5 problems

Let ${\cal H}(2n)$ denote the class of compact convex hypersurfaces $\Sigma$ in $\mathbb{R}^{2n}$, and write $\,^{\#}{\cal T}(\Sigma)$ for the number of geometrically distinct clos…

Let $n\ge 2$ and let $\Sigma$ be a compact convex hypersurface in $\mathbb{R}^{2n}$. A geometrically distinct closed characteristic is a closed characteristic on $\Sigma$ not obtai…

Let $(Y,\xi)$ be a contact manifold. $C^1$-genericity conjecture. Convex hypersurfaces are $C^1$-generic in $(Y,\xi)$. This conjecture asserts that convexity is the generic behavio…

Let $(Y,\xi)$ be a contact manifold of dimension at least $5$, and let $\Sigma$ be a closed oriented hypersurface in $(Y,\xi)$. Approximation conjecture. The hypersurface $\Sigma$…

Let $(Y,\xi)$ be a contact manifold and let $\Sigma$ be a closed hypersurface. Write $\Sigma_\xi$ for its characteristic foliation. A robust heterodimensional cycle is a cycle of h…