29 problems

Let $\Phi_p^{[j]}(q)^\vee$ be the vector of mock modular forms dual under $q \to 1/q$ to the linear combination of false theta functions $\Phi^{[j]}_p(q)$ associated to the…

Let $(Y,D)$ be the rational elliptic surface mirror to $\mathbb{P}^2$. For each $r\in\mathbb{Z}_{>0}$ and each integer $c_1$ with $-r<c_1\leq 0$, let $h^{GW}_{r,c_1}(\tau)$ be the…

Let $\mathfrak Y$ be a compactification with higher supersymmetry, let $\gamma=(p^0,p^A,q_A,q_0)$ be an electromagnetic charge vector, and let … After restricting to the sublattice…

Let $\boldsymbol\gamma=(\gamma_1,\ldots,\gamma_n)$ be an ordered collection of charges, let $\hat{\boldsymbol\gamma}$ denote the corresponding hatted charge data, and let…

Rational-coefficient representation conjecture. The coefficients $e_{\mathcal{T}}$ are rational numbers depending only on the topology of $\mathcal{T}$. This proposed representatio…

Let $M_3$ be a three-manifold such that … where $\tau\in\mathbb H$, $c\in\mathbb Q$, $\widetilde\vartheta$ is the Eichler integral of a weight-$3/2$ unary theta function…

Weak monotonicity conjecture. The coefficients of $R^{(k)}_1$ are weakly increasing for all $k\geq 3$; that is,

Let $\mathcal F$ be an $\mathcal N=(0,1)$ SQFT with cylindrical ends and boundary $\partial\mathcal F$, satisfying $Z_{RR}(\partial\mathcal F)=0$, and suppose $\partial\mathcal F$…

Let $M_3$ be a three-manifold whose quantum invariants have the form … where $c\in\mathbb{Q}$, $\widetilde g(\tau)$ is the Eichler integral of a weight-$\frac{3}{2}$ theta function…

Let $f_5(\boldsymbol{z};\tau)$ be the generating function defined by … where the parameters $\alpha_k$, for $k\in\{1,2,3,4\}$, satisfy…

Let $\mathcal{G}$ be the instanton generating function, expanded in terms of the completed generating functions $\widehat h_{p_i,\mu_i}$ and theta series…

For each of the 23 umbral cases, let $G^X$ be the corresponding finite group, let $m$ be its associated positive integer, and let $I^X\subset\{1,2,\dots,m-1\}$ be the specified sub…

Let $\eta_{4}(n)$ denote the fourth symmetrized moment of the odd rank. Conjecture. For every integer $n\geq 0$, the congruences … and … hold. These congruences are presented as nu…

Holomorphy and shadow conjecture. The value $E_{1/2}^*(\tau,0)$ is a holomorphic modular form of weight $1/2$; moreover, it is a linear combination of the shadows of the mock Eisen…

Let $X$ be a Niemeier root system and let $g\in G^X$. Let $\check{H}^X_g$ denote the vector formed by the components of $H^X_g$ indexed by $0<r<m$, and let…

Umbral moonshine module conjecture. There is a naturally defined module $\check{K}^X$ such that, for $g\in G^X$ and $r\in I^X$,

Let $X$ be a Niemeier root system with Coxeter number $m$, let $G^X$ be its associated group, and let $S^X_g$ be the attached cusp form for $g\in G^X$. For $g\in G^X$, let $n_g$ be…

Let … where $q=e^{2\pi i\tau}$ with $\operatorname{Im}(\tau)>0$, $H(n)$ is the Hurwitz class number, and $\lambda_{2k+1}(n)$ denotes the coefficients occurring in Cohen's formal po…

Hurwitz's conjecture. For a prime $\ell$ and $a\in\mathbb{Z}$,

Hurwitz's conjecture. If $L=3,5,6$ and $a\in\mathbb{Z}$, or if $(a,L)\equiv(\pm1,1)\pmod{7}$, then there exist constants $c_1,c_2\in\mathbb{Q}$ such that for every prime…

Hurwitz's conjecture. For every $a,L\in\mathbb{Z}$, there exist constants $c_1,c_2\in\mathbb{Q}$, given explicitly by the following cases, such that for every prime…

Let $H(\tau)$ be a mock modular form with Fourier coefficients $A_n$, and let $M_{24}$ be the simple sporadic Mathieu group. Mathieu moonshine conjecture. The Fourier coefficients…

Let $0<s<1$ and let $C_1,C_2,\ldots$ be independent events with probabilities … Let $A_k$ be the event … so that no sequence of $k$ consecutive events $C_i$ fails to occur. Andrews…

Let $\Lambda=\{2,3,4,5,7,13\}$, let $G^{(\ell)}$ be the corresponding umbral group, and let $H^{(\ell)}_g$ be its vector-valued mock modular McKay–Thompson series. For each…

Let $\Lambda=\{2,3,4,5,7,13\}$, let $g\in G^{(\ell)}$, and let $H^{(\ell)}_g=(H^{(\ell)}_{g,r})$ be the vector-valued mock modular form obtained from the graded supertrace function…