Preface -- Contents -- Introduction -- I. -- II. -- III. -- IV. -- V. -- Chapter 1 Commutative geometry -- 1.1 Commutative algebra -- 1.2 Differential operators on modules and rings -- 1.3 Connections on modules and rings -- 1.4 Homology and cohomology of complexes -- 1.5 Homology and cohomology of groups and algebras -- A. Homology and cohomology of groups -- B. The Koszul complex -- C. Hochschild cohomology -- D. Chevalley-Eilenberg cohomology -- 1.6 Differential calculus over a commutative ring -- 1.7 Sheaf cohomology -- 1.8 Local-ringed spaces -- 1.9 Algebraic varieties -- Chapter 2 Classical Hamiltonian system -- 2.1 Geometry and cohomology of Poisson manifolds -- A. Symplectic manifolds -- B. Presymplectic manifolds -- C. Poisson manifolds -- D. Symplectic and Poisson reductions -- E. Koszul-Brylinski-Poisson homology -- F. Lichnerowicz-Poisson cohomology -- 2.2 Geometry and cohomology of symplectic foliations -- 2.3 Hamiltonian systems -- A. Poisson and symplectic Hamiltonian systems -- B. Presymplectic Hamiltonian systems -- C. Hamiltonian systems with symmetries -- D. Partially integrable Hamiltonian systems -- 2.4 Hamiltonian time-dependent mechanics -- A. Fibre bundles over R -- B. Lagrangian time-dependent mechanics -- C. Hamiltonian time-dependent mechanics -- D. Time-dependent completely integrable systems -- E. Degenerate systems -- F. Systems with time-dependent parameters -- G. The vertical extension of Lagrangian and Hamiltonian mechanics -- 2.5 Constrained Hamiltonian systems -- A. Constrained Hamiltonian systems -- B. Dirac constrained systems -- C. Time-dependent constraints -- D. Lagrangian constraints -- 2.6 Geometry and cohomology of Kähler manifolds -- A. Complex structure on a vector space -- B. Almost complex manifolds -- C. Hermitian manifolds -- D. Kähler manifolds -- E. Cohomology of Kähler manifolds.

F. Hyper-Kähler manifolds -- 2.7 Appendix. Poisson manifolds and groupoids -- Chapter 3 Algebraic quantization -- 3.1 GNS construction I. C*-algebras of quantum systems -- A. Involutive algebras -- B. Hilbert spaces -- C. Countably Hilbert spaces and nuclear spaces -- D. Operators in Hilbert spaces -- E. Representations of involutive algebras -- F. The GNS representation -- 3.2 GNS construction II. Locally compact groups -- 3.3 Coherent states -- 3.4 GNS construction III. Groupoids -- 3.5 Example. Algebras of infinite qubit systems -- 3.6 GNS construction IV. Unbounded operators -- 3.7 Example. Infinite canonical commutation relations -- 3.8 Automorphisms of quantum systems -- Chapter 4 Geometry of algebraic quantization -- 4.1 Banach and Hilbert manifolds -- 4.2 Dequantization -- 4.3 Berezin's quantization -- 4.4 Hilbert and C*-algebra bundles -- 4.5 Connections on Hilbert and C*-algebra bundles -- 4.6 Example. Instantwise quantization -- 4.7 Example. Berry connection -- Chapter 5 Geometric quantization -- 5.1 Leafwize geometric quantization -- A. Prequantization -- B. Polarization -- C. Quantization -- 5.2 Example. Quantum completely integrable systems -- 5.3 Quantization of time-dependent mechanics -- 5.4 Example. Non-adiabatic holonomy operators -- 5.5 Geometric quantization of constrained systems -- 5.6 Example. Quantum relativistic mechanics -- 5.7 Geometric quantization of holomorphic manifolds -- Chapter 6 Supergeometry -- 6.1 Graded tensor calculus -- 6.2 Graded differential calculus and connections -- 6.3 Geometry of graded manifolds -- 6.4 Lagrangian formalism on graded manifolds -- 6.5 Lagrangian supermechanics -- 6.6 Graded Poisson manifolds -- 6.7 Hamiltonian supermechanics -- 6.8 BRST complex of constrained systems -- 6.9 Appendix. Supermanifolds -- A. Superfunctions -- B. Supermanifolds -- C. DeWitt supermanifolds.

D. Supervector bundles -- E. Superconnections -- F. Principal superconnections -- 6.10 Appendix. Graded principal bundles -- 6.11 Appendix. The Ne'eman-Quillen superconnection -- Chapter 7 Deformation quantization -- 7.1 Gerstenhaber's deformation of algebras -- A. Formal deformation -- B. Deformation of associative algebras -- C. Relative deformation -- D. Commutative deformation -- E. Deformation of Lie algebras -- 7.2 Star-product -- 7.3 Fedosov's deformation quantization -- 7.4 Kontsevich's deformation quantization -- A. Differential graded Lie algebras -- B. L -algebras -- C. Formality theorem -- D. Kontsevich's formula -- E. Globalization of Kontsevich's deformation -- F. Deformation of algebraic varieties -- 7.5 Deformation quantization and operads -- 7.6 Appendix. Monoidal categories and operads -- Chapter 8 Non-commutative geometry -- 8.1 Modules over C*-algebras -- 8.2 Non-commutative differential calculus -- 8.3 Differential operators in non-commutative geometry -- 8.4 Connections in non-commutative geometry -- 8.5 Connes' non-commutative geometry -- 8.6 Landsman's quantization via groupoids -- 8.7 Appendix. K-Theory of Banach algebras -- 8.8 Appendix. The Morita equivalence of C*-algebras -- 8.9 Appendix. Cyclic cohomology -- 8.10 Appendix. KK-Theory -- Chapter 9 Geometry of quantum groups -- 9.1 Quantum groups -- 9.2 Differential calculus over Hopf algebras -- 9.3 Quantum principal bundles -- Chapter 10 Appendixes -- 10.1 Categories -- 10.2 Hopf algebras -- 10.3 Groupoids and Lie algebroids -- 10.4 Algebraic Morita equivalence -- 10.5 Measures on non-compact spaces -- A. Infinite-dimensional topological vector spaces -- B. Measures on locally compact spaces -- C. Haar measures -- D. Measures on Hausdorff spaces -- E. Measures on infinite-dimensional vector spaces -- 10.6 Fibre bundles I. Geometry and connections -- A. Fibre bundles.

B. Vector bundles -- C. Differential forms and multivector fields -- D. Regular distributions -- E. Differential geometry of Lie groups -- F. First order jet manifolds -- G. Connections -- H. Composite connections -- 10.7 Fibre bundles II. Higher and infinite order jets -- 10.8 Fibre bundles III. Lagrangian formalism -- 10.9 Fibre bundles IV. Hamiltonian formalism -- 10.10 Fibre bundles V. Characteristic classes -- 10.11 K-Theory of vector bundles -- 10.12 Elliptic complexes and the index theorem -- Bibliography -- Index.