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Unité Mixte de Recherche CNRS 5582 Université Grenoble I

UFR de Mathématiques

Institut Fourier 100 rue des maths, BP 74, 38402 St Martin d'Hères cedex, (France)

Téléphone : (+33/0) 4.76.51.46.56 Fax : (+33/0) 4.76.51.44.78



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Summer School 2013

NUMBER THEORY AND DYNAMICS

June 17th - July 5th, 2013

(Videos of Summer School 2013)

Practical Information

The summer school reception desk will be located on room 06 on the Institut Fourier’s groundfloor.

Schedule :

1st week

2nd week

3rd week

Titles and abstracts

LECTURE NOTES :

Shigeki Akiyama :

lecture notes 1

lecture notes 2

Karma Dajani

Mike Boyle :

lecture notes 1

lecture notes 2

lecture notes 3

lecture notes 4

Marie-José Bertin :

lecture notes 1

lecture notes 2

lecture notes 3

lecture notes 4

Fabien Durand

lecture notes 1

lecture notes 2

Alexander Gorodnik

lecture notes 1

lecture notes 2

lecture notes 3

Pierre Liardet

Mark Pollicott

Number Theory has old origins (Diophante, Fermat, ...) and became enriched with numerous theories over the last century, as those coming form Algebraic Geometry and Dynamical Systems. The main thema of this Summer School is concerned with the mathematics about dynamical systems of numeration, in a broad sense, and the related various methods in interaction with them: Diophantine Approximation, Mahler measures, automaticity and Cobham Theories, beta-expansions and substitutions, Pisot numbers, Salem numbers, dynamical numbers, symbolic dynamics and ergodicity, group actions, dynamical zeta function, limit equidistribution of conjugates, geometrical representation and Rauzy fractals, multidimensional continued fractions, Perron-Frobenius theory and its recent developments.

The organizer of this Summer School is Jean-Louis Verger-Gaugry (CR, CNRS researcher)

The Summer School will present the recent developments of these theories, as an active crossfertilization domain, where many works find their origin, not only in Mathematics but also in Theoretical computer science and Cryptology.

Week 1 : Pisot numbers, Salem numbers, Mahler measures, multidimensionnal continued fractions, Symbolic Dynamics, Perron-Frobenius theory and positivity, Diophantine Approximation, Ergodic Theory of numbers

Speakers :

Shigeki Akiyama, University of Tsukuba (Japon)

Pierre Arnoux, Institut de Mathématiques de Luminy

Marie-José Bertin, Institut de Mathématiques de Jussieu

Valérie Berthé, LIAFA

Mike Boyle, University of Maryland (USA)

Karma Dajani, University of Utrecht (The Netherlands)

Week 2 : same topics as Week 1 (following) with : Cobham theory, beta-expansions, limit Equidistribution, dynamical zeta Fonction, Applications to Cryptology

Speakers :

Marie-José Bertin, Institut de Mathématiques de Jussieu

Mike Boyle, University of Maryland (USA)

Karma Dajani, University of Utrecht (The Netherlands)

Fabien Durand, LAMFA, Université de Picardie Jules Verne

Christiane Frougny, LIAFA Paris

Alexander Gorodnik, University of Bristol (UK)

Pierre Liardet, Université d’Aix-Marseille

Mark Pollicott, University of Warwick

Jean-Louis Verger-Gaugry, Institut Fourier

Week 3 :

Contributing talks : the participants are encouraged to submit a contributing talk. These (20 minutes) contributing talks have to be announced by an abstract deposited on ATLAS-CONFERENCES here.

They will be scheduled as special afternoon sessions of the 28e Journées Arithmétiques JA2013.

View all the abstracts of the JA2013.

The deadline for abstracts is April 15, 2013.

The spaekers and the participants during the first and the second week :

Titles and abstracts

Shigeki AKIYAMA

Title: Numeration systems and associated tilings dynamical systems

Abstract: We shall discuss numeration systems as dynamical systems, called tiling dynamical systems. In this manner, many number theoretical problems are embed into dynamical problems, in particular, spectral property of tilings. One of the goal of this lecture is to discuss Pisot conjecture in the beta numeration system.

Pierre ARNOUX

title: Continued fractions

Abstract:

Valérie BERTHE

Title: Multidimensional continued fractions and dynamics

Abstract: The aim of this lecture is to discuss multidimensional continued fractions and Euclidean algorithms from the viewpoint of dynamical systems, by focusing on their connections with numeration systems and substitutions. We will consider mainly two types of generalizations of continued fractions, namely, multidimensional continued fractions algorithms defined by piecewise linear fractional transformations (this includes the most classical transformations such as Jacobi-Perron algorithm) and continued fractions associated with lattice reduction algorithms.

Marie-José BERTIN

Title: From Salem numbers to the Mahler measure of K3 surfaces

Abstract: McMullen’s recent paper « Dynamics with small entropy on projective K3 surfaces » puts a new light on Salem numbers. These algebraic integers remain however very mysterious. All Salem numbers can be obtained from Pisot numbers by Salem’s construction (Boyd (1977)) but the existence of a Salem number < 1.1762.. is still an open problem. After the review of the Salem’s construction and Boyd’s theorem, we shall define the logarithmic Mahler measure of a multivariate polynomial and prove that the Mahler measure of such a polynomial in two variables is the limit of an infinite sequence of Mahler measures of polynomials in one variable (Boyd (1981)). Then the course will focuse on explicit formulas obtained for certain types of polynomials in 2 and 3 variables. More precisely in some cases of 3 variables, we shall present two aspects of this Mahler measure: an arithmetic aspect as a Hecke L-series of an imaginary quadratic number field and a geometric one as a L-series of the K3 surface defined by the polynomial expressed as a L-series of a modular form of weight 3 and rational coefficients. We shall conclude by an evocation of more geometric problems of elliptic fibrations on algebraic K3 surfaces.

Mike BOYLE

Title: Nonnegative matrices: Perron Frobenius theory and related algebra

Abstract: 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.

Karma DAJANI

Title: Title: An introduction to Ergodic Theory of Numbers

Abstract: 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.

Fabien DURAND

Title: On Cobham’s Theorem

Abstract:

Christiane FROUGNY

Title:Numeration systems and automata

Abstract: • Finite automata and rational languages of finite words • Finite automata and infinite words • Real base numeration systems • Pisot numbers, Parry numbers and Perron numbers • Numeration systems defined by a basis

Alexander GORODNIK

Title: Diophantine approximation and flows on homogeneous spaces

Abstract: 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.

Pierre LIARDET

Title: Randomness and Cryptography with a dynamical point of view

Abstract:

Mark POLLICOTT

Title: Dynamical Zeta functions

Abstract:

Jean-Louis VERGER-GAUGRY

Title: Limit Equidistribution of conjugates

Abstract:

Titres et résumés

Shigeki AKIYAMA

Title: Numeration systems and associated tilings dynamical systems

Abstract: We shall discuss numeration systems as dynamical systems, called tiling dynamical systems. In this manner, many number theoretical problems are embed into dynamical problems, in particular, spectral property of tilings. One of the goal of this lecture is to discuss Pisot conjecture in the beta numeration system.

Pierre ARNOUX

titre : Fractions continues

Résumé :

Valérie BERTHE

Titre : Fractions continues multidimensionnelles et dynamique

Résumé : Le but de cet exposé est de présenter des généralisations multidimensionnelles des fractions continues et de l’algorithme d’Euclide d’un point de vue systèmes dynamiques, en nous concentrant sur les liens avec la numération et les substitutions. Nous allons considérer principalement deux types de généralisations, à savoir, les algorithmes définis par homographies, comme l’algorithme de Jacobi-Perron, et les fractions continues associées aux algorithmes de réduction dans les réseaux.

Marie-José BERTIN

Titre : Des nombres de Salem à la mesure de Mahler de surfaces K3

Résumé : Le récent article de McMullen « Dynamics with small entropy on projective K3 surfaces » éclaire d’un jour nouveau les nombres de Salem. Ces entiers algébriques gardent cependant tout leur mystère. On peut tous les obtenir grâce à la construction de Salem (Boyd (1977)) et cependant on ignore s’il en existe un inférieur à 1,1762... Après avoir rappelé la construction de Salem et le théorème de Boyd, on définira la mesure de Mahler logarithmique d’un polynôme de plusieurs variables. On prouvera que la mesure de Mahler d’un polynôme de deux variables est la limite d’une suite de mesures de Mahler de polynômes d’une variable (Boyd (1981)). On donnera des mesures explicites de mesures de Mahler de certaines classes de polynômes de 2 et 3 variables. En particulier dans le cas de 3 variables on présentera deux aspects de l’expression de cette mesure, un aspect arithmétique comme série L de Hecke d’un corps quadratique imaginaire et un aspect géométrique comme série L de la surface K3 définie par le polynôme qui s’exprime comme série L d’une forme modulaire de poids 3 à coefficients rationnels. Pour terminer, on évoquera des problèmes plus géométriques de fibrations elliptiques sur les surfaces K3 algébriques.

Mike BOYLE

Title: Nonnegative matrices: Perron Frobenius theory and related algebra

Abstract: 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.

Karma DAJANI

Title: Title: An introduction to Ergodic Theory of Numbers

Abstract: 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.

Fabien DURAND

Titre : Sur le Théorème de Cobham

Résumé :

Christiane FROUGNY

Titre : Systèmes de numération et automates

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

Alexander GORODNIK

Title: Diophantine approximation and flows on homogeneous spaces

Abstract: 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.

Pierre LIARDET

Titre : Aléa et Cryptographie avec un point de vue dynamique

Résumé :

Mark POLLICOTT

Title: Dynamical Zeta functions

Abstract:

Jean-Louis VERGER-GAUGRY

Titre : Equidistribution limite de conjugués

Résumé :


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