14 problems

Alternating-group monodromy conjecture. All of the $[1,1]$-origamis in Theorem have monodromy group isomorphic to $\operatorname{Alt}_{2g-1}$.

Let $n$ be an integer greater than $8$, and consider primitive $n$-squared origamis in $\mathcal{H}(4)$, with $\mathcal{H}^{\mathrm{odd}}(4)$ denoting the odd component and…

Quantum modularity conjecture. The $q$-bracket $\langle f\rangle_q$ is a quantum modular form of weight $k$.

Let $\mathcal{H}:=\mathcal{H}_{g}(k_{1},\dots,k_{s})$ be a stratum, and let $(\mathcal{G}_{n})_{n}$ be a family of $\operatorname{SL}(2,\mathbb{Z})$-orbit graphs of primitive $n$-s…

Let $n$ be a positive integer, and let $\mathcal{A}_n(1,1)$ and $\mathcal{S}_n(1,1)$ denote the cardinalities of the orbits of primitive $n$-squared origamis in $\mathcal{H}(1,1)$…

Let $(\mathcal{G}_{n})_{n}$ be the family of $\operatorname{SL}(2,\mathbb{Z})$-orbit graphs of primitive $n$-squared origamis in the stratum $\mathcal{H}(2)$. McMullen's expander c…

Let $(0,b,1,e)$ be a reduced prototype with $D\equiv 1\pmod 4$, and let the target prototype be $(0,(D-1)/4,1,\pm1)$, with the sign determined by the spin invariant when…

A primitive origami is a square-tiled surface whose covering of the unit square torus does not factor through a nontrivial isogeny. The associated $\operatorname{SL}(2,\mathbb{Z})$…

Persistence conjecture. This behaviour persists in all genera.

SL(2,Z)-orbit conjecture. For odd genus $g\geq 5$, the origamis constructed by Aougab-Menasco-Nieland lie in exactly two $\operatorname{SL}(2,\mathbb{Z})$-orbits inside the odd com…

Jack-polynomial interpolation conjecture. The generating series $\mathcal{R}_\mu(t,\mathbf{p};b)$ interpolates between the quasimodular forms given by the generating functions for…

Let $\mathcal{O}$ be an origami of degree $d$ and genus $2$, let $\Gamma\subseteq\operatorname{SL}_2(\mathbb{Z})$ be its Kontsevich–Zorich monodromy, and let $\mathcal{B}_d$ denote…

Let $N_g$ denote the number of distinct $SL(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2g-2)$ with $2g-1$ squares, such that each of their horizontal and vertical cylinder d…

Let $n$ be a positive integer. Using the generating set $\{P,R\}$ for $\operatorname{SL}(2,\mathbb{Z})$, where … let $\overline{\mathcal{G}}_{n}$, $\overline{\mathcal{G}}_{n}^{A}$,…