Symplectic Geometry Seminar

A hypersurface in a contact manifold is convex if there is a contact vector field that is transverse to it. Convex surfaces have long been a fundamental tool in 3-dimensional contact topology. In higher dimensions, convex hypersurfaces remained poorly understood until recent groundbreaking approximation results of Honda-Huang. 

In this talk, I will explain recent works aimed at further refining the work of Honda-Huang via the introduction of techniques from smooth and hyperbolic dynamics. I will discuss recent solo work proving that convex hypersurfaces are not smoothly generic in higher dimensions, in contrast to the 3-dimensional case by Giroux. I will also touch on ongoing joint work with Yasha Eliashberg and Dishant Pancholi to study convex contactomorphisms, give a dynamical characterization of convexity and to improve the regularity of Honda-Huang's approximation result to the C1 topology.