19 problems

Let $\Sigma$ be a star-shaped hypersurface in $\mathbb{R}^{2n}$, equipped with the standard contact form, and call its closed Reeb orbits closed characteristics. A closed character…

Let $E$ be a compact odd-dimensional manifold with a free circle action, let $B=E/S^1$, and let $\operatorname{pr}:E\to B$ be the associated principal circle bundle. Let $\eta$ be…

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

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 $(M,\omega)$ be a symplectic manifold, and let $\alpha_0,\alpha_\infty\in H_*(M)$ be homology classes. A hypersurface $S\subset M$ is separating the homology classes $\alpha_0$…

Let $\Sigma$ be a strictly convex hypersurface in $\mathbb{R}^{2n}$, equipped with the standard contact form, and call its closed Reeb orbits closed characteristics. A closed chara…

Let $\Sigma$ be a star-shaped hypersurface in $\mathbb{R}^{2n}$, equipped with the standard contact form, and call its closed Reeb orbits closed characteristics. A closed character…

Let $\Sigma\in\mathcal{H}(2n)$ be a compact convex hypersurface, and let $^{\#}\widetilde{\mathcal{J}}(\Sigma)$ denote the number of geometrically distinct closed characteristics o…

Let $\Sigma\in\mathcal{H}(2n)$ be a compact convex hypersurface, and let $^{\#}\widetilde{\mathcal{J}}(\Sigma)$ denote the number of geometrically distinct closed characteristics o…

Let $\Sigma\in\mathcal{H}(2n)$ be a compact convex hypersurface in $\mathbb{R}^{2n}$, where $n\geq2$. A closed characteristic is elliptic when all Floquet multipliers lie on the un…

Star-shaped closed-characteristics conjecture. The same lower bound should hold for every strictly star-shaped hypersurface:

Let $n\geq 1$ and let $\Sigma\in{\cal H}_{st}(2n)$ be a compact strictly star-shaped hypersurface in ${\bf R}^{2n}$ enclosing the origin. Write $\,^{\#}{\cal T}(\Sigma)$ for the nu…

For an integer $n\geq2$, let ${\cal H}_{st}(2n)$ be the class of compact strictly star-shaped hypersurfaces in ${\bf R}^{2n}$, and let $\,^{\#}{\cal T}(\Sigma)$ denote the number o…

Star-shaped closed-characteristic count conjecture. The same lower bound should hold for every $\Sigma\in{\cal H}_{st}(2n)$:

Let $\Sigma$ be a compact star-shaped hypersurface in $\mathbb{R}^{2n}$, and suppose that it has only finitely many geometrically distinct closed characteristics. A closed characte…

Let $\Sigma$ be a compact convex hypersurface in $\mathbb{R}^{2n}$. A closed characteristic on $\Sigma$ is a periodic characteristic trajectory on $\Sigma$. Multiplicity conjecture…

Let $\Sigma$ be a compact convex hypersurface in $\mathbb{R}^{2n}$. A closed characteristic is an embedded loop in $\Sigma$ tangent to the characteristic foliation…

Let $n\geq 2$, and let $\Sigma$ be a $C^2$ compact convex hypersurface in ${\bf R}^{2n}$. Write $\tilde{\mathcal{J}}(\Sigma)$ for the set of geometrically distinct closed character…