Convex hypersurface theory in higher-dimensional contact homology

Contact 3-manifolds occupy a central role in low-dimensional topology partly because of their interactions with Floer-theoretic invariants.  Convex surface theory and bypasses are extremely powerful tools for analysing contact 3-manifolds, and have been successfully applied in particular to many classification problems.  After reviewing convex surface theory in dimension 3, we explain how to generalize many of the properties to higher dimensions.  Joint work with Yang Huang.