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Explore the mathematical intricacies of gravitational scattering theory in this 29-minute conference lecture that examines how classical integrability structures transform when moving from Cauchy to hyperboloidal spacetime foliations. Delve into the well-established 'KdV-Virasoro-Lax' triangular structure that characterizes Schwarzschild gravitational scattering on Cauchy foliations and discover how this framework becomes fundamentally altered in hyperboloidal contexts. Investigate the challenges posed by non-selfadjoint dynamics generators that cleanly separate bulk and boundary contributions, making traditional Lax-pair formulations problematic. Learn about an innovative approach to preserve integrability through phase space extension via source terms that act linearly on new degrees of freedom while influencing original variables. Follow the application of the Antonowicz-Fordy scheme to demonstrate how this extension recreates Cauchy-like conditions where the spectral problem exhibits isospectral multi-Hamiltonian structure. Understand how the resulting dynamics can be elegantly described through semi-direct action from bulk to boundary components, providing new insights into the mathematical foundations of hyperboloidal scattering theory.
