'This book describes the interesting mathematical structures behind scattering amplitudes, which have led to important simplifications for explicit computations. A must-read for people interested in this active and fast-moving field.' Juan Maldacena, Institute for Advanced Study, Princeton

• Acknowledgements; 1. Introduction; 2. Introduction to on-shell functions and diagrams; 3. Permutations and scattering amplitudes; 4. From on-shell diagrams to the Grassmannian; 5. Configurations of vectors and the positive Grassmannian; 6. Body configurations, graphs, and permutations; 7. The invariant top-form and the positroid stratification; 8. (Super) conformal and dual conformal invariance; 9. Positive diffeomorphisms and Yangian invariance; 10. The kinematical support of physical on-shell forms; 11. Homological identities among Yangian-invariants; 12. (Relatively) orienting canonical coordinate charts on positroids; 13. Classification of Yangian-invariants and their relations; 14. The Yang–Braxter relation and ABJM theories; 15. On-shell diagrams for theories with N<4 supersymmetries; 16. Dual graphs and cluster algebras; 17. On-shell representations of scattering amplitudes; 18. Outlook; References.

'This book describes the interesting mathematical structures behind scattering amplitudes, which have led to important simplifications for explicit computations. A must-read for people interested in this active and fast-moving field.' Juan Maldacena, Institute for Advanced Study, Princeton

• Acknowledgements; 1. Introduction; 2. Introduction to on-shell functions and diagrams; 3. Permutations and scattering amplitudes; 4. From on-shell diagrams to the Grassmannian; 5. Configurations of vectors and the positive Grassmannian; 6. Body configurations, graphs, and permutations; 7. The invariant top-form and the positroid stratification; 8. (Super) conformal and dual conformal invariance; 9. Positive diffeomorphisms and Yangian invariance; 10. The kinematical support of physical on-shell forms; 11. Homological identities among Yangian-invariants; 12. (Relatively) orienting canonical coordinate charts on positroids; 13. Classification of Yangian-invariants and their relations; 14. The Yang–Braxter relation and ABJM theories; 15. On-shell diagrams for theories with N<4 supersymmetries; 16. Dual graphs and cluster algebras; 17. On-shell representations of scattering amplitudes; 18. Outlook; References.

'This book describes the interesting mathematical structures behind scattering amplitudes, which have led to important simplifications for explicit computations. A must-read for people interested in this active and fast-moving field.' Juan Maldacena, Institute for Advanced Study, Princeton

• Acknowledgements; 1. Introduction; 2. Introduction to on-shell functions and diagrams; 3. Permutations and scattering amplitudes; 4. From on-shell diagrams to the Grassmannian; 5. Configurations of vectors and the positive Grassmannian; 6. Body configurations, graphs, and permutations; 7. The invariant top-form and the positroid stratification; 8. (Super) conformal and dual conformal invariance; 9. Positive diffeomorphisms and Yangian invariance; 10. The kinematical support of physical on-shell forms; 11. Homological identities among Yangian-invariants; 12. (Relatively) orienting canonical coordinate charts on positroids; 13. Classification of Yangian-invariants and their relations; 14. The Yang–Braxter relation and ABJM theories; 15. On-shell diagrams for theories with N<4 supersymmetries; 16. Dual graphs and cluster algebras; 17. On-shell representations of scattering amplitudes; 18. Outlook; References.