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This monograph develops a semi-global scattering theory for a broad class of quasilinear wave equations in a neighbourhood of spacelike infinity, including both past and future null infinity. Scattering data are prescribed on an ingoing null cone and at past null infinity.

The authors establish weighted, optimal-in-decay energy estimates and prove the propagation of polyhomogeneous asymptotics from past to future null infinity. They further introduce an explicit algorithm for computing the coefficients in the resulting expansions and apply it to several linear and nonlinear models. A key consequence is the summability in the spherical-harmonic index ? of fixed-mode estimates previously obtained in the series "The Case Against Smooth Null Infinity."

The framework extends beyond finite-energy solutions and applies directly to systems such as the Einstein vacuum equations in harmonic gauge. A novel ansatz accommodating the stronger-than-Schwarzschildean divergence of light cones enables the treatment of slowly decaying data, thereby enlarging the regime of known stability results for Minkowski space in harmonic gauge.

This book is intended for researchers and graduate students in partial differential equations, mathematical relativity, and geometric analysis who seek a precise and versatile framework for understanding asymptotics near null and spacelike infinity.