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Chapter 1.Introduction 1.Topological aspects of Hamiltonian group actions 2.Hamiltonian cobordism 3.The linearization theorem and non-compact cobordisms 4.Abstract moment maps and non-degeneracy 5.The quantum linearization theorem and its applications 6.Acknowledgements Part 1.Cobordism Chapter 2.Hamiltonian cobordism 1.Hamiltonian group actions 2.Hamiltonian geometry 3.Compact Hamiltonian cobordisms 4.Proper Hamiltonian cobordisms 5.Hamiltonian complex cobordisms Chapter 3.Abstract moment maps 1.Abstract moment maps: definitions and examples 2.Proper abstract moment maps 3.Cobordism 4.First examples of proper cobordisms 5.Cobordisms of surfaces 6.Cobordisms of linear actions Chapter 4.The linearization theorem 1.The simplest case of the linearization theorem 2.The Hamiltonian linearization theorem 3.The linearization theorem for abstract moment maps 4.Linear torus actions 5.The right-hand side of the linearization theorems 6.The Duistermaat-Heckman and Guillemin-Lerman-Sternberg formulas Chapter 5.Reduction and applications 1.(Pre-)symplectic reduction 2.Reduction for abstract moment maps 3.The Duistermaat-Heckman theorem 4.Kaihler reduction 5.The complex Delzant construction 6.Cobordism of reduced spaces 7.Jeffrey-Kirwan localization 8.Cutting Part 2.Quantization Chapter 6.Geometric quantization 1.Quantization and group actions 2.Pre-quantization 3.Pre-quantization of reduced spaces 4.Kirillov-Kostant pre-quantization 5.Polarizations, complex structures, and geometric quantization 6.Dolbeault Quantization and the Riemann-Roch formula 7.Stable complex quantization and Spinc quantization 8.Geometric quantization as a push-forward Chapter 7.The quantum version of the linearization theorem 1.The quantization of Cd 2.Partition functions 3.The character of Q(Ca) 4.A quantum version of the linearization theorem Chapter 8.Quantization commutes with reduction 1.Quantization and reduction commute 2.Quantizatioa of stable complex toric varieties 3.Linearization of [Q,R]=0 4.Straightening the symplectic and complex structures 5.Passing to holomorphic sheaf cohomology 6.Computing global sections; the lit set 7.The Cech complex 8.The higher cohomology 9.Singular [Q,R]=0 for non-symplectic Hamiltonian G-manifolds 10.Overview of the literature Part 3.Appendices Appendix A.Signs and normalization conventions 1.The representation of G on C¡Þ(M) 2.The integral weight lattice 3.Connection and curvature for principal torus bundles 4.Curvature and Chern classes 5.Equivariant curvature; integral equivariant cohomology Appendix B.Proper actions of Lie groups 1.Basic definitions 2.The slice theorem 3.Corollaries of the slice theorem 4.The Mostow-Palais embedding theorem 5.Rigidity of compact group actions Appendix C.Equivariant cohomology 1.The definition and basic properties of equivariant cohomology 2.Reduction and cohomology 3.Additivity and localization 4.Formality 5.The relation between H* G and H* T 6.Equivariant vector bundles and characteristic classes 7.The Atiyah-Bott-Berline-Vergne localization formula 8.Applications of the Atiyah-Bott-Berline-Vergne localization formula 9.Equivariant homology Appendix D.Stable complex and Spine-structures 1.Stable complex structures 2.Spine-structures 3.Spine-structures and stable complex structures Appendix E.Assignments and abstract moment maps 1.Existence of abstract moment maps 2.Exact moment maps 3.Hamiltonian moment maps 4.Abstract moment maps on linear spaces are exact 5.Formal cobordism of Hamiltonian spaces Appendix F.Assignment cohomology 1.Construction of assignment cohomology 2.Assignments with other coefficients 3.Assignment cohomology for pairs 4.Examples of calculations of assignment cohomology 5.Generalizations of assignment cohomology Appendix G.Non-degenerate abstract moment maps 1* GDefinitions and basic examples 2.Global properties of non-degenerate abstract moment maps 3.Existence of non-degenerate two-forms Appendix H.Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization 1.The Hamiltonian cobordism ring and characteristic classes 2.Characteristic numbers 3.Characteristic numbers as a full system of invariants 4.Non-degenerate cobordisms 5.Geometric quantization Appendix I.The Kawasaki Riemann-Roch formula 1.Todd classes 2.The Equivariant Riemann-Roch Theorem 3.The Kawasaki Riemann-Roch formula I: finite abelian quotients 4.The Kawasaki Riemann-Roch formula II: torus quotients Appendix J.Cobordism invariance of the index of a transversally elliptic operator by Maxim Braverman 1.The SpinC-Dirac operator and the SpinC-quantization 2.The summary of the results 3.Transversally elliptic operators and their indexes 4.Index of the operator Ba 5.The model operator 6.Proof of Theorem 1 Bibliography Index