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Contents
Part I Geometry
Lecture 1 Erlangen Programme: Preview
1.1 Make a Guess in Three Attempts
1.2 Covariance of FSCc
1.3 Invariants: Algebraic and Geometric
1.4 Joint Invariants: Orthogonality
1.5 Higher-order Joint Invariants: Focal Orthogonality
1.6 Distance, Length and Perpendicularity
1.7 The Erlangen Programme at Large
Lecture 2 Groups and Homogeneous Spaces
2.1 Groups and Transformations
2.2 Subgroups and Homogeneous Spaces
2.2.1 From a Homogeneous Space to the Isotropy Subgroup
2.2.2 From a Subgroup to the Homogeneous Space
2.3 Differentiation on Lie Groups and Lie Algebras
2.3.1 One-parameter Subgroups and Lie Algebras
2.3.2 Invariant Vector Fields and Lie Algebras
2.3.3 Commutator in Lie Algebras
2.4 Integration on Groups
Lecture 3 Homogeneous Spaces from the Group
SL
2
(ℝ)
3.1 The Affine Group and the Real Line
3.2 One-dimensional Subgroups of
SL
2
(ℝ)
3.3 Two-dimensional Homogeneous Spaces
3.3.1 From the Subgroup
K
3.3.2 From the Subgroup
N
′
3.3.3 From the Subgroup
3.3.4 Unifying All Three Cases
3.4 Elliptic, Parabolic and Hyperbolic Cases
3.5 Orbits of the Subgroup Actions
3.6 Unifying EPH Cases: The First Attempt
3.7 Isotropy Subgroups
3.7.1 Trigonometric Functions
Lecture 4 The Extended Fillmore–Springer–Cnops Construction
4.1 Invariance of Cycles
4.2 Projective Spaces of Cycles
4.3 Covariance of FSCc
4.4 Origins of FSCc
4.4.1 Projective Coordiantes and Polynomials
4.4.2 Co-Adjoint Representation
4.5 Projective Cross Ratio
Lecture 5 Indefinite Product Space of Cycles
5.1 Cycles: an Appearance and the Essence
5.2 Cycles as Vectors
5.3 Invariant Cycle Product
5.4 Zero-radius Cycles
5.5 Cauchy–Schwarz Inequality and Tangent Cycles
Lecture 6 Joint Invariants of Cycles: Orthogonality
6.1 Orthogonality of Cycles
6.2 Orthogonality Miscellanea
6.3 Ghost Cycles and Orthogonality
6.4 Actions of FSCc Matrices
6.5 Inversions and Reflections in Cycles
6.6 Higher-order Joint Invariants: Focal Orthogonality
Lecture 7 Metric Invariants in Upper Half-Planes
7.1 Distances
7.2 Lengths
7.3 Conformal Properties of Möbius Maps
7.4 Perpendicularity and Orthogonality
7.5 Infinitesimal-radius Cycles
7.6 Infinitesimal Conformality
Lecture 8 Global Geometry of Upper Half-Planes
8.1 Compactification of the Point Space
8.2 (Non)-Invariance of The Upper Half-Plane
8.3 Optics and Mechanics
8.3.1 Optics
8.3.2 Classical Mechanics
8.3.3 Quantum Mechanics
8.4 Relativity of Space-Time
Lecture 9 Invariant Metric and Geodesics
9.1 Metrics, Curves’ Lengths and Extrema
9.2 Invariant Metric
9.3 Geodesics: Additivity of Metric
9.4 Geometric Invariants
9.5 Invariant Metric and Cross-Ratio
Lecture 10 Conformal Unit Disk
10.1 Elliptic Cayley Transforms
10.2 Hyperbolic Cayley Transform
10.3 Parabolic Cayley Transforms
10.4 Cayley Transforms of Cycles
10.4.1 Cayley Transform and FSSc
10.4.2 Geodesics on the Disks
Lecture 11 Unitary Rotations
11.1 Unitary Rotations—an Algebraic Approach
11.2 Unitary Rotations—a Geometrical Viewpoint
11.3 Rebuilding Algebraic Structures from Geometry
11.3.1 Modulus and Argument
11.3.2 Rotation as Multiplication
11.4 Invariant Linear Algebra
11.4.1 Tropical form
11.4.2 Exotic form
11.5 Linearisation of the Exotic Form
11.6 Conformality and Geodesics
11.6.1 Retrospective: Parabolic Conformality
11.6.2 Perspective: Parabolic Geodesics
Lecture 12 Cycles Cross Ratio: an Invitation
12.1 Preliminaries: Projective space of cycles
12.2 Fractional linear transformations and the invariant product
12.3 Cycles cross ratio
12.4 Discussion and generalisations
Lecture 13 Extension of Lie Geometry: Ensembles and their Implementation
13.1 Introduction
13.2 Möbius–Lie Geometry and the
cycle
Library
13.2.1 Möbius–Lie geometry and FSC construction
13.2.2 Clifford algebras, FLT transformations, and Cycles
13.3 Ensembles of Interrelated Cycles and the
figure
Library
13.3.1 Connecting quadrics and cycles
13.3.2 Figures as families of cycles—functional approach
13.4 Mathematical Usage of the Library
13.5 To Do List
Lecture 14 Poincaré Extension of Möbius Transformations
14.1 Introduction
14.2 Geometric construction
14.3 Möbius transformations and Cycles
14.4 Extending cycles
14.5 Homogeneous spaces of cycles
14.6 Triples of intervals
14.7 Geometrisation of cycles
14.8 Concluding remarks
Lecture 15 Conformal Parametrisation of Loxodromes
15.1 Introduction
15.2 Preliminaries: Fractional Linear Transformations and Cycles
15.3 Fractional Linear Transformations and Loxodromes
15.4 Three-cycle Parametrisation of Loxodromes
15.5 Applications of Three-Cycle Parametrisation
15.6 Discussion and Open Questions
Lecture 16 Continued Fractions, Möbius Transformations and Cycles
16.1 Introduction
16.2 Continued Fractions
16.3 Möbius Transformations and Cycles
16.4 Continued Fractions and Cycles
16.5 Multi-dimensional Möbius maps and cycles
16.6 Continued fractions from Clifford algebras and horocycles
Part II Harmonic Analysis
Lecture 1 Representation Theory
1.1 Representations
1.2 Decomposition of Representations
1.3 Schur’s Lemma
1.4 Induced Representations
Lecture 2 Wavelets on Groups
2.1 Wavelet Transform on Groups
2.2 Square Integrable Representations
2.3 Fundamentals of Wavelets on Homogeneous Spaces
Lecture 3 Hypercomplex Linear Representations
3.1 Hypercomplex Characters
3.2 Induced Representations
3.3 Similarity and Correspondence: Ladder Operators
Lecture 4 Covariant Transform
4.1 Extending Wavelet Transform
4.2 Examples of Covariant Transform
4.3 Symbolic Calculi
4.4 Contravariant Transform
4.5 Composing the Co- and Contravariant Transforms
4.5.1 Real and Complex Technique in Harmonic Analysis
Lecture 5 Analytic Functions
5.1 Induced Covariant Transform
5.2 Induced Wavelet Transform and Cauchy Integral
5.3 The Cauchy-Riemann (Dirac) and Laplace Operators
5.4 The Taylor Expansion
5.5 Wavelet Transform in the Unit Disk and Other Domains
Lecture 6 Affine Group: the Real and Complex Techniques in Harmonic Analysis
6.1 Introduction
6.2 Two approaches to harmonic analysis
6.3 Affine group and its representations
6.4 Covariant transform
6.5 The contravariant transform
6.6 Intertwining properties of covariant transforms
6.7 Composing the covariant and the contravariant transforms
6.8 Transported norms
Part III Functional Calculi and Spectra
Lecture 1 Covariant and Contravariant Calculi
1.1 Functional Calculus as an Algebraic Homomorphism
1.2 Intertwining Group Actions on Functions and Operators
1.3 Jet Bundles and Prolongations
1.4 Spectrum and Spectral Mapping Theorem
1.5 Functional Model and Spectral Distance
1.6 Covariant Pencils of Operators
Part IV The Heisenberg Group and Physics
Lecture 1 Preview: Quantum and Classical Mechanics
1.1 Axioms of Mechanics
1.2 “Algebra” of Observables
1.3 Non-Essential Noncommutativity
1.4 Quantum Mechanics from the Heisenberg Group
1.5 Classical Noncommutativity
1.6 Summary
Lecture 2 The Heisenberg Group
2.1 The Symplectic Form and the Heisenberg group
2.2 Lie algebra of the Heisenberg group
2.3 Automorphisms of the Heisenberg group
2.4 Subgroups of ℍ
n
and Homogeneous Spaces
Lecture 3 Representations of the Heisenberg Group
3.1 Left Regular Representations and Its Subrepresentations
3.2 Induced Representations on Homogeneous Spaces
3.3 Co-adjoint Representation and Method of Orbits
3.4 Stone–von Neumann Theorem
3.5 Shale–Weil Representation
Lecture 4 Harmonic Oscillator and Ladder Operators
4.1 p-Mechanic Formalism
4.1.1 Convolutions (Observables) on ℍ
n
and Commutator
4.1.2 States and Probability
4.2 Elliptic characters and Quantum Dynamics
4.2.1 Fock–Segal–Bargmann and Schrödinger Representations
4.2.2 Commutator and the Heisenberg Equation
4.2.3 Quantum Probabilities
4.3 Ladder Operators and Harmonic Oscillator
4.3.1 Ladder Operators from the Heisenberg Group
4.3.2 Symplectic Ladder Operators
4.4 Hyperbolic Quantum Mechanics
4.4.1 Hyperbolic Representations of the Heisenberg Group
4.4.2 Hyperbolic Dynamics
4.4.3 Hyperbolic Probabilities
4.4.4 Ladder Operators for the Hyperbolic Subgroup
4.4.5 Double Ladder Operators
4.5 Parabolic (Classical) Representations on the Phase Space
4.5.1 Classical Non-Commutative Representations
4.5.2 Hamilton Equation
4.5.3 Classical probabilities
4.5.4 Ladder Operator for the Nilpotent Subgroup
4.5.5 Similarity and Correspondence
Lecture 5 Wavelet Transform, Uncertainty Relation and Analyticity
5.1 Induced Wavelet (Covariant) Transform
5.2 The Uncertainty Relation
5.3 The Gaussian
5.4 Right Shifts and Analyticity
5.5 Uncertainty and Analyticity
5.6 Hardy Space on the Real Line
5.7 Contravariant Transform and Relative Convolutions
5.8 Norm Estimations of Relative Convolutions
Appendix A Open Problems
A.1 Geometry
A.2 Analytic Functions
A.3 Functional Calculus
A.4 Quantum Mechanics
Appendix B Supplementary Material
B.1 Dual and Double Numbers
B.2 Classical Properties of Conic Sections
B.3 Comparison with Yaglom’s Book
B.4 Other Approaches and Results
B.5 FSCc with Clifford Algebras
Appendix C How to Use the Software
C.1 Viewing Colour Graphics
C.2 Installation of CAS
C.2.1 Booting from the DVD Disk
C.2.2 Running a Linux Emulator
C.2.3 Recompiling the CAS on Your OS
C.3 Using the CAS and Computer Exercises
C.3.1 Warming Up
C.3.2 Drawing Cycles
C.3.3 Library figure
C.3.4 Further Usage
C.4 Library for Cycles
C.5 Predefined Objects at Initialisation
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Last modified: October 28, 2024.
