55 problems

Integrality conjecture. $X$ is integral over $\Lambda$ whenever no such subset sum vanishes.

Gromov–Witten integrality conjecture. The element

Let $\mathbb H_{u_1,u_2,\ldots,u_g}[X_1,X_2,\ldots,X_n;q,t,T]$ be the logarithmic HLV kernel, and let $\mathbb H_{u_1,u_2,\ldots,u_g,\lambda}(q,t)$ denote the coefficient of…

Atkin–Swinnerton-Dyer conjecture. One has $N=1$ if and only if the subgroup contains a congruence subgroup. This conjecture concerns the integrality of Fourier coefficients for mod…

Let $b$ be an odd positive integer, let $cmathcal M_b$ be the class of 3-manifolds under consideration, and let $cmathcal Gamma_b$ be the subring of unified invariants whose evalua…

Zudilin's conjecture. For any positive integers $N_1,N_2,\dots,N_k$,

Let $L$ be a boundary link, and let $\phi_n(L)$ denote the associated link invariant. Factorial integrality conjecture. For every boundary link $L$ and every $n\geq 1$, … The autho…

Ohtsuki's factorial integrality conjecture. For every $n\geq 1$ and every integral homology 3-sphere $M$,

Rozansky's integrality conjecture. The invariant satisfies

For $n\ge 0$, let ${\bf u}_n$ and ${\bf v}_n$ be the sequences defined by the Apéry-type hypergeometric series ${\bf S}_{n,4,2,1,1}(1)={\bf u}_n\zeta(4)-{\bf v}_n$. Define … where…

Property $T$ integrality conjecture. The representation $\varrho$ is integral.

Integral mirror-map conjecture. These are the $\nabla_{AV}$-flat coordinates on $\tilde H^2$, and every coefficient in their expansions as power series in the $\nabla_{AP}$-flat co…

Let $F_{\leq k}\mathcal{M}_{\mathbb{Z}}:=F_{\leq k}\mathcal{M}\cap\mathbb{Z}[[q]]$ be the integral quasimodular forms of filtered weight at most $k$, and let $g(\lambda)$ be the Sc…

Let $\mathcal{M}_{\mathbb{Z}}:=\mathcal{M}\cap\mathbb{Z}[[q]]$ be the integral lattice of quasimodular forms, and let $\mathcal{G}_{\mathbb{Z}}:=\sum_\lambda\mathbb{Z}\,g(\lambda)$…

Let $u_{-1}=0$ and $u_0=1$, and for $n\ge0$ define … Here $b$ and $\lambda$ are parameters arising from the three-term recurrence for the even Taylor coefficients of $f(z)f(-z)$ wh…

Conjecture 2. The degree of integrality in Conjecture 1 should be $m-1$, and the number of traces in Conjecture 2 should be $m$.

Connected Gromov–Witten integrality conjecture. The quantity

Let $\mathcal{M}_\Lambda$ be a super Macdonald polynomial and define its integral form by … where … and $\mathcal{B}(\Lambda)$ is the set of boxes in the diagram of $\Lambda$ that…

Let $R\subset\mathbb{C}$ be a finitely generated $\mathbb{Z}$-algebra, let $X$ be a smooth $R$-scheme, let $(\mathscr{E},\nabla)$ be a flat vector bundle on $X/R$, let…

Let $R\subset\mathbb{C}$ be an integral domain finitely generated over $\mathbb{Z}$, let $(X,\mathscr{F})$ be a smooth scheme with a foliation over $R$, and let $x\in X(R)$. The le…

Let $n\ge1$, let $\mathbf d=(d_1,\dots,d_n)\in(\mathbb Z_{\ge0})^n$, and set $g=|\mathbf d|+1$. For $r\ge0$, let $p_r(\mathbf d)$ denote the multiplicity of $r$ among the entries o…

Let $K_{ij}$ be the monoid defined by allowing the $(i,j)$-coordinate to be negative subject to the stated sum condition, let $\tau_{ij}$ be the associated formal series, and defin…

Let … where $\operatorname{A001850}$ and $\operatorname{A001003}$ denote the sequences identified by those OEIS entries. Zhi-Wei Sun's conjecture. The number $a(n)$ is an odd integ…

Let … where $D_0,D_1,\ldots$ are the central Delannoy numbers. Zhi-Wei Sun's conjecture. The number $a(n)$ is an integer for every positive integer $n$. The source says that Sun pr…

For integers $\ell,r\geqslant1$, let $\gamma^2(\ell,r)$ denote the asymptotic variance associated with the tree permutation determined by the parameters $\ell$ and $r$. The compute…