Symmetries and Laplacians

Introduction to Harmonic Analysis, Group Representations and Applications

D. Gurarie

• Publisher: North Holland
• Year: 1992
• Print ISBN: 9780444886125
• eBook ISBN: 9780080872858
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

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...it 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.

Table of contents

• Front Cover
• Symmetries and Laplacians: Introduction to Harmonic Analysis, Group Representations and Applications
• Copyright Page
• Table of Contents
• Introduction
• Chapter 1. Basics of representation theory
• Chapter 2. Commutative Harmonic analysis
• Chapter 3. Representations of compact and finite groups
• Chapter 4. Lie groups SU(2) and SO(3)
• Chapter 5. Classical compact Lie groups and algebras
• Chapter 6. The Heisenberg group and semidirect products
• Chapter 7. Representations of SL2
• Chapter 8. Lie groups and hamiltonian mechanics
• Appendices:
• References
• List of frequently used notations
• Index