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In its traditional form, Clifford analysis provides the function theory for solutions of the Dirac equation. From the beginning, however, the theory was used and applied to problems in other fields of mathematics, numerical analysis, and mathematical physics. recently, the theory has enlarged its scope considerably by incorporating geometrical methods from global analysis on manifolds and methods from representation theory. New, interesting branches of the theory are based on conformally invariant, first-order systems other than the Dirac equation, or systems that are invariant with respect to a group other than the conformal group. This book represents an up-to-date review of Clifford analysis in its present form, its applications, and directions for future research.
Readership: Mathematicians and theoretical physicists interested in Clifford analysis itself, or in its applications to other fields.
List of contents
Riemann-Hilbert Problems in Clifford Analysis.- The Continuous Wavelet Transform in Clifford Analysis.- Automated Symbolic Computation in Spin Geometry.- Monogenic Forms of the Polynomial Type.- On Beltrami Equations in Clifford Analysis and Its Quasi-Conformal Solutions.- Parallel transport of Algebraic Spinors on Clifford Manifolds.- A Correspondence of Hyperholomorphic and Monogenic Functions in ?4.- On Weighted Spaces of Monogenic Quaternion-valued Functions.- Plane waves in Premetric Electrodynamics.- Contact Symplectic Geometry in Parabolic Invariant Theory and Symplectic Dirac Operator.- Communication via Holomorphic Green Functions.- Hyper-holomorphic Cells and Riemann-Hubert Problems.- Nilpotent Lie Groups in Clifford Analysis and Mathematical Physics.- A Quaternionic Generalization of the Riccati Differential Equation.- Invariant Operators for Quaternionic Structures.- On Generalized Clifford Algebras - a Survey of Applications.- Is the Visual Cortex a "Clifford Algebra Quantum Computer"?.- The Cln-Valued Robin Boundary Value Problem on Lipschitz Domains in ?n.- Quaternionic Analysis in ?3 versus Its Hyperbolic Modification.- Contributions to a Geometric Function Theory in Higher Dimensions by Clifford Analysis Methods: Monogenic Functions and M-conformal Mappings.- The Dirac Type Tensor Equation in Riemannian Space.- Harmonic Analysis of Dirac Operators on Lipschitz Domains.- Weight Problems for Higher Dimensional Singular Integrals via Clifford Analysis.- The Conformal Laplacian on Spheres and Hyperbolas via Clifford Analysis.- Combinatorics and Clifford Analysis.- The Spherical X-Ray Transform of Texture Goniometry.- Quaternionic Complexes in Clifford Analysis.- Clifford Analysis on the Level of Abstract Vector Variables.- Clifford Analysis as aStudy of Invariant Operators.- Teodorescu Type Transforms in Applications.- Combinatorics and Clifford Analysis.- Double Covers of Pseudo-orthogonal Groups.- Higher Spin Fields on Smooth Domains.- Bergman Type Spaces on the Unit Disk.- List of Participants.
Summary
In its traditional form, Clifford analysis provides the function theory for solutions of the Dirac equation. From the beginning, however, the theory was used and applied to problems in other fields of mathematics, numerical analysis, and mathematical physics. recently, the theory has enlarged its scope considerably by incorporating geometrical methods from global analysis on manifolds and methods from representation theory. New, interesting branches of the theory are based on conformally invariant, first-order systems other than the Dirac equation, or systems that are invariant with respect to a group other than the conformal group. This book represents an up-to-date review of Clifford analysis in its present form, its applications, and directions for future research.
Readership: Mathematicians and theoretical physicists interested in Clifford analysis itself, or in its applications to other fields.
