56 problems

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 $M_{24}$ be the Mathieu group, and let $H^{(2)}(\tau)$ be the mock-modular form arising from the decomposition of the elliptic genus of a K3 surface into irreducible characters…

For each $g\in M_{24}$, let $n_g$ be the order of $g$, let $h_g$ be the length of a minimal cycle in its induced permutation, and let $H_g$ denote the corresponding McKay–Thompson…

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 $\phi:Y\to B$ be an elliptic fibration and let $D\subset Y$ be a normal crossing divisor such that $\phi(D)$ is a divisor in $B$. Assume that the generic fiber of…

Xiong's conjecture.

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…

Let … where $(a;q)_n$ denotes the $q$-Pochhammer symbol and $J_a$ and $J_{a,b}$ denote the theta-product notation used for these functions. The fifth-order mock theta conjecture fo…

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 Seifert manifold with three singular fibres. For $b\in{\rm Spin}^c(M_3)$ and defect labels $\boldsymbol\nu\in\mathbb N^3$, define … Identify $q$-series up to multipl…

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…

Let a Borel integral be one of the integrals considered in the paper, and let the resurgent analytic continuation procedure be the decomposition procedure whose uniqueness is prove…

Dual-pair existence conjecture. There exists a unique dual pair. The supplied span ends before specifying the objects comprising the pair or the property that characterizes it, so…

Let $m$ be a positive integer and $j\in\{0,1\}$. Define … For each $m_2\in\mathbf Z^{[m]}_j$, let ${\rm ch}^{(+)}_{H(\Lambda^{[K(m),m_2]})}(\tau,z)$ be the corresponding character,…

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

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 $E_m(\mathcal{M}^{(0)};0)$ be the generalized error-function expression associated with the matrix $\mathcal{M}^{(0)}$ defined in the paper. Generalized error-function identity…

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 $D$ range over fundamental discriminants. For the values of the Klein modular invariant $j$ at real quadratic irrationalities, define … where the sum is over classes of indefin…

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…

Let $\widetilde{\mathbb{C}/\mathbb{Z}}_{2}$ denote the squashed version of the $\mathbb{Z}_2$ quotient of the complex plane, and let the $SL(2,\mathbb{R})_{k}/U(1)$ coset theory be…