Note: Intro -- Preface -- Contents -- Contributors -- 1 Prelude: A General Overview -- 1.1 Introduction -- 1.2 Poisson Geometry and Classical Dynamics -- 1.3 Geometric and Deformation Quantization -- 1.3.1 Geometric Quantization -- 1.3.2 Deformation Quantization -- 1.4 Non-commutative Geometry and Quantum Groups -- 1.5 Quantum Fields -- References -- 2 Deformation Quantization and Group Actions -- 2.1 What Do We Mean by Quantization? -- 2.1.1 Classical Mechanics -- 2.1.2 Quantum Mechanics -- 2.2 Deformation Quantization -- 2.2.1 Definition and Examples of Star Products -- 2.2.2 Existence of Star Products -- 2.2.3 The Notion of States -- 2.3 Fedosov's Star Products on a Symplectic Manifold -- 2.3.1 The Weyl Bundle -- 2.3.2 Flat Connections on the Weyl Bundle -- 2.3.3 Fedosov's Star Products -- 2.4 Classification of Poisson Deformations and Star Products -- 2.4.1 Hochschild Cohomology -- 2.4.2 Equivalence of Star Products -- 2.5 Star Products on Poisson Manifolds and Formality -- 2.5.1 The DGLA of Polydifferential Operators -- 2.5.2 The DGLA of Multivector Fields -- 2.5.3 Linfty-Algebras, Linfty-Morphism and Formality -- 2.5.4 Formality for mathbbRd -- 2.6 Group Actions in Deformation Quantization -- 2.6.1 In a Classical Setting -- 2.6.2 In the Deformation Quantization Setting -- 2.6.3 Classification of Invariant Star Products -- 2.6.4 Invariance of Fedosov's Star Product -- 2.7 Reduction in Deformation Quantization -- 2.7.1 The Classical Koszul Resolution -- 2.7.2 The Quantized Koszul Complex -- 2.7.3 The Reduced Star Product -- 2.8 Some Remarks About Convergence -- References -- 3 Principal Fiber Bundles in Non-commutative Geometry -- 3.1 Introduction -- 3.2 Review of Principal Fiber Bundles -- 3.2.1 Fiber Bundles -- 3.2.2 Pull-Backs -- 3.2.3 Principal Fiber Bundles -- 3.2.4 Functoriality and Classification. , 3.3 Basic Ideas of Non-commutative Geometry -- 3.3.1 Two Classical Dualities Between Spaces and Algebras -- 3.3.2 Non-commutative Algebras -- 3.3.3 Extending Basic Operations to Non-commutative Spaces -- 3.4 From Groups to Hopf Algebras -- 3.4.1 Algebraic Groups -- 3.4.2 Bialgebras -- 3.4.3 Hopf Algebras -- 3.4.4 Examples of Hopf Algebras from Finite Groups -- 3.4.5 The Heyneman--Sweedler Sigma Notation -- 3.5 Quantum Groups Associated with SL2(mathbbC) -- 3.5.1 The Quantum Coordinate Algebra of SL2(mathbbC) -- 3.5.2 A Quotient of SLq(2) -- 3.5.3 The Quantum Enveloping Algebra of mathfraksl(2) -- 3.5.4 A Finite-Dimensional Quotient of Uq mathfraksl(2) -- 3.6 Group Actions in Non-commutative Geometry -- 3.6.1 Comodule-Algebras -- 3.6.2 Group-Graded Algebras -- 3.6.3 Algebras with Group Actions -- 3.6.4 The Quantum Plane and Its SLq(2)-coaction -- 3.6.5 Quantum Homogeneous Spaces -- 3.7 Hopf Galois Extensions -- 3.7.1 Definition and Examples -- 3.7.2 The Classification Problem -- 3.7.3 The Set GalH(mathbbC) May Be Non-trivial -- 3.7.4 Push-Forward of Central Hopf Galois Extensions -- 3.7.5 Universal Central Hopf Galois Extensions -- 3.8 Flat Deformations of Hopf Algebras -- 3.8.1 A Universal Construction by Takeuchi -- 3.8.2 The Generic Hopf Galois Extension Associated with a Hopf Algebra -- 3.8.3 Multiparametric Deformations of Uq mathfraksl(2) and of mathfrakud -- References -- 4 An Introduction to Nichols Algebras -- 4.1 Preliminaries -- 4.1.1 Conventions -- 4.1.2 Groups -- 4.1.3 The Tensor Algebra -- 4.1.4 The Symmetric Algebra -- 4.1.5 Coalgebras and Hopf Algebras -- 4.1.6 The Tensor Coalgebra -- 4.1.7 Gelfand--Kirillov Dimension -- 4.2 Braided Tensor Categories -- 4.2.1 Braided Vector Spaces -- 4.2.2 Braided Tensor Categories -- 4.3 Nichols Algebras -- 4.3.1 Hopf Algebras in Braided Tensor Categories -- 4.3.2 Bosonization. , 4.3.3 Nichols Algebras: Definitions -- 4.3.4 Nichols Algebras: Techniques -- 4.4 Classes of Nichols Algebras -- 4.4.1 Symmetries and Hecke Type -- 4.4.2 Diagonal Type -- 4.4.3 Triangular Type -- 4.4.4 Rack Type, Infinite Dimension -- 4.4.5 Rack Type, Finite Dimension -- References -- 5 Quantum Field Theory in Curved Space-Time -- 5.1 Introduction -- 5.2 Quantum Field Theory in Minkowski Space-Time -- 5.3 Quantum Field Theory in Curved Space-Time -- 5.3.1 Quantization -- 5.3.2 Representations -- 5.4 Cosmology -- 5.4.1 General Remarks -- 5.4.2 Inflation -- References -- 6 An Introduction to Pure Spinor Superstring Theory -- 6.1 Introduction -- 6.2 Particle and Superparticle -- 6.2.1 Brink--Schwarz Superparticle -- 6.2.2 Pure Spinor Superparticle -- 6.3 Pure Spinor Superstring -- 6.3.1 General Issues -- 6.3.2 Some Symmetries -- 6.3.3 OPEs and Anomaly -- 6.3.4 Massless States -- 6.3.5 Tree-Level Scattering Amplitudes -- 6.4 Appendix -- 6.4.1 Cartan and Chevalley Definitions -- 6.4.2 Pure Spinor Parameterization -- References -- 7 Introduction to Elliptic Fibrations -- 7.1 Introduction -- 7.2 Elliptic Curves over mathbbC -- 7.2.1 Modular Group and Complex Tori -- 7.2.2 The Weierstrass Equation -- 7.2.3 Moduli Space of Smooth Elliptic Curves -- 7.3 Elliptic Fibrations -- 7.3.1 Weierstrass Models for Elliptic Curves -- 7.3.2 Preparation for the Relative Case -- 7.3.3 Weierstrass Models for Elliptic Fibrations -- 7.3.4 The j-Invariant -- 7.3.5 Deligne's Formulaire -- 7.4 Kodaira--Néron Classification of Singular Fibers -- 7.4.1 Monodromy -- 7.4.2 Fiber Type -- 7.4.3 Tate's Algorithm -- 7.5 Miranda Models -- 7.5.1 Fibers at the Collisions of a Miranda Model -- 7.5.2 Szydlo's Generalization of Miranda Models -- References -- 8 Batalin--Vilkovisky Formalism as a Theory of Integration for Polyvectors -- 8.1 Motivations and Program -- 8.2 BV Integral. , 8.2.1 BV Laplacian -- 8.2.2 Definition of the Integral -- 8.2.3 Advantages of the BV Formalism -- 8.3 Gauge Fixing -- 8.3.1 Gauge Fixing in BV Formalism -- 8.3.2 Schouten--Nijenhuis Bracket -- 8.4 Master Equations -- 8.4.1 BV Laplacians in Coordinate -- 8.4.2 Quantum Master Equations -- 8.4.3 Observables in BV Formalism -- 8.5 Conclusion -- References -- 9 Split Chern--Simons Theory in the BV-BFV Formalism -- 9.1 Introduction -- 9.2 Overview of the BV and BV-BFV Formalisms -- 9.2.1 Perturbative Quantisation of Lagrangian Field Theories -- 9.2.2 Perturbative Quantisation of Gauge Theories -- 9.2.3 On Manifolds with Boundary -- 9.2.4 The BV-BFV Formalism -- 9.2.5 The Quantum BV-BFV Formalism -- 9.2.6 Abelian BF Theory in the Quantum BV-BFV Formalism -- 9.3 Chern--Simons Theory as a BF-like Theory -- 9.3.1 Split BV Chern--Simons Theory -- 9.3.2 Perturbative Expansion -- 9.3.3 Feynman Graphs and Rules -- 9.4 Split Chern--Simons Theory on the Solid Torus -- 9.4.1 Effective Action on the Solid Torus -- 9.4.2 mQME -- 9.4.3 Change of Data -- 9.5 Conclusions and Outlook -- References -- 10 Weighted Direct Product of Spectral Triples -- 10.1 Introduction and Motivation -- 10.2 Weighted Direct Product of Spectral Triples -- 10.3 Example of Weighted Direct Product with Toeplitz Operators -- References -- Index.