Progress in Mathematics
6 primary works • 8 total works
Book 99
Sympletic Geometry and Mathematical Physics
by P. Donato, etc., C. Duval, J. Elhaded, and G. M. Tuynman
Published 1 December 1991
This volume contains the proceedings of the conference "Colloque de Goometrie Symplectique et Physique Mathematique" which was held in Aix-en-Provence (France), June 11-15, 1990, in honor of Jean-Marie Souriau. The conference was one in the series of international meetings of the Seminaire Sud Rhodanien de Goometrie, an organization of geometers and mathematical physicists at the Universities of Avignon, Lyon, Mar- seille, and Montpellier. The scientific interests of Souriau, one of the founders of geometric quantization, range from classical mechanics (symplectic geometry) and quantization problems to general relativity and astrophysics. The themes of this conference cover "only" the first two of these four areas. The subjects treated in this volume could be classified in the follow- ing way: symplectic and Poisson geometry (Arms-Wilbour, Bloch-Ratiu, Brylinski-Kostant, Cushman-Sjamaar, Dufour, Lichnerowicz, Medina, Ouzilou), classical mechanics (Benenti, Holm-Marsden, Marle), particles and fields in physics (Garcia Perez-Munoz Masque, Gotay, Montgomery, Ne'eman-Sternberg, Sniatycki) and quantization (Blattner, Huebschmann, Karasev, Rawnsley, Roger, Rosso, Weinstein). However, these subjects are so interrelated that a classification by headings such as "pure differential geometry, applications of Lie groups, constrained systems in physics, etc.," would have produced a completely different clustering! The list of authors is not quite identical to the list of speakers at the conference. M. Karasev was invited but unable to attend; C. Itzykson and M. Vergne spoke on work which is represented here only by the title of Itzykson's talk (Surfaces triangulees et integration matricielle) and a summary of Vergne's talk.
Book 107
Loop Spaces, Characteristic Classes and Geometric Quantization
by Jean-Luc Brylinski
Published December 1992
This book examines the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kahler geometry of the space of knots, and Cheeger--Chern--Simons secondary characteristics classes. It also covers the Dirac monopole and Dirac's quantization of the electrical charge.
Book 114
Residue Currents and Bezout Identities
by Carlos A. Berenstein, etc., Roger Gay, Alekos Vidras, and Alain Yger
Published September 1993
The objective of this monograph is to present a coherent picture of the almost mysterious role that analytic methods and, in particular, multidimensional residue have recently played in obtaining effective estimates for problems in commutative algebra. Bezout identifies, i.e.,f1g1+...+fmgm=1, appear naturally in many problems, for example in commutative algebra in the Nullstellensatz, and in signal processing in the deconvolution problem. One way to solve them is by using explicit interpolation formulas in Cn, and these depend on the theory of multidimensional residues. The authors present this theory in detail, in a form developed by them, and illustrate its applications to the effective Nullstellensatz and to the Fundamental Principle for convolution equations.
Book 162
Brieskorn Anniversary
by Vladimir I. Arnold, etc., G.-M. Greuel, and J.H.M. Steenbrink
Published May 1998
In July 1996, a conference was organized by the editors of this volume at the Mathematische Forschungsinstitut Oberwolfach to honour Egbert Brieskorn on the occasion of his 60th birthday. Most of the mathematicians invited to the conference have been influenced in one way or another by Brieskorn's work in singularity theory. It was the first time that so many people from the Russian school could be present at a conference in singularity theory outside Russia. This volume contains papers on singularity theory and its applications, written by participants of the conference. In many cases, they are extended versions of the talks presented there. The diversity of subjects of the contributions reflects singularity theory's relevance to topology, analysis and geometry, combining ideas and techniques from all of these fields, as well as demonstrating the breadth of Brieskorn's own interests. This volume contains papers on singularity theory and its applications, written by participants of the conference. In many cases, they are extended versions of the talks presented there. The diversity of subjects of the contributions reflects singularity theory's relevance to topology, analysis and geometry, combining ideas and techniques from all of these fields, as well as demonstrates the breadth of Brieskorn's own interests.
Book 173
Computational Methods for Representations of Groups and Algebras
by P. Draexler and etc.
Published June 1999
This book presents material from 3 survey lectures and 14 additional invited lectures given at the Euroconference "Computational Methods for Representations of Groups and Algebras" held at Essen University in April 1997. The purpose of this meeting was to provide a survey of general theoretical and computational methods and recent advances in the representation theory of groups and algebras. The foundations of these research areas were laid in survey articles by P. DrAxler and R. NArenberg on "Classification problems in the representation theory of finite-dimensional algebras," R. A. Wilson on "Construction of finite matrix groups" and E. Green on "Noncommutative GrAbner bases, and projective resolutions." Furthermore, new applications of the computational methods in linear algebra to the revision of the classification of finite simple sporadic groups are presented. Computational tools (including high-performance computations on supercomputers) have become increasingly important for classification problems. They are also inevitable for the construction of projective resolutions of finitely generated modules over finite-dimensional algebras and the study of group cohomology and rings of invariants. A major part of this book is devoted to a survey of algorithms for computing special examples in the study of Grothendieck groups, quadratic forms and derived categories of finite-dimensional algebras. Open questions on Lie algebras, Bruhat orders, Coxeter groups and Kazhdan Lusztig polynomials are investigated with the aid of computer programs. The contents of this book provide an overview on the present state of the art. Therefore it will be very useful for graduate students and researchers in mathematics, computer science and physics.
Book 185
Analysis and Geometry on Complex Homogeneous Domains
by J. Faraut, etc., Soji Kaneyuki, Qi-keng Lu, and Adam Koranyi
Published 10 December 1999
A number of important topics in complex analysis and geometry are
covered in this excellent introductory text. Written by experts in
the subject, each chapter unfolds from the basics to the more complex.
The exposition is rapid-paced and efficient, without compromising
proofs and examples that enable the reader to grasp the essentials.
The most basic type of domain examined is the bounded symmetric
domain, originally described and classified by Cartan and Harish-
Chandra. Two of the five parts of the text deal with these domains:
one introduces the subject through the theory of semisimple Lie
algebras (Koranyi), and the other through Jordan algebras and triple
systems (Roos). Larger classes of domains and spaces are furnished by
the pseudo-Hermitian symmetric spaces and related R-spaces. These
classes are covered via a study of their geometry and a presentation
and classification of their Lie algebraic theory (Kaneyuki).
In the fourth part of the book, the heat kernels of the symmetric
spaces belonging to the classical Lie groups are determined (Lu).
Explicit computations are made for each case, giving precise results
and complementing the more abstract and general methods presented.
Also explored are recent developments in the field, in particular, the
study of complex semigroups which generalize complex tube domains and
function spaces on them (Faraut).
This volume will be useful as a graduate text for students of Lie
group theory with connections to complex analysis, or as a self-study
resource for newcomers to the field. Readers will reach the frontiers
of the subject in a considerably shorter time than with existing
texts.
covered in this excellent introductory text. Written by experts in
the subject, each chapter unfolds from the basics to the more complex.
The exposition is rapid-paced and efficient, without compromising
proofs and examples that enable the reader to grasp the essentials.
The most basic type of domain examined is the bounded symmetric
domain, originally described and classified by Cartan and Harish-
Chandra. Two of the five parts of the text deal with these domains:
one introduces the subject through the theory of semisimple Lie
algebras (Koranyi), and the other through Jordan algebras and triple
systems (Roos). Larger classes of domains and spaces are furnished by
the pseudo-Hermitian symmetric spaces and related R-spaces. These
classes are covered via a study of their geometry and a presentation
and classification of their Lie algebraic theory (Kaneyuki).
In the fourth part of the book, the heat kernels of the symmetric
spaces belonging to the classical Lie groups are determined (Lu).
Explicit computations are made for each case, giving precise results
and complementing the more abstract and general methods presented.
Also explored are recent developments in the field, in particular, the
study of complex semigroups which generalize complex tube domains and
function spaces on them (Faraut).
This volume will be useful as a graduate text for students of Lie
group theory with connections to complex analysis, or as a self-study
resource for newcomers to the field. Readers will reach the frontiers
of the subject in a considerably shorter time than with existing
texts.
v. 123
v. 131 & 132
These two volumes contain 18 invited papers in honour of the 80th birthday of mathematician Israel M. Gelfand, who is considered to have played an important role in the development of functional analysis in the latter half of the 20th century. The papers cover areas in which Gelfand has a strong interest, including geometric quantum field theory, representation theory, combinatorial structures underlying various "continuous" constructions, quantum groups and geometry. The second volume contains the more "geometric" papers.