30 problems

Let $z(q)$ be the mirror map for a one-parameter deformation of an algebraic K3 surface arising from an orbifold construction whose Picard–Fuchs equation is third order. A Lian–Yau…

Let $(U,Y,{\bf 1},\Omega)$ be a nice rational enhanced VOA with local algebra $\mathcal{A}(\Omega)\cong\mathcal{A}(\Omega_{\operatorname{\textsl{Ru}}})$. Assume that $U$ has rank…

Let $\Pi^{\natural} \subset \Lambda\otimes\mathbb{R}\subset V^{f\natural}_{tw}$ be a $4$-dimensional subspace, and let $\mathsf{Top}_{\Pi^{\natural}}$ be the full subcategory of to…

Let $f=q^{-1}+\sum_{n=1}^\infty a_nq^n$ be a formal $q$-series with rational coefficients that is replicable. Norton's conjecture. The series $f$ is either a Hauptmodul for a genus…

A vertex operator algebra is strong-rational if it satisfies the paper's strong rationality conditions, and it is holomorphic if its representation category is trivial. The moonshi…

Let $g\in \textsl{Th}$, with $g$ not in the classes $\mathrm{9B}$, $\mathrm{18B}$ or $\mathrm{27ABC}$. Let $\breve{F}_{\mathrm{3C},g}(\tau)$ be the McKay–Thompson series, let…

Let $s(w)=2\pi\Omega^2S(w)$ be the rescaled inverse mirror-map coordinate, and let $g(t)$ and $h(t)$ be the power series associated with the elliptic quartic as defined above. Arou…

Let $V$ be a self-dual integral form of the monster vertex algebra, let $g$ be a Fricke element, and suppose that $g^j$ is a Fricke element for every integer $j>0$. Weaker Borcherd…

Composite-order graded Brauer character conjecture. Each of these series is a linear combination of Hauptmoduls. This predicts a uniform Hauptmodul description for the graded Braue…

Class 3C Apéry-like conjecture.

Thompson-group conjecture. For every prime $p$,

Class 3C supersingular-polynomial conjecture.

A local conformal net is holomorphic when its only irreducible representation is the vacuum representation. Consider a holomorphic local conformal net with central charge $c=24$, a…

Let $g\in O(\Lambda)$ be an element whose frame shape equals the type of an orbit lattice $N(Z)$, and let $\Lambda_g=(\Lambda^g)^\perp$. Choose a lift $\hat g$ to an automorphism o…

Let $V$ be a strongly rational, holomorphic vertex operator algebra of central charge $24$, and let $V_1$ denote its weight-one Lie algebra. Uniqueness conjecture. The Lie algebra…

Let $\mathfrak{m}$ be the Monster Lie algebra, and let $G$ be the group of q-compatible automorphisms of $\mathfrak{m}$. The Monster group $\mathbb{M}$ is a subgroup of $G$. The q-…

Let $N$ range over the 24 Niemeier lattices and let $\Phi(N)$ be the associated set of Conway or umbral twining functions. Realization conjecture. For every element…

Let $\mathcal{T}(\Pi)$ be a $K3$ nonlinear sigma model with symmetry group $G$, let $\Gamma_G$ be its co-invariant lattice, and let $\Phi(N)$ denote the corresponding set of Conway…

Let $V$ be a nice, holomorphic vertex operator algebra of central charge $24$, and let $V_1$ denote its weight-one subspace with its induced Lie algebra structure. Classification c…

Harvey–Rayhaun's Thompson Moonshine conjecture. There exists such a graded $Th$-supermodule $W$ for which, for every $g\in Th$, the series $\mathcal{T}_{[g]}(\tau)$ is a specifical…

Moonshine uniqueness conjecture. Then $V$ is isomorphic to the moonshine module vertex operator algebra.

Let $\Pi=P_X\oplus P_Z$ be the positive plane determining a supersymmetric non-linear K3 sigma model, and let $G_\Pi$ be the corresponding group of automorphisms. For $g\in G_\Pi$,…

Let $\Lambda$ be the Leech lattice, and let $V^{s\natural}$ be the canonically-twisted module considered in the paper. A supersymmetry-preserving automorphism of a supersymmetric n…

Let ${\rm Th}$ be the Thompson sporadic group, let ${\cal F}_3$ be the relevant weight one-half weakly holomorphic modular form, and let $W_m$ be the representation attached to the…

Let ${\rm Th}$ be the Thompson sporadic group, let ${\cal F}_3$ be the relevant weight one-half weakly holomorphic modular form, and let $W_m$ be the representation attached to the…