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Geometry and Nature
Edited by: Hanna Nencka, University of Madeira, Portugal, and Jean-Pierre Bourguignon, IHES, Bures-sur-Yvette, France

Contemporary Mathematics
1997; 296 pp; softcover
Volume: 203
ISBN-10: 0-8218-0607-6
ISBN-13: 978-0-8218-0607-4
List Price: US$79
Member Price: US$63.20
Order Code: CONM/203
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This volume is the outgrowth of a conference devoted to William K. Clifford entitled, "New Trends in Geometrical and Topological Methods", which was held at the University of Madeira in July and August 1995. The aim of the conference was to bring together active workers in fields linked to Clifford's work and to foster the exchange of ideas between mathematicians and theoretical physicists. Divided into 6 one-day sessions, each session was devoted to a specific aspect of Clifford's work.

This volume is an attempt to bring the Clifford legacy in a new perspective to a larger community of mathematicians and physicists. New concepts, ideas, and results stemming from Clifford's work are discussed. Each article in the book is a self-contained paper that was presented at or submitted to the conference.


Advanced graduate students, research mathematicians, mathematical physicists, astrophysicists, cosmologists, and statisticians interested in differential geometry, complex analysis, and algebra.

Table of Contents

  • H. Nencka and J.-P. Bourguignon -- W. K. Clifford as a geometer
Part I. Clifford Algebras
  • A. Trautman -- Clifford and the `square root' ideas
  • B. Iversen -- Dirac's algebra and Brauer-Wall groups
  • D. Kastler and M. Mebkhout -- Linear endomorphisms of Clifford algebras
Part II. Riemannian Surfaces
  • I. Kra -- Uniformizations of Riemann surfaces: Poincaré theta series, Riemann's theta function and theta constants
  • W. Abikoff -- Adapted metrics and Möbius transformations defined over Clifford algebras
Part III. Information Geometry
  • S.-I. Amari -- Information geometry
  • G. Burdet, H. Nencka, and M. Perrin -- An example of dynamical behaviour of the relative entropy
  • P. Combe and H. Nencka -- Information geometry and learning in formal neural networks
  • R. F. Streater -- Statistical dynamics and information geometry
Part IV. Noncommutative Geometry
  • D. Kastler, J. Madore, and T. Masson -- On finite differential calculi
  • M. Dubois-Violette -- Some aspects of noncommutative differential geometry
  • D. Kastler, J. Madore, and D. Testard -- Connections of bimodules in non-commutative geometry
  • B. Iochum, D. Kastler, and T. Schücker -- Riemannian and non-commutative geometry in physics
  • B. Iochum, D. Kastler, and T. Schücker -- Spectral model and fuzzy mass relations
Part V. Cosmology and General Relativity
  • M. Demiański -- Not so simple universe
  • B. Carter -- Extended tensorial curvature analysis for embeddings and foliations
  • T. Papakostas -- Spaces admitting a foliation by isotropic hypersurfaces
  • R. Triay -- An alternative to inflation
Part VI. Symplectic Geometry and Self-Similar Structures
  • P. Bieliavsky, M. Cahen, and S. Gutt -- A class of homogeneous symplectic manifolds
  • J. Harrison -- \(r^{th}\) order conditionally convergent series of fractal domains
Part VII. Field Theory
  • C. Duval and P. A. Horváthy -- Chern-Simons vortices
  • M. Szydłowski -- The generalized local instability criterion from the geodesic deviation equation
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