40 problems

Let $Y$ be a closed manifold and let $\lambda$ be a contact form on $Y$. Its Reeb vector field is the vector field $R$ determined by $\lambda(R)=1$ and $d\lambda(R,\cdot)=0$; a Ree…

Large minimal Chern number conjecture. If $c_B$ is sufficiently large, for example in the range suggested by the source, then a dynamically convex contact form on $M$ carries infin…

Universal-cover conjecture. If $V$ carries a hypertight contact form, then

Prequantization and minimal-orbit conjecture. Every such contact manifold is a prequantization of an orbifold, and the minimal number of simple closed Reeb orbits among its contact…

Finite-orbit prequantization conjecture. Any such contact form is necessarily non-degenerate and lacunary, meaning that the Conley–Zehnder indices of all closed orbits have the sam…

Let $W\subset\mathbb R^{2n}$ be a star-shaped domain and let $\operatorname{SH}^t(W;\mathbb F)$ denote its filtered symplectic homology with coefficients in a field $\mathbb F$. Th…

Let a Reeb pseudo-rotation be a Reeb flow with finitely many prime closed Reeb orbits. A non-degenerate Reeb flow is one for which all closed Reeb orbits are non-degenerate, and it…

Let $W\subset{\mathbb R}^{2n}$ be a star-shaped domain, and consider the Reeb flow on its boundary $\partial W$. A closed Reeb orbit is prime if it is not an iterate of another clo…

Let $\Sigma$ be a star-shaped hypersurface in $\mathbb{R}^{2n}$, equipped with the contact form induced by the standard symplectic structure. A simple Reeb orbit is a Reeb orbit th…

Local symplectic-homology equivariance conjecture. For every $k\in\mathbb{N}$, at least with rational coefficients, one has

Iterated Reeb SDM conjecture. The presence of an iterated Reeb SDM implies that the Reeb flow has infinitely many simple Reeb orbits.

Irrational ellipticity conjecture. All closed Reeb orbits in such a pseudo-rotation are irrationally elliptic.

Multiplicity conjecture. The number of closed Reeb orbits is at least $n$ for star-shaped hypersurfaces in $\mathbb{R}^{2n}$, and at least

Hopf-link conjecture. Every Reeb flow on $(S^3,xi_0)$ possesses a pair of closed orbits transversely isotopic to a Hopf link.

Let $Y$ be a closed manifold equipped with a contact form $\lambda$ whose Reeb vector field is $R_\lambda$. A periodic Reeb orbit is a map $\gamma:S^1\to Y$ for which there is some…

Local homology concentration conjecture. The second condition in the theorem, namely the existence of an integer $d$ satisfying

Let $\Sigma^{2n-1}$ bound a strictly convex domain in $\mathbb{R}^{2n}$. Elliptic orbit conjecture. There exists at least one elliptic closed Reeb orbit on $\Sigma$. The source des…

Let $\Sigma^{2n-1}$ bound a convex domain in $\mathbb{R}^{2n}$, with $n\geq 2$, and let $\mathcal{T}(\Sigma)$ denote the set of prime closed Reeb orbits on $\Sigma$. A closed Reeb…

Let $\Sigma^{2n-1}$ bound a star-shaped domain in $\mathbb{R}^{2n}$. Higher-dimensional two-or-infinity conjecture. For each such $\Sigma$, … The dimension-four case is solved by t…

Let $\Sigma^{2n-1}$ be the boundary of a star-shaped domain in $\mathbb{R}^{2n}$, and let $\mathcal{T}(\Sigma)$ denote the set of prime closed Reeb orbits on $\Sigma$. Multiplicity…

Two-or-infinitely-many periodic Reeb orbits conjecture. The associated Reeb flow has either exactly two or infinitely many periodic orbits.

Reeb 1-curve conjecture. The manifold $M$ has a Reeb 1-curve.

Sphere orbit-count conjecture. Every contact form on $S^{2n+1}$ has at least $n+1$ simple periodic Reeb orbits.

Let $(Y,\xi)$ be a closed contact manifold with contact form $\alpha$. Write $\mathfrak{gap}_\sigma(Y,\alpha)$ for the spectral gap associated with a class $\sigma\in CH(Y,\xi)$. P…

Let $(Y,\xi)$ be a closed contact manifold with a periodic contact form $\alpha$ of period $T$. Let $CH(Y,\xi)$ denote contact homology, $\mathfrak{s}_\sigma(Y,\alpha)$ the spectra…