The HADES seminar on Tuesday, **March 29** will be at **3:30 pm** in **Room 740**.

**Speaker**: Yuchen Mao

**Abstract**: Unlike many other equations, initial data for the Einstein equation have to solve the constraint equations, which makes it an interesting problem to construct asymptotically flat localized initial data. Carlotto and Scheon proved the existence of gluing construction of such initial data supported in a cone through a functional analytic approach. We give a simpler proof by explicitly constructing a solution with conic support that achieves the optimal decay conjectured by Carlotto, and lower regularity. Another conjecture made by Carlotto is whether we can construct initial data localized in a smaller region without violating the positive mass theorem. As an application of our solution operator, we prove this is possible for the case of a degenerate sector. This is a joint work with Zhongkai Tao.