LfEnseignement Mathematique Monograph no.
39
(ISBN 2-940264-06-6)
softbound; 336 pages, 2003;
It was in 1899 that Henri Fehr (Geneva) and
Charles-Ange Laisant
(Paris) founded the international journal
L'Enseignement
Mathematique. The form that the celebratory
symposium took was a
historical survey of developments in mathematics
education at key
periods in the 20th century followed by a
short discussion of the
manner in which mathematics education might
meet the demands of
societies today and in the near future.
The symposium demonstrated how over the century
the emphasis
shifted from discussions of the mathematics
to be taught to an
elite, to the needs of a wider range of students
and of society.
It reminded us of the way in which two generations
had tried to
make enormous changes in the content of school
mathematics and
methods of teaching it. It gave us an opportunity
to see where
these earlier efforts had not been wholly
successful and
challenged us to determine why. With such
an understanding we
should be better equipped to tackle both
the problems that now
face us and those which will arise in the
future.
Table of contents
Introduction: Daniel Coray - Bernard R. Hodgson
L'Enseignement Mathematique: birth and stakes
Fulvia Furinghetti: Mathematical instruction
in an international
perspective: the contribution of the journal
L'Enseignement
Mathematique
Gert Schubring: L'Enseignement Mathematique
and the first
International Commission (IMUK): the emergence
of international
communication and cooperation
Gila Hanna: Journals of mathematics education,
1900-2000
Reaction: Jean-Pierre Bourguignon
Geometry
Rudolph Bkouche: La geometrie dans les premieres
annees de la
revue L'Enseignement Mathematique
Geoffrey Howson: Geometry: 1950-70
Colette Laborde: Geometrie - Periode 2000
et apres
Reaction: Nicolas Rouche
Analysis
Jean-Pierre Kahane: L'enseignement du calcul
differentiel et
integral au debut du vingtieme siecle
Man-Keung Siu: Learning and teaching of analysis
in the mid
twentieth century: a semi-personal observation
Lynn Steen: Analysis 2000: challenges and
opportunities
Reaction: Michele Artigue
Applications of mathematics: mathematics
as a service subject
Philippe Nabonnand: Les debats autour des
applications des
mathematiques dans les reformes de l'enseignement
secondaire au
debut du vingtieme siecle
Helene Gispert: Applications: les mathematiques
comme discipline
de service dans les annees 1950-1960
Mogens Niss: Applications of mathematics
'2000'
Reaction: Gerhard Wanner
Perspectives for mathematics education
Ubiratan D'Ambrosio: Stakes in mathematics
education for the
societies of today and tomorrow
Jeremy Kilpatrick: Scientific solidarity
today and tomorrow
Reaction: Hyman Bass
Description
In this text, the author considers discrete
Laplace operators
defined on lattices based on finitely ramified
self-similar sets
and their continuous analogs defined on the
self-similar sets. He
focuses on the spectral properties of these
operators. The basic
example is the lattice based on the Sierpinski
gasket. He
introduces a new renormalization map that
appears to be a
rational map defined on a smooth projective
variety. (More
precisely, this variety is isomorphic to
a product of three types
of Grassmannians: complex Grassmannians,
Lagrangian Grassmannian,
and orthogonal Grassmannians.) He relates
some characteristics of
the dynamics of its iterates with some characteristics
of the
spectrum of the operator. Specifically, he
gives an explicit
formula for the density of states in terms
of the Green current
of the map, and he relates the indeterminacy
points of the map
with the so-called Neumann-Dirichlet eigenvalues
which lead to
eigenfunctions with compact support on the
unbounded lattice.
Depending on the asymptotic degree of the
map, he can prove
drastically different spectral properties
of the operators. The
formalism is valid for the general class
of finitely ramified
self-similar sets.
Contents
Introduction
Definitions and basic results
Preliminaries
The renormalization map. Expression of the
density of states
Analysis of the psh function $G_{\vert\pi^{-1}(\mathbb{L}^G)}$
Examples
Remarks, questions and conjecture
Appendix
Bibliography
Details:
Series: Memoires de la Societe Mathematique
de France, Number: 92
Publication Year: 2003
ISBN: 2-85629-133-3
Paging: 104 pp.
Binding: Softcover
Expected publication date is August 20, 2003
Description
This book uses material from both complex
differential geometry
and complex algebraic geometry.
The first part of the book presents the fundamental
results of
Hodge theory, including a few preliminary
chapters on Kahler
geometry and sheaf cohomology. It concludes
with the development
of the notion of Hodge structure and the
study of its variation
with respect to the complex structure.
The second part is more advanced and presents
applications of
Hodge theory to complex algebraic geometry.
It starts with a
study of the topology of families of algebraic
varieties, both
from a classical and modern point of view,
and then focuses on
the applications of the theory of infinitesimal
variations of
Hodge structure. The last chapters are devoted
to the
presentation of the relations between Hodge
theory and algebraic
cycles, whose conjectural part is the famous
Bloch-Beilinson
conjecture.
This book will be useful for Ph.D. students
and for researchers
interested in modern methods of differential
and algebraic
geometry. Readers will find here a complete
didactic exposition
and an up-to-date presentation of applications
of Hodge theory to
the study of algebraic cycles.
Contents
Introduction
Partie I. Preliminaires
Fonctions holomorphes de plusieurs variables
Varietes complexes
Metriques kahleriennes
Faisceaux et cohomologie
Partie II. La decomposition de Hodge
Formes harmoniques et cohomologie
Cas des varietes kahleriennes
Structures de Hodge et polarisations
Complexes de de Rham holomorphes et suites
spectrales
Partie III. Variations de structure de Hodge
Familles de deformations
Variation de structure de Hodge
Partie IV. Cycles et classes de cycles
Classes de Hodge
Cohomologie de Deligne-Beilinson et application
d'Abel-Jacobi
Partie V. Topologie des varietes algebriques
Le theoreme de Lefschetz sur les sections
hyperplanes
Etudes pinceaux de Lefschetz
Monodromie
Suite spectrale de Leray
Partie VI. Variation de structure de Hodge
Transversalite et applications
Filtration de Hodge des hypersurfaces
Fonctions normales et invariants infinitesimaux
Travaux de Nori
Partie VII. Cycles algebriques
Groupes de Chow
Le theoreme de Mumford et ses generalisations
La conjecture de Bloch et ses generalisations
Bibliographie
Index
Details:
Series: Cours Specialises--Collection SMF,
Number: 10
Publication Year: 2003
ISBN: 2-85629-129-5
Paging: 595 pp.
Binding: Softcover