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Date
Editeur
Auteur
Titre
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Alexander Gorodnik - Diophantine approximation and flows on homogeneous spaces (Part 2)
/ Fanny Bastien
/ Canal-u.fr
Voir le résumé
Voir le résumé
The
fundamental problem in the theory of Diophantine approximation is to
understand how well points in the Euclidean space can be approximated by
rational vectors with given bounds on denominators. It turns out that
Diophantine properties of points can be encoded using flows on
homogeneous spaces, and in this course we explain how to use techniques
from the theory of dynamical systems to address some of questions in
Diophantine approximation. In particular, we give a dynamical proof of
Khinchin’s theorem and discuss Sprindzuk’s question regarding
Diophantine approximation with dependent quantities, which was solved
using non-divergence properties of unipotent flows. In conclusion we
explore the problem of Diophantine approximation on more general
algebraic varieties. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory
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Jean-Louis Verger-Gaugry - Limit Equidistribution (Part 1)
/ Fanny Bastien
/ 26-06-2013
/ Canal-u.fr
Verger-Gaugry Jean-Louis
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Voir le résumé
indisponible Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, number theory, summer school
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Mark Pollicott - Dynamical Zeta functions (Part 1)
/ Fanny Bastien
/ 24-06-2013
/ Canal-u.fr
Pollicott Mark
Voir le résumé
Voir le résumé
indisponible Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory
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Pierre Liardet - Randomness and Cryptography with a dynamical point of view (Part 1)
/ Fanny Bastien
/ 24-06-2013
/ Canal-u.fr
Liardet Pierre
Voir le résumé
Voir le résumé
indisponible Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory
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Accéder à la ressource
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Alexander Gorodnik - Diophantine approximation and flows on homogeneous spaces (Part 1)
/ Fanny Bastien
/ 24-06-2013
/ Canal-u.fr
Gorodnik Alexander
Voir le résumé
Voir le résumé
The
fundamental problem in the theory of Diophantine approximation is to
understand how well points in the Euclidean space can be approximated by
rational vectors with given bounds on denominators. It turns out that
Diophantine properties of points can be encoded using flows on
homogeneous spaces, and in this course we explain how to use techniques
from the theory of dynamical systems to address some of questions in
Diophantine approximation. In particular, we give a dynamical proof of
Khinchin’s theorem and discuss Sprindzuk’s question regarding
Diophantine approximation with dependent quantities, which was solved
using non-divergence properties of unipotent flows. In conclusion we
explore the problem of Diophantine approximation on more general
algebraic varieties. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory
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Accéder à la ressource
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Christiane Frougny - Systèmes de numération et automates (Part 1)
/ Fanny Bastien
/ 27-06-2013
/ Canal-u.fr
Frougny Christiane
Voir le résumé
Voir le résumé
Automates
finis et langages rationnels de mots finis • Automates finis et mots
infinis • Systèmes de numération à base réelle • Nombres de Pisot,
nombres de Parry et nombres de Perron • Systèmes de numération définis
par une suite Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory, automates, Systèmes de numération
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Fabien Durand - Sur le Théorème de Cobham (Part 1)
/ Fanny Bastien
/ 26-06-2013
/ Canal-u.fr
Durand Fabien
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Voir le résumé
indisponible Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory
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Karma Dajani - An introduction to Ergodic Theory of Numbers (Part 2)
/ Fanny Bastien
/ 19-06-2013
/ Canal-u.fr
Dajani Karma
Voir le résumé
Voir le résumé
In this course we give an introduction to the ergodic theory behind common number expansions, like expansions to integer and non-integer bases, Luroth series and continued fraction expansion. Starting with basic ideas in ergodic theory such as ergodicity, the ergodic theorem and natural extensions, we apply these to the familiar expansions mentioned above in order to understand the structure and global behaviour of different number theoretic expansions, and to obtain new and old results in an elegant and straightforward manner. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory
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Karma Dajani - An introduction to Ergodic Theory of Numbers (Part 1)
/ Fanny Bastien
/ 17-06-2013
/ Canal-u.fr
Dajani Karma
Voir le résumé
Voir le résumé
In
this course we give an introduction to the ergodic theory behind common
number expansions, like expansions to integer and non-integer bases,
Luroth series and continued fraction expansion. Starting with basic
ideas in ergodic theory such as ergodicity, the ergodic theorem and
natural extensions, we apply these to the familiar expansions mentioned
above in order to understand the structure and global behaviour of
different number theoretic expansions, and to obtain new and old results
in an elegant and straightforward manner. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory
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Mike Boyle - Nonnegative matrices : Perron Frobenius theory and related algebra (Part 4)
/ Fanny Bastien
/ 25-06-2013
/ Canal-u.fr
Boyle Mike
Voir le résumé
Voir le résumé
Lecture
I. I’ll give a complete elementary presentation of the essential
features of the Perron Frobenius theory of nonnegative matrices for the
central case of primitive matrices (the "Perron" part). (The "Frobenius"
part, for irreducible matrices, and finally the case for general
nonnegative matrices, will be described, with proofs left to
accompanying notes.) For integer matrices we’ll relate "Perron numbers"
to this and Mahler measures. Lecture II. I’ll describe how the
Perron-Frobenius theory generalizes (and fails to generalize) to 1,2,... x 1,2,...
nonnegative matrices. Lecture III. We’ll see the simple, potent
formalism by which a certain zeta function can be associated to a
nonnegative matrix, and its relation to the nonzero spectrum of the
matrix, and how polynomial matrices can be used in this setting for
constructions and conciseness. Lecture IV. We’ll describe a natural
algebraic equivalence relation on finite square matrices over a semiring
(such as Z, Z_+, R, ... ) which refines the nonzero spectrum and is
related to K-theory. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory
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