19 problems

Let $\mathbf{M}$ be the modular torus, the hyperbolic once-punctured torus described in the paper. A pair of simple closed geodesics is a pair of simple geodesics on $\mathbf{M}$ w…

A once-punctured torus is a hyperbolic surface whose simple closed geodesics have a length spectrum, with multiplicity counting how many distinct simple closed geodesics have a giv…

Bounded-deviation strengthening. There exists $C>0$ such that if an orbit $\widetilde y,\dots,\widetilde f^{n_0}(\widetilde y)$ of $\widetilde f$ crosses $V_C(\widetilde\gamma)$ fo…

Let $F$ be a Finsler metric on $S^2$, and call a closed geodesic prime if it is not an iterate of a shorter closed geodesic. Two-or-infinity conjecture for Finsler metrics. Every F…

Let $n>2$, and let $(S^n,g)$ be an $n$-dimensional Riemannian sphere. A Besicovitch-type geodesic–hypersurface inequality asserts that there is a constant $C(n)>0$, depending only…

Let $n>2$, and let $(S^n,g)$ be an $n$-dimensional Riemannian sphere. A Besicovitch-type product inequality asserts that there is a constant $C(n)>0$, depending only on the dimensi…

Power-saving error-term conjecture. The conclusion of the first theorem holds with a power-saving error term in terms of $q$.

Let $F$ be a Finsler metric on the two-sphere $S^2$, and let a closed geodesic be a periodic geodesic of $F$. Long's elliptic-geodesic conjecture. There exists at least one ellipti…

Let $M_k$ be the infimum of lengths of geodesics with self-intersection number at least $k$ among all finite-type hyperbolic surfaces. A minimal-length geodesic conjecture asserts…

Let $coperatorname{\mathbb{R}P}^2$ be a real projective plane endowed with a Riemannian metric $g$. A closed geodesic is non-contractible if its underlying loop represents a nontri…

Lipnowski–Wright's conjecture. As $g\to\infty$, on most closed hyperbolic surfaces in $\mathcal{M}_g$, most geodesics of length significantly less than $\sqrt g$ are simple and non…

Let $\Sigma$ be a closed surface, and consider the finite-dimensional parameter space of real affine structures on $\Sigma$. A real affine structure carries a closed geodesic if it…

Let $D$ be a squarefree fundamental discriminant. For $0\leq r<R$, let $A_{r,R}(w)$ be the hyperbolic annulus centered at $w$ with inner radius $r$ and outer radius $R$, and let…

Intersection-angle distribution conjecture. As $D\rightarrow\infty$, the multiset $\theta_q(D)$ tends towards the probability distribution with density

Intersection equidistribution conjecture. As $D\rightarrow\infty$, the multiset $I_q(D)$ is equidistributed on $\tilde{\ell}_q$ with respect to the hyperbolic metric.

Finiteness conjecture. The diameter of the closed geodesic norm is finite.

The infinite-dimensionality conjecture. If $-1\notin G$, then the moduli space of Besse metrics on $S^2/G$ is infinite-dimensional.

Let $M$ be a compact manifold, let $\{b_i(t)\}_{i\in \mathbb{Z}/n}$ be pursuit curves, and let $l_i(t)$ and $\theta_i(t)$ denote the associated edge lengths and angles. Angle conve…

Let $M$ be a compact Riemannian manifold. For cyclic pursuit curves $b_i(t)$, let $\beta^t$ be the piecewise geodesic loop obtained by concatenating the shortest geodesics between…