Thesis DefenseFriday August 16th, 2019 Time: 11:00 AM Title: The Charged Penrose Inequality for Manifolds with Cylindrical Ends and Related Inequalities Speaker: Jaroslaw Jaracz, Stony Brook University Location: Math Tower 5-127 Abstract: We prove the charged Penrose inequality for manifolds with asymptotically cylindrical ends in the time-symmetric setting via a doubling argument. This comes with an appropriate rigidity statement. The techniques we developed are then used to give a quick proof of the positive mass theorem with charge for manifolds with asymptotically cylindrical ends, again in the time-symmetric setting. We conclude with some results related to Bekenstein bounds and the Hoop conjecture via a coupling of the generalized Jang equation to inverse mean curvature flow.