46 problems

For a finite type surface $S$, let $f\colon S\to S$ be a pseudo-Anosov homeomorphism, and let $(X,q)$ be the associated complex structure and holomorphic quadratic differential. Ca…

Polynomial growth conjecture. The constant $L(t)$ above can be a polynomial of $t$.

Polynomial growth conjecture. The constant $D(t)$ above can be a polynomial of $t$.

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…

Zero-intersection characterization. For any $F\in\mathcal{PMF}$,

Let $S$ be a closed oriented surface, let $\rho$ be a Hitchin representation, and let $e_\rho$ denote its energy functional on Teichmüller space. Nondegeneracy conjecture. The mini…

Let $\widehat{\Sigma}$ be the algebraic universal cover of a closed surface $\Sigma$, let $\ell$ be a base leaf, and let ${\sf M}(\widehat{\Sigma},\ell)$ be its mapping class group…

Let $\widehat{\Sigma}$ be the algebraic universal cover of a closed surface $\Sigma$, with Teichmüller space $\mathcal{T}(\widehat{\Sigma})$. Let ${\sf M}(\widehat{\Sigma},\ell)$ b…

Let $\Sigma$ be a fixed compact surface of genus greater than $1$, let $\mathcal{T}(\Sigma)$ be its Teichmüller space, and let $d_{\mathcal{T}(\Sigma)}$ be the Teichmüller metric.…

Hitchin–Goldman conjecture. The function $e$ has a unique minimum.

Bers density conjecture. If $M$ is singly degenerate, then $\Gamma\in\overline{B}_X$, where $X$ is the conformal boundary of $M$.

Let $B_b(\tau)$ be the universal integral means spectrum for bounded univalent functions, let $S_f$ denote the Schwarzian derivative of an extremal function $f$, and let…

Let $B_b(\tau)$ be the universal integral means spectrum for bounded univalent functions, let $N_f$ denote the Pre-Schwarzian derivative of an extremal function $f$, and let…

Let $\mathcal T_g$ be the Teichmüller space of genus $g\geq 2$ compact Riemann surfaces, with universal family $\mathcal X/\mathcal T_g$. For an isomonodromic section…

Let $S$ be a closed, oriented surface of genus at least $2$. An admissible weighted graph $(\Gamma,\theta)$ in $S$ is a graph embedded in $S$ that is the 1-skeleton of a cell decom…

For a set of minima, let $C_{\mathrm{max}}$ be the largest set of curves containing $C$ for which every nonfilling subset $C'$ of $C_{\mathrm{max}}$ gives…

For each route to chaos $C$, let $[C]$ be its boundary, namely the collection of limit dilatations of sequences drawn from the dilatation sets of the stages of $C$. Let $T_{chaos}$…

Let $S$ be the underlying surface, let $\mathcal{T}$ be its Teichmüller space, let $\mathcal{T}^\ast$ denote the corresponding space with the opposite orientation, and let $\chi$ b…

Let $\Sigma_g$ be a closed surface of genus $g$, let $\sigma\in\mathscr T_g$ be a hyperbolic structure, and let $v$ be a vector in the stretch unit sphere of $T_\sigma\mathscr T_g$…

Let $\mathcal{T}_g$ be Teichmüller space and let $T\mathcal{T}_g$ be its tangent bundle, equipped with the complex structure induced from the holomorphic tangent bundle. The Teichm…

Let $q_0\in\mathcal{Q}_g$ be generic, meaning that $q_0$ lies in the principal stratum. Let $T^H_{q_0}\mathcal{Q}_g$ denote a horizontal lift determined by a guiding frame, and let…

Uniqueness conjecture. Let $\phi\in\overline{\mathbf{S}}_q$. Then the function $f_\phi$ obtained in this way is unique, so that the IMS functional

Boundary extremal-function conjecture. For each $t\in\mathbb{R}$, $B_b(t)$ has at least one extremal function whose Pre-Schwarzian derivative lies in $\partial\mathcal{T}$.

Nonexistence conjecture. There is no asymmetric metric on Teichmüller space which is mapping class group invariant and for which left earthquakes are geodesics.

Let $S_g$ be a closed surface of genus $g$, let $E$ be the edge set of a triangulation of $S_g$, and let $\Theta:E\to[0,\pi)$ be a Delaunay angle structure. Let $P(\Theta)$ be the…