Yvan Saint-Aubin - Böcker

Les auteurs présentent des concepts mathématiques de base, comme les chaînes de Markov, les fonctions itérées, la structure de corps fini, et les développent dans le contexte de problèmes réels tels que la navigation internet, la compression de données et les codes correcteurs d'erreurs. Un des leitmotivs du texte est de montrer comment la modélisation mathématique, la puissance des outils mathématiques et l'abstraction sont toutes cruciales pour l'innovation technologique. Les divers sujets sont présentés avec clarté et les mathématiques toujours discutées à partir de connaissances de base. Chaque chapitre suggère de nombreux exercices, certains élémentaires pour renforcer la compréhension, d'autres plus avancés pour explorer de nouvelles problématiques. Une mise en contexte historique de certains concepts mathématiques ou de l'évolution d'une technologique enrichit le texte.

Of what use is mathematics? Hasn’t everything in mathematics already been disc- ered? Thesearenaturalquestionsoftenaskedbyundergraduates. Theanswersprovided by their professors are often quite brief. Most university courses, pressed for time and rigidly structured, o?er little opportunity to present and study actual applications and real-world examples. Evenmorehigh-schoolstudentsaskthesamequestionswithmoreinsistence. Tea- ers in these schools generally work under even tighter constraints than university prof- sors. If they are able to competently respond to these questions it is probably because they received good answers from their teachers and professors. And if they do not have the answers, then whose fault is it? The genesis of this text It is impossible to introduce this text without ?rst discussing the course in which it originated. The course “Mathematics and Technology” was created at the Universit´ e de Montr´ eal and taught for the ?rst time in the winter semester of 2001. It was created after observing that most courses in the department neglect to present real applications. Sinceitscreationthecoursehasbeenopentobothundergraduatemathematicsstudents and future high-school teachers. Since no appropriate text or manual for the course we envisioned existed, we were led to write our own course notes, from which we taught. We got so caught up in writing these notes that they quickly grew to the size of a textbook, containing much more material than could possibly be taught in one semester. Despite the two of us being career mathematicians, we must admit that we both knew little or nothing about most of the applications presented in the following chapters.

This book pays tribute to two pioneers in the field of Mathematical physics, Jiri Patera and Pavel Winternitz of the CRM. Each has contributed more than forty years to the subject of mathematical physics, particularly to the study of algebraic methods.

This volume provides a snapshot of topics engaging theoretical physicists at the end of the twentieth century and the beginning of the twenty-first. Based on seven of the courses given at the 1999 CRM Summer School in Banff, the chapters provide timely, diverse and exciting views of their fields. The contributions include:- Supersymmetric Yang-Mills theory and integrable systems (E. D'Hoker and D.H. Phong) - Branes, black holes and anti-De Sitter space (M. J. Duff)- Turbulence (K. Gawedzki)- Bose--Einstein condensation and coherent matter waves (A. Griffin) - Deformed Virasoro and elliptic algebras (S. Odake) - Mesoscopic physics (B. Simons, A. Atland) - QCD in extreme conditions (F. Wilczek) Young physicists will find in these chapters pedagogical introductions to subjects currently active in theoretical physics, and more seasoned physicists will find a chance to share the excitement of fields outside their immediate research interests.

The Centre de recherches mathematiques (CRM) was created in 1968 by the Universite de Montreal to promote research in the mathematical sciences. It is now a national institute that hosts several groups and holds special theme years, summer schools, workshops, and a post-doctoral program. The focus of its scientific activities ranges from pure to applied mathematics and includes statistics, theoretical computer science, mathemat­ ical methods in biology and life sciences, and mathematical and theoretical physics. The CRM also promotes collaboration be­ tween mathematicians and industry. It is subsidized by the Nat­ ural Sciences and Engineering Research Council of Canada, the Fonds FCAR of the Province of Quebec, the Canadian Institute for Advanced Research, and has private endowments. Current activities, fellowships, and annual reports can be found on the CRM Web page at www. CRM. UMontreal. CA. The CRM Series in Mathematical Physics includes mono­ graphs, lecture notes, and proceedings based on research pur­ sued and on events held at the CRM. Yvan Saint-Aubin Montreal Preface This monograph is derived from a series of six lectures which were given at the Centre de recherches mathematiques (CRM) in Montreal, in March and June 2000, while the author was holder of the Aisenstadt Chair. Precis During the March 2000 meeting of the Workshop on Strings, Du­ ality, and Geometry in Montreal, Canada, three lectures were delivered on topics in fluid mechanics, while the author was holder of the Aisenstadt Chair.

Of what use is mathematics? Hasn’t everything in mathematics already been disc- ered? Thesearenaturalquestionsoftenaskedbyundergraduates. Theanswersprovided by their professors are often quite brief. Most university courses, pressed for time and rigidly structured, o?er little opportunity to present and study actual applications and real-world examples. Evenmorehigh-schoolstudentsaskthesamequestionswithmoreinsistence. Tea- ers in these schools generally work under even tighter constraints than university prof- sors. If they are able to competently respond to these questions it is probably because they received good answers from their teachers and professors. And if they do not have the answers, then whose fault is it? The genesis of this text It is impossible to introduce this text without ?rst discussing the course in which it originated. The course “Mathematics and Technology” was created at the Universit´ e de Montr´ eal and taught for the ?rst time in the winter semester of 2001. It was created after observing that most courses in the department neglect to present real applications. Sinceitscreationthecoursehasbeenopentobothundergraduatemathematicsstudents and future high-school teachers. Since no appropriate text or manual for the course we envisioned existed, we were led to write our own course notes, from which we taught. We got so caught up in writing these notes that they quickly grew to the size of a textbook, containing much more material than could possibly be taught in one semester. Despite the two of us being career mathematicians, we must admit that we both knew little or nothing about most of the applications presented in the following chapters.

The Centre de recherches matMmatiques (CRM) was created in 1968 by the Universite de Montreal to promote research in the mathematical sciences. It is now a national institute that hosts several groups and holds special theme years, summer schools, workshops, and a postdoctoral program. The focus of its scientific activities ranges from pure to applied mathematics and includes statistics, theoretical computer science, mathematical methods in biology and life sciences, and mathematical and theoretical physics. The CRM also promotes collaboration between mathematicians and industry. It is subsidized by the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR of the Province de Quebec, and the Canadian Institute for Advanced Research and has private endowments. Current ac­ tivities, fellowships, and annual reports can be found on the CRM Web page at www.CRM.UMontreal.CA. The CRM Series in Mathematical Physics includes monographs, lecture notes, and proceedings based on research pursued and events held at the Centre de recherches matMmatiques.

This book pays tribute to two pioneers in the field of Mathematical physics, Jiri Patera and Pavel Winternitz of the CRM. Each has contributed more than forty years to the subject of mathematical physics, particularly to the study of algebraic methods.