Geometry, Topology and Operator Algebras: Global Analysis, Invariants and Their Significance in Theoretical Physics

Preface
Contents
Contributors
Foliations and Operator Algebras
	1 Introduction
	2 Some General Notions
		2.1 Manifolds
		2.2 C^*C*-Algebras
		2.3 KK-Theory of Operator Algebras
		2.4 Lie Groupoids
		2.5 C^*C*-Algebras of Lie Groupoids
		2.6 Differential and Pseudodifferential Operators
	3 Foliations
		3.1 Foliations: Definitions and Examples
		3.2 Holonomy
		3.3 Topological Index; Index Theorem for Foliations
	References
Lectures on the Euler Characteristic of Affine Manifolds
	1 Introduction
		1.1 Affine Manifolds
		1.2 Complete Manifolds and the Developing Map
	2 Some Classical Results
		2.1 Milnor's Inequalities and the Conjecture in Dimension d=2d=2
		2.2 The Counterexamples of Smillie
		2.3 The Case of Complete Manifolds, After Sullivan and Kostant
	3 The Case of Special Affine Manifolds, After Klingler
		3.1 Local Product Structures and Affine Manifolds
		3.2 The Spectral Sequence and the Euler Class
		3.3 A Lemma About Invariant Forms
		3.4 Spectral Sequences of Sheaves
	A  Some Differential Geometry
		A.1  Connections
		A.2  Geodesics and the Exponential Map
		A.3  Curvature and the Chern–Gauss–Bonnet Theorem
	B  Spectral Sequences
	C  Sheaves
		C.1  Basic Definitions and Results
		C.2  Cohomology of Sheaves
		C.3  The Functors Rh_*,Rh_!Rh*,Rh! and the Induced Spectral Sequence
		C.4  Acyclic, Flabby, Soft, Fine and Constructible Sheaves
	References
Homogeneous Spaces of Operators
	1 Introduction
	2 Homogeneous Spaces
		2.1 The Geometry of G^+G+
		2.2 Polar Decomposition of Reflections
		2.3 A Space of Idempotents Related to a Symmetry
		2.4 The Unitary Group of a Symmetry
		2.5 \q_\rho mathcalQ ρ as a Homogeneous Space of \u_\rho mathcalU ρ
		2.6 \d mathcalD  as a Homogeneous Space of \u_\rho mathcalU ρ
		2.7 Geodesics of \d mathcalD
		2.8 \h mathcalH  as a Reductive Homogeneous Space
		2.9 The Borel Subgroup of \u_{\rho'} mathcalU ρ'
		2.10 The Covariant Derivative in \h mathcalH
	3 Metric Structure
		3.1 Finsler Metrics
		3.2 Immersions in G_{M_2}^+GM2+
		3.3 The Tangent Spaces of \k_\rho mathcalK ρ
	References
The Poincaré Half-Space of a C^*C*-Algebra
	1 Vector Bundles
		1.1 The Coefficient Bundle
	2 Kähler Structure
		2.1 The Complex Structure of \q_\rho mathcalQ ρ
		2.2 The Hilbertian Product in \q_\rho mathcalQ ρ
		2.3 The Symplectic Form \omegaω
		2.4 The Moment Map
		2.5 The Invariant Finsler Structure
		2.6 The Scalar Case
		2.7 Valuations
		2.8 The Liouville 11-Form
	3 Projective Geometry
		3.1 The Hyperbolic Part of \pa mathcalA mathbbP1
		3.2 The Invariant Metric in \pa^\rho mathcalA mathbbP1ρ
		3.3 Limit Points of Geodesics
		3.4 Operator Cross Ratio in the Hyperbolic Part of the Projective Line
		3.5 The Cross Ratio, the Logarithm and the Exponential
		3.6 An Example
	References
An Introduction to the Non-harmonic Analysis of Pseudo-Differential Operators
	1 Introduction
	2 Nonharmonic Fourier Series
		2.1 Biorthogonal Systems and Riesz Basis in Hilbert Spaces
		2.2 An Example: Nonharmonic Fourier Series
	3 Harmonic Analysis and Pseudo-Differential Operators on Spaces with Symmetries
		3.1 Sobolev Spaces and the Spectrum of the Laplacian
		3.2 Pseudo-Differential Operators
	4 Nonharmonic Analysis of Boundary Value Problems
		4.1 Basic Assumptions
		4.2 Test Functions for \mathfrak{L}mathfrakL and \mathfrak{L}^{*}mathfrakL*
		4.3 \mathfrak{L}mathfrakL-Fourier Transform and \mathfrak{L}^{*}mathfrakL*-Fourier Transform
		4.4 Plancherel Formula and Sobolev Spaces Associated to \mathfrak{L}mathfrakL and \mathfrak{L}^*mathfrakL*
		4.5 Full Symbols and Global Pseudo-Differential Operators Associated to \mathfrak{L}mathfrakL and \mathfrak{L}^*mathfrakL*
		4.6 Difference Operators and a Global Symbolic Calculus for Pseudo-Differential Operators Associated to Boundary Value Problems
	References
Mathematical Foundations of Topological Matter
	1 Introduction
	2 Quantum Phase Transitions and Level Crossing
	3 Topology of Metallic Bands and Bloch–Floquet Theorems
	4 Edge States
	5 Dirac Operator
	6 Edge States of Dirac Hamiltonians
	7 Integer Hall Effect
	8 Integer Hall Effect on Graphene
	9 Hall Conductivity and Edge States
	10 Hall Effect in Graphene Strips
		10.1 APS Boundary Conditions
		10.2 Chiral Boundary Conditions
	11 Bulk-Edge Correspondence
	References
Index