Lectures on Clifford (Geometric) Algebras and Applications

Lecture 1: Introduction to Clifford Algebras.- 1.1 Introduction.- 1.2 Clifford algebra of the Euclidean plane.- 1.3 Quaternions.- 1.4 Clifford algebra of the Euclidean space ?3.- 1.5 The electorn spin in a magnetic field.- 1.6 From column spinors to spinor operators.- 1.7 In 4D: Clifford algebra C?4 of ?4.- 1.8 Clifford algebra of Minkowski spacetime.- 1.9 The exterior algebra and contractions.- 1.10 The Grassmann-Cayley algebra and shuffle products.- 1.11 Alternative definitions of the Clifford algebra.- 1.12 References.- Lecture 2: Mathematical Structure of Clifford Algebras.- 2.1 Clifford algebras.- 2.2 Conjugation.- 2.3 References.- Lecture 3: Clifford Analysis.- 3.1 Introduction.- 3.2 Foundations of Clifford analysis.- 3.3 Other types of Clifford holomorphic functions.- 3.4 The equation Dkf=0.- 3.5 Conformal groups and Clifford analysis.- 3.6 Conformally flat spin manifolds.- 3.7 Boundary behavior and Hardy spaces.- 3.8 More on Clifford analysis on the sphere.- 3.9 The Fourier transform and Clifford analysis.- 3.10 Complex Clifford analysis.- 3.11 References.- Lecture 4: Applications of Clifford Algebras in Physics.- 4.1 Introduction.- 4.2 Three Clifford algebras.- 4.3 Paravectors and relativity.- 4.4 Eigenspinors.- 4.5 Maxwell’s equation.- 4.6 Quantum theory.- 4.7 Conclusions.- 4.8 References.- Lecture 5: Clifford Algebras in Engineering.- 5.1 Introduction.- 5.2 Quaternions.- 5.3 Biquaternions.- 5.4 Points, lines, and planes.- 5.5 Computer vision example.- 5.6 Robot kinematics.- 5.7 Concluding remarks.- 5.8 References.- Lecture 6: Clifford Bundles and Clifford Algebras.- 6.1 Spin geometry.- 6.2 Conformal structure.- 6.3 Tractor constructions.- 6.4 References.- 211.