Marcus Berg & Cecile DeWitt-Morette (University of Texas, Austin, USA)

ON CERTAIN UNITARY REPRESENTATIONS OF AN INFINITE GROUP OF TRANSFORMATIONS : Thesis by Le'on Van Hove

On April 20, 1951, Le'on Van Hove presented his thesis "Sur certaines repre'sentations unitaires d'un groupe infini de transformations " to the Universite' libre de Bruxelles (Free University of Brussels), two days before the University of Grenoble had approved the creation of L'Ecole d'e'te' de physique the'orique at Les Houches (Haute Savoie, France). The first session of the "Ecole des Houches" began on July 15, 1951, with a month-long course by Van Hove on quantum mechanics. The lecture notes for this course were written for the benefit of physicists who -- like most of their colleagues outside the US, Canada, and England at that time -- did not know quantum mechanics but wanted to learn it seriously. Van Hove's course met their expectations fully. The physics course benefitted from the mathematical expertise of the lecturer, which is also apparent in this thesis. Without his own research as scaffolding, Van Hove could not have built the short and beautiful course which provided the participants with a solid, useful foundation in modern physics.

The lecture notes are in French. If they had been in English they would have been published together with the translation of the thesis. The first three pages of the notes are reproduced at the end of this book. The set of notes was reproduced by stencils and distributed to the participants at the beginning of the course.

The translation of Le'on Van Hove's thesis was initiated in late 2000, when Bob Hermann, formerly in the Department of Mathematics at MIT, sent to Van Hove's son Michel his view on the thesis: "I would consider it as one of the most important mathematical physics papers of the past fifty years, containing the key ideas for what has become known as 'geometric quantization.'" Indeed, the thesis is interesting both to historians of science and to theoretical physicists and mathematicians exploring the relationships between quantum and classical physics, based on the Hilbert-space approach to classical mechanics.

Contents:
Introduction and Review
The Group of Transformations That Leaves Invariant the Pfaff Form ds -- Spjdqj
Unitary Representations R and R(a) of the Group G
Infinitesimal Transformations in the Representations R and R(a)
Irreducibility of the Representations R(a)
Comparison between Quantum Mechanical and Classical Operators

Readership: Mathematicians and physicists.

100pp (approx.) Pub. date: Scheduled Summer 2001
ISBN 981-02-4643-9(pbk)

edited by J H Kwah, S Hong (Pohang University of Science & Technology, South Korea),
K H Kim & F W Roush (Alabama State University, USA)

COMBINATORIAL AND COMPUTATIONAL MATHEMATICS:
PRESENT AND FUTURE Pohang, South Korea 15 - 17 February 2000

This book describes and summarizes past work in important areas of combinatorics and computation, as well as gives directions for researchers working in these areas in the 21st century. It contains primarily survey papers and presents original research by Peter Fishburn, Jim Ho Kwak, Jaeun Lee, K H Kim, F W Roush and Susan Williams. The papers deal with some of the most exciting and promising developments in the areas of coding theory in relation to number theory, lattice theory and its applications, graph theory and its applications, topological techniques in combinatorics, symbolic dynamics and mathematical social science.


Readership: Researchers, graduate students and advanced undergraduates in combinatorics and computational mathematics.

300pp (approx.) Pub. date: Scheduled Fall 2001
ISBN 981-02-4678-1

by Zhang Wei-Ping (Nankai Institute of Mathematics, P R China)

LECTURES ON CHERN-WEIL THEORY AND WITTEN DEFORMATIONS

This book is based on the notes of a graduate course which the author gave at the Nankai Institute of Mathematics. It presents some carefully chosen fundamental results and methods in global differential geometry as well as global analysis on manifolds. It will be of great value to graduate students and researchers in mathematical physics and differential geometry.

Contents:
Chern-Weil Geometric Theory for Characteristic Classes
Bott Vanishing Theorem on Foliations
Bott Localization Theorem for Characteristic Numbers on Manifolds with Circle Actions
Mathai-Quillen's Proof of the Gauss-Bonnet-Chern Theorem
Witten's Analytic Proof of the Poincare-Hopf Index Theorem as Well as Morse Inequalities
Atiyah Vanishing Theorem for the Kervaire Semi-Characteristic

Readership: Graduate students and researchers in differential geometry and mathematical physics.

110pp (approx.) Pub. date: Scheduled Fall 2001
ISBN 981-02-4685-4
ISBN 981-02-4686-2(pbk)

by Huaxin Lin (University of Oregon, USA)

AN INTRODUCTION TO THE CLASSIFICATION OF
AMENABLE C*- ALGEBRAS

The theory and applications of C*-algebras are related to fields ranging from operator theory, group representations and quantum mechanics, to non-commutative geometry and dynamical systems. By Gelfand transformation, the theory of C*-algebras is also regarded as non-commutative topology. About a decade ago, George A. Elliott initiated the program of classification of C*-algebras (up to isomorphism) by their K-theoretical data. It started with the classification of AT-algeras with real rank zero. Since then great efforts have been made to classify amenable C*-algebras, a class of C*-algebras that arises most naturally. For example, a large class of simple amenable C*-algebras is discovered to be classifiable. The application of these results to dynamical systems has been established.

This book introduces the recent development of the theory of the classification of amenable C*-algebras -- the first such attempt. The first three chapters present the basics of the theory of C*-algebras which are particularly important to the theory of the classification of amenable C*-algebras. Chapter 4 shows the classification of the so-called AT-algebras of real rank zero. The first four chapters are self-contained, and can serve as a text for a graduate course on C *-algebras. The last two chapters contain more advanced material. In particular, they deal with the classification theorem for simple AH-algebras with real rank zero, the work of Elliott and Gong. The book contains many new proofs and some original results related to the classification of amenable C*-algebras. Besides being as an introduction to the theory of the classification of amenable C*-algebras, it is a comprehensive reference for those more familiar with the subject.

Contents:
The Basics of C*-Algebras
Amenable C*-Algebras and K-Theory
AF-Algebras and Ranks of C*-Algebras
Classification of Simple AT-Algebras
C*-Algebras Extensions
Classification of Simple Amenable C*-Algebras

Readership: Researchers and graduate students in algebra.

300pp (approx.) Pub. date: Scheduled Winter 2001
ISBN 981-02-4680-3

by Tomotoda Ohtsuki (Tokyo Institute of Technology, Japan)

QUANTUM INVARIANTS
A Study of Knot, 3-Manifolds, and Their Sets

Series on Knots and Everything

This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The Chern-Simons field theory and the Wess-Zumino-Witten model are described as the physical background of the invariants.

Contents:
Knots and Polynomial Invariants
Braids and Representations of the Braid Groups
Operator Invariants of Tangles via Sliced Diagrams
Ribbon Hopf Algebras and Invariants of Links
Monodromy Representations of the Braid Groups Derived from the Knizhnik-Zamolodchikov Equation
The Kontsevich Invariant
Vassiliev Invariants
Quantum Invariants of 3-Manifolds
Perturbative Invariants of Knots and 3-Manifolds
The LMO Invariant
Finite Type Invariants of Integral Homology 3-Spheres

Readership: Researchers, lecturers and graduate students in geometry, topology and mathematical physics.

480pp (approx.) Pub. date: Scheduled Fall 2001
ISBN 981-02-4675-7