m

F Nous contacter

0

Documents  Parusinski, Adam | enregistrements trouvés : 7

O
     

-A +A

P Q

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Research talks;Dynamical Systems and Ordinary Differential Equations;Algebraic and Complex Geometry

Classifying regular systems of first order linear ordinary equations is a classical subject going back to Poincare and Dulac. There is a gauge group whose action can be described and an integrable normal form produced. A similar problem for higher order differential equations was never addressed, perhaps because the corresponding equivalence relationship is not induced by any group action. Still one can develop a reasonable classification theory, largely parallel to the classical theory. This is a joint work with Shira Tanny from the Weizmann Institiute, see http://arxiv.org/abs/1412.7830. Classifying regular systems of first order linear ordinary equations is a classical subject going back to Poincare and Dulac. There is a gauge group whose action can be described and an integrable normal form produced. A similar problem for higher order differential equations was never addressed, perhaps because the corresponding equivalence relationship is not induced by any group action. Still one can develop a reasonable classification ...

34C20 ; 34M35

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Research talks;Dynamical Systems and Ordinary Differential Equations;Algebraic and Complex Geometry;Logic and Foundations

It is by now well known that collections of compact (real-)analytic vector fields and locally connected trajectories thereof are mutually well behaved in a way that can be made precise via notions from mathematical logic, namely, by saying that the structure on the real field generated by the collection is o-minimal (that is, every subset of the real numbers definable in the structure is a finite union of points and open intervals). There are also many examples known where the assumption of analyticity or compactness can be removed, yet o-minimality still holds. Less well known is that there are examples where o-minimality visibly fails, but there is nevertheless a well-defined notion of tameness in place. In this talk, I will: (a) make this weaker notion of tameness precise; (b) describe a class of examples where the weaker notion holds; and (c) present evidence for conjecturing that there might be no other classes of examples of “non-o-minimal tameness”. (Joint work with Patrick Speissegger.)
A few corrections and comments about this talk are available in the PDF file at the bottom of the page.
It is by now well known that collections of compact (real-)analytic vector fields and locally connected trajectories thereof are mutually well behaved in a way that can be made precise via notions from mathematical logic, namely, by saying that the structure on the real field generated by the collection is o-minimal (that is, every subset of the real numbers definable in the structure is a finite union of points and open intervals). There are ...

03C64 ; 34E05

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Research talks;Algebraic and Complex Geometry;Logic and Foundations;Number Theory

In the spirit of famous papers by Pila & Bombieri and Pila & Wilkie, I will explain how to bound the number of rational points, with respect to their height, in various kinds of sets, such as transcendental sets definable in some o-minimal - or even not o-minimal - structure over the real field. I will emphazise the role played by bounds on derivatives and on sets of zeroes in this context.

03C98 ; 11D88 ; 14G05

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Research talks;Dynamical Systems and Ordinary Differential Equations;Algebraic and Complex Geometry;Number Theory

Smooth parametrization consists in a subdivision of mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the talk is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently separated domains: Smooth Dynamics, Diophantine Geometry, and Approximation Theory. The structure of the results, open problems, and conjectures in each of these domains shows in many cases a remarkable similarity, which I’ll try to stress. Sometimes this similarity can be easily explained, sometimes the reasons remain somewhat obscure, and it motivates some natural questions discussed in the talk. I plan to present also some new results, connecting smooth parametrization with “Remez-type” (or “Norming”) inequalities for polynomials restricted to analytic varieties. Smooth parametrization consists in a subdivision of mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the talk is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently separated domains: Smooth Dynamics, Diophantine Geometry, and Approximation ...

37C05 ; 11Gxx ; 41A46

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Research talks;Dynamical Systems and Ordinary Differential Equations;Algebraic and Complex Geometry

Ecalle’s resurgent functions appear naturally as Borel transforms of divergent series like Stirling series, formal solutions of differential equations like Euler series, or formal series associated with many other problems in Analysis and dynamical systems. Resurgence means a certain property of analytic continuation in the Borel plane, whose stability under con- volution (the Borel counterpart of multiplication of formal series) is not obvious. Following the analytic continuation of the convolution of several resurgent functions is indeed a delicate question, but this must be done in an explicit quan- titative way so as to make possible nonlinear resurgent calculus (e.g. to check that resurgent functions are stable under composition or under substitution into a convergent series). This can be done by representing the analytic continuation of the convolution product as the integral of a holomorphic n-form on a singular n-simplex obtained as a suitable explicit deformation of the standard n-simplex. The theory of currents is convenient to deal with such integrals of holomorphic forms, because it allows to content oneself with little regularity: the deformations we use are only Lipschitz continuous, because they are built from the flow of non-autonomous Lipschitz vector fields. Ecalle’s resurgent functions appear naturally as Borel transforms of divergent series like Stirling series, formal solutions of differential equations like Euler series, or formal series associated with many other problems in Analysis and dynamical systems. Resurgence means a certain property of analytic continuation in the Borel plane, whose stability under con- volution (the Borel counterpart of multiplication of formal series) is not obvious. ...

30D05 ; 37FXX

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

- 125p.
ISBN 978-2-85629-236-5

Panoramas et synthèses , 0024

Localisation : Collection 1er étage

ensemble algébrique réel # espace des arcs # ensemble symétrique par arc # équivalence analytique après éclatement # intégration motivique # fonction algébriquement constructible # invariant topologique

14Pxx ; 14P05 ; 14P10 ; 14P25 ; 32B20 ; 32C05 ; 58A07

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

- xvi; 120 p.
ISBN 978-2-85629-844-2

Astérisque , 0383

Localisation : Périodique 1er étage

décomposition d'un cylindre # D-module # catégorie dérivée # module filtré # objet filtré # filtration # topologie de Grothendieck # cohomologie modérée # catégorie quasi-abélienne # faisceau # espace de Sobolev # ensemble sous-analytique # topologie sous-analytique

16E35 ; 16W70 ; 18A25 ; 18D10 ; 18D35 ; 18F20 ; 32B20 ; 32C05 ; 32C38 ; 32S60 ; 46E35 ; 58A03

... Lire [+]

Z