The book provides a self-contained and accessible introduction to elementary twistor theory; a technique for solving differential equations in applied mathematics and theoretical physics. It starts with an introduction to integrability of ordinary and partial differential equations. Subsequent chapters explore symmetry analysis, gauge theory, gravitational instantons, twistor transforms, and anti-self-duality equations. The three appendices cover basic differential geometry, complex manifold theory and the exterior differential system.

Preface 1. Integrability in classical mechanics 2. Soliton equations and the Inverse Scattering Transform 3. The hamiltonian formalism and the zero-curvature representation 4. Lie symmetries and reductions 5. The Lagrangian formalism and field theory 6. Gauge field theory 7. Integrability of ASDYM and twistor theory 8. Symmetry reductions and the integrable chiral model 9. Gravitational instantons 10. Anti-self-dual conformal structures Appendix A: Manifolds and Topology Appendix B: Complex analysis Appendix C: Overdetermined PDEs Index

Maciej Dunajski , University of Cambridge, UK