52 problems

Let $m$-periodic variables $\theta_i^t$ satisfy the generalized discrete Toda equations referred to as, and define $q_{j,i}^t$ from them via. Theta-to-q conjecture. The quantities…

Let $k,q\in\mathbb{N}$, $v\in\mathbb{C}$, and $\omega,\lambda\in\mathbb{R}_{>0}^2$. Let $H^{q,k}(z,v,\omega,\lambda)$ be the integral defined by … where $J$ and $\mathfrak{g}$ are…

Let $X$ be a hyperelliptic Riemann surface of genus $g\geq 2$, let $\tau$ be a period matrix associated to $X$, and let … be a fundamental system, where $\mathcal{F}$ is the set of…

Full strange duality conjecture. The morphism $SD$ is an isomorphism. This is the global formulation of the full strange-duality statement for the generalized Verlinde bundles; the…

Affine theta-function expansion conjecture. The $A_{N-1}$ theta function in the blow-up formula satisfies

Positivity conjecture. For all integers $k\ge1$, $d\ge2$, and $n\ge0$, the Laurent polynomial $\mathscr{L}_{k,n}^{d}(z)$ has positive coefficients. This conjecture is presented as…

Merca's conjecture. This theta series has non-negative coefficients. The conjecture refines the established positivity theorem for averaged truncations of Jacobi's triple product i…

Let $V$ be a cluster variety, let $V^{\vee}(\mathbb{Z}^t)$ be the tropical integral points on its dual cluster variety, let $\Theta$ be the proposed theta-function index set, and l…

For positive integers $h$ and $k$, define … where $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, and define … Let $C_1$ and $C_2$ be any fixed positive numbe…

Stronger linear lower-bound conjecture. For each prime $k>3119$,

Let $\Theta_v$ be the theta functions indexed by the relevant tropical points, and let $a_v$ be non-zero coefficients. For a linear combination $\sum a_v\Theta_v$, write…

Let $(X,E)$ be a smooth Looijenga pair with affine interior $U=X\setminus E$, let $u:\mathcal T\to X$ be the universal torsor, and set $G=\mathcal T|_U$. Let $V$ be the mirror to…

Let $(X,E)$ be a smooth Looijenga pair with affine interior $U=X\setminus E$, let $u:\mathcal T\to X$ be the universal torsor, and set $G=\mathcal T|_U$. Let $U^{\vee}$ and $V$ den…

The theta-block crank conjecture. The function $C_{k,j}(\textbf{z},\tau)$ defines a crank for the desired $(k+j)$-colored partitions and explains most of the Ramanujan-like congrue…

Schlosser–Zeng's conjecture. For every $k\in\mathbb N_0$ and $n\in\mathbb N$, the sequence

Merca's conjecture. This theta series has non-negative coefficients.

Andrews–Merca's conjecture. The coefficient of $q^m$ in this series is non-negative.

Non-negativity conjecture. This series has non-negative coefficients.

Let $\operatorname{DT}$ denote the DT transformation, and let theta functions be the theta functions associated with the cluster or scattering data under consideration. Theta-funct…

Let $q$ be a formal variable, let $m_1,m_2\geq 0$, and define the theta-product notation by $\theta(a_1,\ldots,a_r;Q)=\prod_{i=1}^r(a_i;Q)_\infty(Q/a_i;Q)_\infty$. The mod-$9$ iden…

Span conjecture. Equation $({\mathcal{M}}_F^\theta)^\Delta={\mathcal{M}}_\mathbb{Q}^{(4d/d_2)}$ holds for any totally real field $F$ of degree $d$. This predicts that diagonal rest…

Hamming-cube decay conjecture. For every fixed $c>2$, the quantity

Let $L$ be a two-dimensional lattice of area $A$, and let $E_f(L)$ denote its lattice energy for a potential $f$. A potential is completely monotone if it satisfies…

Let $d(n)$ be the integer sequence defined by … where $\theta_3(z):=\sum_{n=-\infty}^{\infty}e^{\pi i n^2z}$. Romik's conjecture. Let $p$ be an odd prime. Then: 1. If…

Let $U$ be a positive log Calabi–Yau variety with maximal boundary, and let $N_U(\mathbb{Z})$ denote its integral tropical points. A mirror positive log Calabi–Yau variety is a pos…