Paolo Piazza, Sapienza Universita Di Roma, Italy, Thomas Schick, Georg-August Universitat Gottingen, Germany.Vito Felice Zenobi, Instituto Nazionale di Alta Matematica, Roma, Italy.

• Chapters1. Introduction2. The relation between the structure algebra $D^*(\widetilde {M})^\Gamma $ and $\mathrm {\Psi }^0_\Gamma (\widetilde {M})$3. Rho classes4. Non-commutative de Rham homology and Chern characters5. Pseudodifferential operators and smooth subalgebras6. Mapping analytic surgery to homology7. Higher rho numbers associated to elements in $HC^* (\mathbb {C}\Gamma ,\langle x \rangle )$8. Higher rho numbers from relative cohomology $H^*(M\to B\Gamma )$9. Delocalized higher Atiyah-Patodi-Singer indices10. The space of metrics of positive scalar curvature and higher rho-invariants11. Naturality for the action by the diffeomorphism group12. The role of real K-theory13. Geometric applications of delocalized cyclic cohomology14. Geometric applications of relative cohomology15. Geometric applications: A topological approach16. Concluding remarks

Paolo Piazza, Sapienza Universita Di Roma, Italy, Thomas Schick, Georg-August Universitat Gottingen, Germany.Vito Felice Zenobi, Instituto Nazionale di Alta Matematica, Roma, Italy.

• Chapters1. Introduction2. The relation between the structure algebra $D^*(\widetilde {M})^\Gamma $ and $\mathrm {\Psi }^0_\Gamma (\widetilde {M})$3. Rho classes4. Non-commutative de Rham homology and Chern characters5. Pseudodifferential operators and smooth subalgebras6. Mapping analytic surgery to homology7. Higher rho numbers associated to elements in $HC^* (\mathbb {C}\Gamma ,\langle x \rangle )$8. Higher rho numbers from relative cohomology $H^*(M\to B\Gamma )$9. Delocalized higher Atiyah-Patodi-Singer indices10. The space of metrics of positive scalar curvature and higher rho-invariants11. Naturality for the action by the diffeomorphism group12. The role of real K-theory13. Geometric applications of delocalized cyclic cohomology14. Geometric applications of relative cohomology15. Geometric applications: A topological approach16. Concluding remarks

Paolo Piazza, Sapienza Universita Di Roma, Italy, Thomas Schick, Georg-August Universitat Gottingen, Germany.Vito Felice Zenobi, Instituto Nazionale di Alta Matematica, Roma, Italy.

• Chapters1. Introduction2. The relation between the structure algebra $D^*(\widetilde {M})^\Gamma $ and $\mathrm {\Psi }^0_\Gamma (\widetilde {M})$3. Rho classes4. Non-commutative de Rham homology and Chern characters5. Pseudodifferential operators and smooth subalgebras6. Mapping analytic surgery to homology7. Higher rho numbers associated to elements in $HC^* (\mathbb {C}\Gamma ,\langle x \rangle )$8. Higher rho numbers from relative cohomology $H^*(M\to B\Gamma )$9. Delocalized higher Atiyah-Patodi-Singer indices10. The space of metrics of positive scalar curvature and higher rho-invariants11. Naturality for the action by the diffeomorphism group12. The role of real K-theory13. Geometric applications of delocalized cyclic cohomology14. Geometric applications of relative cohomology15. Geometric applications: A topological approach16. Concluding remarks