46 problems

Let $\Delta_k$ denote the optimal asymptotic MSE constant within the generalized grouping (GG) family for groups of size $k$. Monotonicity conjecture. … i.e., the optimal asymptoti…

For each positive genus $g$, let $S_{n,k}^{(g)}$ be the coefficients indexed by positive integers $n$ and $k$, and let $\chi^{(g)}$ be a triangular array of constants. Let the expr…

Let $S_{n,k}^{(g)}$ denote the coefficients indexed by $n$, $k$, and genus $g$, and let the expression referred to as … , with $g=3$, gives $S_{n,k}^{(3)}$ for all $n$ and $k$. The…

Let $\operatorname{Sing}_{n,q}^{\mathrm{per}}$ and $\operatorname{Sing}_{n,q}$ be the singular sets for the quantum permanent and quantum determinant, respectively. For…

Let $r\ge2$, and let $P_r(n)$ and $\widetilde P_r(n)$ denote the two partition-counting functions defined earlier in the paper. Qin's punctual-partition generating-function conject…

Reconstruction conjecture. If

Let $p_r(n)$ denote the $r$-colored partition function, and define $L_{p_r}(d)$ and $M_{p_r,k}(d)$ analogously to the corresponding quantities for the partition and overpartition f…

Let $pl(n)$ denote the plane partition function, counting plane partitions of $n$. A perfect power is an integer of the form $a^b$ with integers $a,b>1$. The plane-partition perfec…

For each positive integer $n$, let $p(n)$ denote the partition function, and let $\{r\}$ denote the fractional part of a real number $r$. The equidistribution conjecture. For each…

For each fixed integer $d\geq 0$, define … and let $M_k(d)$ be the largest $n$ for which $p(n)$ is within $d$ of a $k$th power. Merca–Ono–Tsai's stabilization conjecture. There is…

The asymptotic conjecture for $M_k(d)$. For $k\geq 2$,

Merca–Ono–Tsai's finiteness conjecture. For fixed integers $k>1$ and $d\geq 0$, there are at most finitely many $n$ for which $\Delta_k(n)\leq d$. This finiteness assertion is used…

Fixed-order MLDE conjecture. For any given 4d ${\cal N}=2$ superconformal field theory, ${\cal I}_k(q)$ solves a modular linear differential equation of fixed order and Wronskian i…

Sun's perfect-power repulsion conjecture. The partition function $p(n)$ repels perfect powers in the following senses: (i) for every $n\ge2$ and $k\ge2$, $p(n)\neq m^k$ for all int…

Let $Z_{\bm{G}}(\beta)$ denote the partition function of the Sherrington–Kirkpatrick model, let $\varepsilon>0$, and let $\beta_{\mathsf{2nd}}$ be the second-moment threshold. A co…

Consider a family of unitary conformal field theories with fixed central charge $c$, parameterized by $\varepsilon\to 0$, and closed surfaces $\Sigma$ in the family described by lo…

Let $\mathcal{Z}_{\alpha}(\boldsymbol{\theta})$ and $\mathcal{Z}_{\alpha}^{\eta}(\boldsymbol{\theta})$ denote the zero-spin and spin-$\eta$ pure partition functions, respectively,…

Let $\mathrm{LP}(n,m)$ be the set of radial link patterns, let $\mathcal{Z}_{\alpha}(\boldsymbol{\theta})$ be the pure partition functions, let…

Initial log-concavity threshold conjecture. For $k\leq6$, the smallest such values are

Square power partition function inequality conjecture. The inequality

Let $\alpha$ and $a$ be as in the preceding construction, let $\beta\in\{0,1\}$, let $\varphi$ denote the relevant theta function, let $q$ be the nome, let $W\tau$ denote the trans…

The localization method is applied to the functions and congruences described above, with $L_{\alpha}=L_{\alpha}^{(0)}+L_{\alpha}^{(1)}$ lying in $\mathcal{M}(\mathrm{X}_0(10))$ an…

Fix parameters $\beta$ and $\gamma$, not both zero, and let $Z^{\sigma_\Lambda}_{G}(\lambda)$ denote the partition function of the 2-spin system with boundary condition…

Threshold conjecture. For $1\leq k\leq 5$, $\Delta^k p(n)$, respectively $\Delta^k\bar{p}(n)$, satisfies the third-order Turán inequality whenever $n\geq T_p(k)$, respectively…

Threshold conjecture. For $1\leq m\leq 11$ and $1\leq k\leq 5$, $\Delta^k p(n)$, respectively $\Delta^k\bar{p}(n)$, satisfies the Laguerre inequality of order $m$ whenever…