30 problems

Let a closed manifold carry a geodesic flow that is analytically integrable, meaning integrable by real-analytic first integrals in the sense used in the source, and let…

Let $M$ be a manifold, let $\sigma$ be a symplectic form on $M$, and let $H$ be the twisted geodesic Hamiltonian on $T^*M$. Magnus conjecture. For every $M$ and any symplectic form…

Let $G$ be a compact Lie group, let $H$ be a subgroup, and let $\mathfrak v$ denote the corresponding tangent-space representation used to form the invariant polynomial algebra…

Let $(T^1M,f^t)$ be the geodesic flow on a compact rank $1$ surface of nonpositive curvature. For $\alpha\geq 0$, let $L(\alpha)$ be the level set of Lyapunov exponents, and write…

Let $g$ be a Riemannian metric whose geodesic flow is robustly transitive, meaning that $g$ admits a $C^2$-neighbourhood such that every metric in that neighbourhood has topologica…

Periodic-orbit Lyapunov rigidity conjecture. If the unstable Lyapunov exponents are constant across all periodic orbits, then $M$ has constant negative sectional curvature.

Let the geodesic flow be the Hamiltonian flow of a Riemannian metric on the 2-torus. A polynomial first integral is a polynomial in the momenta that Poisson-commutes with the Hamil…

Let $X(\{\ell_n\},\{t_n\})$ be a flute surface whose cuff lengths form a non-decreasing sequence $\{\ell_n\}$, and let each twist parameter satisfy $t_n\in\{0,1/2\}$. Strong Kahn–M…

Flat-geodesic finiteness conjecture. All flat geodesics are closed, and there are only finitely many homotopy classes of such geodesics. In particular,

Entropy-volume conjecture. For all $n\geq 3$ and all $\alpha\in(0,1)$,

Calta–Kraaikamp–Schmidt conjecture. For all $n\geq 3$ and all $\alpha\in(0,1)$, the natural extension of the first pointwise expansive power of $T_{n,\alpha}$ is given by the first…

Hyperbolic closed-geodesic characterization conjecture. The characterization theorem for Finsler metrics, and for Riemannian metrics on closed surfaces, should also hold for revers…

Let $Q$ be a closed surface, let $g$ be a Riemannian metric on $Q$, and write $h_{\rm top}(g)$ for the topological entropy of its geodesic flow. Say that $h_{\rm top}(g)$ is robust…

Constant-norm geodesic conjecture. Shifted exponential solutions of the form $Be^{tC}$ are the only geodesics with constant norm when viewed as curves in the vector space…

Nonexistence conjecture. If $n$ is odd, then $\mathsf{SL}(n)$ has no bounded geodesics.

Let $(N,g)$ be a compact Riemannian manifold. A geodesic Swiss cheese model is the complement $M_{\mathcal B}=N\setminus\bigl(\bigcup_\alpha\operatorname{int}(B_\alpha)\bigr)$ of a…

Lyapunov volume conjecture.

Let $(M,g)$ be a connected compact Riemannian manifold, and define $\mathcal D(g)$ as the infimum of the defect $\delta(g^\bullet)$ over Riemannian metrics $g^\bullet$ on $SM$ of v…

Invariant-measure periodic-orbit conjecture. If $\mu(S^*H)>0$, then $\mu|_{S^*H}$ is supported on a union of periodic geodesics $\gamma$ such that $\gamma\cap S^*H$ has positive ar…

Let $(M,g)$ be a smooth, connected and closed surface of genus $d524\geq 2$ with nonpositive curvature. Let $SM$ be its unit tangent bundle, let $\Lambda$ denote the set of flat ge…

Let $T^2$ be the two-dimensional torus, and let a natural Hamiltonian system on it have a nonconstant potential. A linear, quadratic, or higher-degree polynomial integral is an int…

Let $T^2$ be the two-dimensional torus with a Riemannian metric, and consider its geodesic flow. A polynomial integral of degree $3$ in momenta is a nontrivial function on the cota…

Metric-invariance conjecture. The homology groups of these differential complexes depend only on the connected component of the space of traversally generic metrics on $M$ containi…

On the two-dimensional sphere, consider Riemannian metrics whose geodesic flows are integrable by means of an integral of degree $n>4$ in the momenta. Kozlov and Fomenko's conjectu…

Let $M$ be an open smooth $n$-manifold with a Riemannian metric $g$, and let $\alpha:N\subset M$ be a smooth compact codimension-one submanifold. An isotopy $A:N\times[0,1]\to M$ i…