Geometric Multivector Analysis

From Grassmann to Dirac
1. Auflage 2019
ISBN: 978-3-030-31413-2
Verlag: Springer International Publishing

Buch, Englisch, 465 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 721 g

Reihe: Birkhäuser Advanced Texts Basler Lehrbücher

This book presents a step-by-step guide to the basic theory of multivectors and spinors, with a focus on conveying to the reader the geometric understanding of these abstract objects. Following in the footsteps of M. Riesz and L. Ahlfors, the book also explains how Clifford algebra offers the ideal tool for studying spacetime isometries and Möbius maps in arbitrary dimensions.

The book carefully develops the basic calculus of multivector fields and differential forms, and highlights novelties in the treatment of, e.g., pullbacks and Stokes’s theorem as compared to standard literature. It touches on recent research areas in analysis and explains how the function spaces of multivector fields are split into complementary subspaces by the natural first-order differential operators, e.g., Hodge splittings and Hardy splittings. Much of the analysis is done on bounded domains in Euclidean space, with a focus on analysis at the boundary. The book also includes a derivation of new Dirac integral equations for solving Maxwell scattering problems, which hold promise for future numerical applications. The last section presents down-to-earth proofs of index theorems for Dirac operators on compact manifolds, one of the most celebrated achievements of 20-century mathematics.

The book is primarily intended for graduate and PhD students of mathematics. It is also recommended for more advanced undergraduate students, as well as researchers in mathematics interested in an introduction to geometric analysis.

Zielgruppe

Graduate

Autoren/Hrsg.

Rosén, Andreas

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Algebra Lineare und multilineare Algebra, Matrizentheorie
• Mathematik | Informatik Mathematik Mathematische Analysis Integralrechnungen- und -gleichungen

Inhaltsverzeichnis

Prelude: Linear algebra.- Exterior algebra.- Clifford algebra.- Mappings of inner product spaces.- Spinors in inner product spaces.- Interlude: Analysis.- Exterior calculus.- Hodge decompositions.- Hypercomplex analysis.- Dirac equations.- Multivector calculus on manifolds.- Two index theorems.

Über Autor(innen)

Andreas Rosén is a Professor at the Chalmers University of Technology and the University of Gothenburg, Sweden. His research mostly concerns Partial Differential Equations, and uses techniques from harmonic analysis and operator theory.