Contents -- Preface -- A Walk in the Noncommutative Garden A. Connes and M. Marcolli -- Contents -- 1. Introduction -- 2. Handling Noncommutative Spaces in the Wild: Basic Tools -- 3. Phase Spaces of Microscopic Systems -- 4. Noncommutative Quotients -- 5. Spaces of Leaves of Foliations -- 6. The Noncommutative Tori -- 7. Duals of Discrete Groups -- 8. Brillouin Zone and the Quantum Hall Effect -- 9. Tilings -- 10. Noncommutative Spaces from Dynamical Systems -- 11. Noncommutative Spaces from String Theory -- 12. Groupoids and the Index Theorem -- 13. Riemannian Manifolds, Conical Singularities -- 14. Cantor Sets and Fractals -- 15. Spaces of Dimension z and DimReg -- 16. Local Algebras in Supersymmetric QFT -- 17. Spacetime and the Standard Model of Elementary Particles -- 18. Isospectral Deformations -- 19. Algebraic Deformations -- 20. Quantum Groups -- 21. Spherical Manifolds -- 22. Q-lattices -- 23. Modular Hecke Algebras -- 24. Noncommutative Moduli Spaces, Shimura Varieties -- 25. The Ad`ele Class Space and the Spectral Realization -- 26. Thermodynamics of Endomotives and the Tehran Program -- References -- Renormalization of Noncommutative Quantum Field Theory H. Grosse and R. Wulkenhaar -- Contents -- 1. Introduction -- 1.1. Noncommutative geometry -- 2. Some Models for Noncommutative Space(-Time) -- 2.1. The Moyal plane -- 2.2. The noncommutative torus -- 2.3. Fuzzy spaces -- 3. Classical Field Theory on Noncommutative Spaces -- 3.1. Field theory on the noncommutative torus -- 3.2. Classical action functionals on the Moyal plane -- 4. Regularization -- 5. Renormalization -- 5.1. Quantum field theory on the noncommutative torus -- 5.2. Quantum field theories on the Moyal plane -- 5.3. The power-counting analysis of Chepelev and Roiban -- 5.4. -expanded field theories -- 5.5. Noncommutative space-time.

6. Renormalization of Noncommutative 4-theory to All Orders -- 6.1. The 4-action in the matrix base -- 6.2. Renormalization group approach to dynamical matrix models -- 6.3. Power-counting behavior of the noncommutative 4-model -- Acknowledgements -- References -- Lectures on Noncommutative Geometry M. Khalkhali -- Contents -- 1. Introduction -- 2. From C -algebras to noncommutative spaces -- 2.1. Gelfand-Naimark theorems -- 2.2. GNS, KMS, and the ow of time -- 2.3. From groups to noncommutative spaces -- 2.4. Continuous fields of C -algebras -- 2.5. Noncommutative tori -- 3. Beyond C -algebras -- 3.1. Algebras stable under holomorphic functional calculus -- 3.2. Almost commutative and Poisson algebras -- 3.3. Deformation theory -- 4. Sources of noncommutative spaces -- 4.1. Noncommutative quotients -- 4.2. Hopf algebras and quantum groups -- 5. Topological K-theory -- 5.1. The K functor -- 5.2. The higher K-functors -- 5.3. Bott periodicity theorem -- 5.4. Further results -- 5.5. Twisted K-theory -- 5.6. K-homology -- 6. Cyclic Cohomology -- 6.1. Cyclic cocycles -- 6.2. Connes' spectral sequence -- 6.3. Topological algebras -- 6.4. The deformation complex -- 6.5. Cyclic homology -- 6.6. Connes-Chern character -- 6.7. Cyclic modules -- 6.8. Hopf cyclic cohomology -- References -- Noncommutative Bundles and Instantons in Tehran G. Landi and W. D. van Suijlekom -- Contents -- 1. Introduction -- 2. Elements of Gauge Theories -- 2.1. Connections on principal and vector bundles -- 2.2. Connections on noncommutative vector bundles -- 2.3. Gauge transformations -- 2.4. Noncommutative principal bundles -- 3. Toric Noncommutative Manifolds -- 3.1. Deforming along a torus -- 3.2. Examples: planes and spheres -- 3.3. Vector bundles on M -- 3.4 Differential calculus on M -- 3.5. Local index formula on toric noncommutative manifolds.

4. The Hopf Fibration on S4 -- 4.1. Let us roll noncommutative spheres -- 4.2. The SU(2) principal fibration on S -- 4.3. The SU(2) principal fibration on S4 -- 4.4. The noncommutative instanton bundle -- 4.5. Associated modules and their properties -- 4.6. The adjoint bundle -- 4.7. Index of twisted Dirac operators -- 4.8. The noncommutative principal bundle structure -- 5. Yang-Mills Theory on Toric Manifolds -- 5.1. Yang-Mills theory on S4 -- 5.2. On a generic four-dimensional M -- 6. Let Us Twist Symmetries -- 6.1. Twisting Hopf algebras and their actions -- 6.2. The rotational symmetry of S -- 7. Instantons from Twisted Conformal Symmetries -- 7.1. The basic instanton -- 7.2. The in nitesimal conformal symmetry -- 7.3. The construction of instantons -- 7.4. Instantons on R4 -- 7.5. Moduli space of instantons -- 7.6. Dirac operator associated to the complex -- 8. Final Remarks -- Acknowledgments -- References -- Lecture Notes on Noncommutative Algebraic Geometry and Noncommutative Tori S. Mahanta -- Contents -- 0. Introduction -- 1. Some Preliminaries -- 1.1. Twisted homogeneous coordinate rings -- 1.2. A cursory glance at Grothendieck categories -- 1.3. Justification for bringing in Grothendieck categories -- 1.4. A brief discussion on construction of quotient categories -- 2. Noncommutative Projective Geometry -- 2.1. ProjR -- 2.2. Characterization of ProjR -- 2.3. Cohomology of ProjR -- 2.4. Dimension of ProjR -- 2.5. Some examples (mostly borrowed from [AZ94]) -- 3. Algebraic Aspects of noncommutative Tori -- 3.1. t-structures on Db(X) depending on -- 3.2. ASIDE on Serre duality -- Acknowledgments -- References -- Lectures on Derived and Triangulated Categories B. Noohi -- Contents -- Lecture 1: Abelian Categories -- 1. Products and Coproducts in Categories -- 2. Abelian Categories -- 3. Categories of Sheaves.

4. Abelian Category of Quasi-Coherent Sheaves on a Scheme -- 5. Morita Equivalence of Rings -- 6. Appendix: Injective and Projective Objects in Abelian Categories -- Lecture 2: Chain Complexes -- 1. Why Chain Complexes? -- 2. Chain Complexes -- 3. Constructions on Chain Complexes -- 4. Basic Properties of Cofiber Sequences -- 5. Derived Categories -- 6. Variations on the Theme of Derived Categories -- 7. Derived Functors -- Lecture 3: Triangulated Categories -- 1. Triangulated Categories -- 2. Cohomological Functors -- 3. Abelian Categories Inside Triangulated Categories -- t-Structures -- 4. Producing New Abelian Categories -- 5. Appendix I: Topological Triangulated Categories -- 6. Appendix II: Di erent Illustrations of TR4 -- Acknowledgment -- References -- Examples of Noncommutative Manifolds: Complex Tori and Spherical Manifolds J. Plazas -- Contents -- 0. Introduction -- 1. Noncommutative Tori -- 1.1. Noncommutative tori as topological spaces -- 1.2. Smooth functions on noncommutative tori -- 1.3. Morita equivalences and real multiplication -- 1.4. Complex structures on tori and holomorphic connections -- 1.5. Homogeneous coordinate rings -- 1.6. Arithmetic structures -- 2. Spherical Manifolds -- 2.1. Chern characters -- 2.2. The two sphere S2 -- 2.3. The noncommutative spheres S4 -- 2.4. Noncommutative 3-spheres -- 3. Epilogue -- References -- D-Branes in Noncommutative Field Theory R. J. Szabo -- Contents -- 1. Introduction and Background from String Theory -- 2. Euclidean D-branes -- 2.1. Moyal spaces -- 2.2. Fock modules -- 2.3. Deformation quantization -- 2.4. Derivations -- 3. Solitons on -- 3.1. Projector solitons -- 3.2. Soliton moduli spaces -- 3.3. Partial isometry solitons -- 3.4. Topological charges -- 3.5. Worldvolume construction -- 4. Gauge Theory on -- 4.1. Projective modules -- 4.2. Yang-Mills theory -- 4.3. Fluxons.

5. Toroidal D-branes -- 5.1. Solitons on the noncommutative torus -- 5.2. Gauge theory -- 5.3. Instantons -- 5.4. Instanton moduli spaces -- 5.5. Decompacti cation -- 6. D-branes in Group Manifolds -- 6.1. Symmetric D-branes -- 6.2. Untwisted D-branes -- 6.3. Example: D-branes in SU(2) -- 6.4. Twisted D-branes -- Acknowledgments -- References.