation : Łojasiewicz-type inequalities, global Łojasiewicz inequalities with or without parameters. The rationality of Łojasiewicz’s exponents for this class is also proved.
Manifolds with a unique embedding
We show that if X, Y are smooth, compact k-dimensional submanifolds of ℝⁿ and 2k+2 ≤ n, then each diffeomorphism ϕ: X → Y can be extended to a diffeomorphism Φ: ℝⁿ → ℝⁿ which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping ϕ is analytic, then we can choose Φ to be also analytic. We extend this result to some interesting categories of closed (not necessarily compact) subsets of ℝⁿ, namely, to the category of Nash submanifolds...
Manis valuations and Prüfer extensions
Manis valuations and Prüfer extensions. I.
Matching local Witt invariants
The starting point of this note is the observation that the local condition used in the notion of a Hilbert-symbol equivalence and a quaternion-symbol equivalence — once it is expressed in terms of the Witt invariant — admits a natural generalisation. In this paper we show that for global function fields as well as the formally real function fields over a real closed field all the resulting equivalences coincide.
Michael's theorem for Lipschitz cells in o-minimal structures
A version of Michael's theorem for multivalued mappings definable in o-minimal structures with M-Lipschitz cell values (M a common constant) is proven. Uniform equi-LCⁿ property for such families of cells is checked. An example is given showing that the assumption about the common Lipschitz constant cannot be omitted.
Minimal generation of basic open semianalytic sets.
Modified Nash triviality of a family of zero-sets of real polynomial mappings
In this paper we introduce the notion of modified Nash triviality for a family of zero sets of real polynomial map-germs as a desirable one. We first give a Nash isotopy lemma which is a useful tool to show triviality.Then, using it, we prove two types of modified Nash triviality theorem and a finite classification theorem for this triviality. These theorems strengthen similar topological results.
Moduli Spaces for Real Algebraic Curves and Real Abelian Varieties.
Moduli spaces of stable real algebraic curves