Adjunction and inversion of adjunction in positive characteristic

In this dissertation, we prove a characteristic p>0 analogue of the log terminal inversion of adjunction and show the equality of the two technical terms F-Different and Different conjectured by Schwede. We also prove a special version of the (relative) Kawamata-Viehweg vanishing theorem for 3-folds, normality of minimal log canonical centers, Kodaira's Canonical Bundle formula for family of rational curves, and the Adjunction Formula on Q-factorial 3-folds in characteristic p>5.