From Penrose to Melrose - Computing Scattering Amplitudes at Infinity

Explore a mathematical physics lecture that presents a novel approach to wave scattering problems by formulating them as boundary-value problems at null infinity. Learn how conformal compactification along null directions makes past and future null infinity numerically accessible boundaries, where smooth incoming radiation profiles with finite energy serve as mathematically natural data. Discover how this method addresses fundamental issues with standard frequency-domain solvers that prescribe plane-wave incident fields at obstacle surfaces and enforce approximate outgoing conditions toward infinity, which are neither truly incoming nor outgoing and fail to provide admissible smooth data at infinity. Understand the inconsistencies that arise in variable media where constant-index plane waves do not solve the background operator. Examine how this approach, grounded in Melrose's geometric scattering framework, enables solutions to induce radiation fields at null infinity while the scattering map carries incoming to outgoing data. See applications to Helmholtz equations with unbounded, variable media where radiation conditions are enforced at null infinity, eliminating artificial truncation while directly computing scattering amplitudes and far-field patterns for obstacles and media. Gain insight into how this method bridges Penrose's global geometric picture with Melrose's microlocal analysis, offering a mathematically rigorous foundation for wave scattering computations.