22 problems

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

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 $R\subset\mathbb{C}$ 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 $K_{ij}$ 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 $X$ be a holomorphic symplectic $4$-fold, let $\beta\beta\in H_2(X,\mathbb{Z})$ be a primitive curve class, and let $n_{0,\beta}(\gamma_1,\ldots,\gamma_n)$,…

Integrality conjecture. The ordered multi-degree contributions satisfy

Revised framed LMOV conjecture. For every partition $\mu$,

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…

Let $X$ be a smooth projective variety, and let $L$ be a rigid local system on $X$. A local system is integral when the traces of its monodromy matrices are algebraic integers. The…

Marino's conjecture. The reformulated quantities

Let the series … has … The conjecture is suggested by known integrality results for the coefficients $a_{m,n}$ in the expansion of the sigma function and analogous results for sigm…

Let $L/K$ be a Galois extension of number fields with Galois group $G$, let $r<0$, and let $\mu_{1-r}(L):=(\mathbb{Q}/\mathbb{Z})(1-r)^{G_L}$, where $G_L$ is the absolute Galois gr…

Let $\mathcal{L}$ be a link with $L$ components, and let $Z_{CS}^{SO}(\mathcal{L},q,t)$ be its orthogonal Chern–Simons partition function. Write … and define … where $\mu(k)$ is th…

Product-expansion conjecture. There are integers $\alpha^{(i)}_{m_0,m_1}$ and $\beta^{(i)}_{m_0,m_1}$ such that

Integrality conjecture. The Taylor series of $q_0,q_1,\dots,q_N$ in $x_0,x_1,\dots,x_N$ have integer coefficients, and the inverse Taylor series likewise have integer coefficients.

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

Let $N\geq 4$ and consider the $\mathfrak{sl}(N)$-link homology defined in the preceding sections over the rational theory, with its relations and homomorphisms suggested to admit…