Symmetries and Laplacians

Introduction to Harmonic Analysis, Group Representations and Applications


  • D. Gurarie, Department of Mathematics and Statistics, Case Western University, Cleveland, OH, USA

Designed as an introduction to harmonic analysis and group representations,this book covers a wide range of topics rather than delving deeply into anyparticular one. In the words of H. Weyl is primarily meant forthe humble, who want to learn as new the things set forth therein, rather thanfor the proud and learned who are already familiar with the subject and merelylook for quick and exact information....

The main objective is tointroduce the reader to concepts, ideas, results and techniques that evolvearound symmetry-groups, representations and Laplacians. Morespecifically, the main interest concerns geometrical objects and structures{X}, discrete or continuous, that possess sufficiently large symmetrygroup G, such as regular graphs (Platonic solids), lattices, andsymmetric Riemannian manifolds. All such objects have a natural Laplacian&Dgr;, a linear operator on functions over X, invariant underthe group action. There are many problems associated with Laplacians onX, such as continuous or discrete-time evolutions, on X,random walks, diffusion processes, and wave-propagation. This book containssufficient material for a 1 or 2-semester course.

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Book information

  • Published: May 1992
  • Imprint: NORTH-HOLLAND
  • ISBN: 978-0-444-88612-5

Table of Contents

Basics of Representation Theory. Commutative Harmonic Analysis.Representations of Compact and Finite Groups. Lie Groups SU(2) and SO(3).Classical Compact Lie Groups and Algebras. The Heisenberg Group and SemidirectProducts. Representations of SL2. Lie Groups and HamiltonianMechanics. Appendices: Spectral Decomposition of Selfadjoint Operators.Integral Operators. A Primer on Riemannian Geometry: Geodesics, Connection,Curvature. References. List of Frequently Used Notations. Index.