Julian Chaidez, a mathematician at the University of Southern California, presents this Joint IAS/Princeton Symplectic Geometry Seminar talk on contact topology. He defines a closed hypersurface in a contact manifold as convex if it admits a transverse contact vector field, and robustly non-convex if no smooth approximation can restore that convexity. Chaidez recounts how he resolved the longstanding open problem of whether such robustly non-convex hypersurfaces exist, using tools from partially hyperbolic dynamics, and explains the connection between non-convexity and a heterodimensional cycle structure in the hypersurface's characteristic foliation. He then shows, in joint work with his PhD student Michael Huang, that any closed hypersurface in any contact manifold can be perturbed in the C0 topology into a robustly non-convex one, a result that generalizes and simplifies his earlier work. The talk assumes familiarity with contact geometry and is aimed at researchers in the field.