Appears in collection : Not Only Scalar Curvature Seminar

In this talk, I will introduce a positive mass theorem for asymptotically flat manifolds with fibers (like ALF and ALG manifolds) under an additional but necessary incompressible condition. I will also make a discussion on its connection with surgery theory as well as quasi-local mass and present some new results in these fields. This talk is based on my recent work joint with J. Chen, P. Liu and Y. Shi.