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Accueil du site > Événements scientifiques > Journées MIAS 25-29 septembre 2017 - More Invariants from Arc Spaces > Résumés des conférences

Résumés des conférences

William Sit (CUNY)

Introduction To Computational Differential Algebra

The goal of this mini-course is to introduce the Ritt-Kolchin theory for differential polynomials. This theory has been further developed and successfully applied to many computational problems where the main objects of study can be described by a set of algebraic differential equations. Besides the obvious applications to "solving" (or more appropriately, “simplifying”) systems of differential equations, differential algebra has interacted with difference algebra, model theory, arithmetic number theory, symbolic computation, Painlevé theory, combinatorics, and more.

These talks will concentrate on fundamental concepts and results with emphasis on the relationship between classical theory and the modern symbolic computation approach.

The course is based on my tutorial paper "The Ritt-Kolchin Theory for Differential Polynomials". It will cover :

(a) basic set up : triangular sets (algorithms), rankings, characteristic sets ;

(b) radical and prime differential ideals, Ritt-Raudenbush Basis Theorem, Rosenfeld’s property and lemma (relating differential polynomial ideals to polynomial ideals) ;

(c) illustration of some advanced theorems such as Ritt’s Component Theorems (on general component and singular components), the Low Power Theorem ;

(d) (time permitting) sketches of some proofs, more advanced results and open problems ;

(e) the ideas of singularities and singular components of solutions and other topics of interest to participants will be discussed either in the course (where appropriate) and/or in informal sessions later.

The prerequisite is a first year graduate course on algebra.

Mercedes Haiech (Rennes I)

On noncomplete completions

Dans cet exposé nous examinerons les diverses topologies naturelles dont on peut munir le complété d’un anneau A lorsque A est muni d’une topologie I-adique. Plus précisément, le complété est muni d’une topologie naturelle pour lequel il est complet qui peut être définie par la filtration des (K_n), où K_n est le noyau du morphisme canonique du complété de A vers le quotient de A par l’idéal I^n. Cependant le complété peut aussi être muni de la topologie I-adique ou de la topologie K_1-adique. Dans le cas où l’anneau A est noethérien, il est connu que toutes ces topologies coïncident. Ce n’est en général plus le cas lorsque A n’est pas noethérien. L’exposé consistera à détailler un exemple d’anneau non noethérien où ces topologies adiques ne font pas du complété un espace complet.

Orlando Villamayor (U. Autónoma Madrid)

Some properties of the multiplicity and of blow ups at equimultiple centers

In this introductory presentation we will discuss general properties of the stratification of singularities of a variety defined by the multiplicity ; particularly about blow ups along equimultiple centers.

Angélica Benito (U. Autónoma Madrid)

Singular locus, Jacobian Criterion and derivations

In this talk we will remind some concepts related to the calculation of the singular locus of a given variety. In particular, we will focus our attention on the Jacobian Criterion and the ways to define basis of the derivations. Time permitting, we will discuss the relation between higher order differential operators and multiplicity and the connection of these operators with the jet spaces and the space of arcs.

Monique Lejeune Jalabert (Versailles)

Espaces d’arcs et multiplicités de Nash

Après avoir rappelé la structure de schéma de l’espace des arcs, j’énoncerai le théorème d’Artin-Greenberg. J’énoncerai ensuite les résultats de M.Hickel le concernant pour une hypersurface à singularité isolée définie sur un corps de caractéristique 0, puis sa définition de la suite des multiplicités de Nash d’un arc tracé sur une variété algébrique et quelques résultats sur ces suites.

Olivier Piltant (Versailles)

Small irreductible components of arc spaces in positive characteristic

Over ground fields of characteristic zero, the arc space of an irreducible algebraic variety is irreducible, a consequence of Kolchin’s Irreducibility Theorem. In positive characteristic, certain subvarieties of the singular locus may induce "small" irreducible components of the arc space. In these two talks, I will report on small irreducible components : examples, criteria for existence and connection with Resolution of Singularities.

David Bourqui (Rennes I)

Deformations of arcs : the Drinfeld-Grinberg-Kazhdan theorem

The Drinfeld-Grinberg-Kazdhan theorem asserts, in some sense, that the generic analytic type of the singularities of the arc scheme is tame, more precisely of finite dimension, even though the arc scheme is not. It may also be interpreted as a finiteness statement about the infinitesimal deformations of non-degenerate arcs. This interpretation plays a crucial part in Drinfeld’s proof of the theorem. We will give a detailed account of the arguments, since the involved technics are likely to be useful to achieve the scientific goals of the MIAS project.

Beatriz Pascual Escudero (U. Autónoma Madrid - ICMAT)

Invariants of singularities via arcs and the Nash multiplicity sequence

Arcs have shown to shed light on many properties of algebraic varieties. As an example, they are useful when one tries to obtain information about singular points. In particular, the Nash multiplicity sequence, which was defined by M. Lejeune-Jalabert for germs of hypersurfaces and generalized later by M. Hickel, allows us to define some invariants for points of maximum multiplicity. These invariants reflect the contact that the arcs centered at a point can have with the set of maximum multiplicity.

In this talk, we will define these invariants and give some results that evidence their nature. For instance, we will show their connection with invariants from constructive Resolution of Singularities.

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