Watch a General Relativity Seminar presentation by Philippe G. LeFloch from Sorbonne University and CNRS exploring advances in asymptotically Euclidian initial data sets and the Corvino-Schoen variational method. Discover the concept of a "localized seed-to-solution" map that parametrizes initial data sets satisfying Einstein constraints near reference data sets. Learn about sharp behavior solutions at infinity that can be localized in asymptotic cones, following Carlotto and Schoen's work. Examine how solutions demonstrate sharp decay estimates at harmonic and super-harmonic levels, and understand newly introduced concepts like 'asymptotic modulators' or 'correctors' to ADM invariants. The research presented draws from collaborative work with The-Cang Nguyen and Bruno Le Floch.