35 problems

Let $X$ be a smooth projective complex surface, and let $X^{[n]}$ be the Hilbert scheme of $n$ points on $X$. For line bundles $L_1,\ldots,L_N$ on $X$, let ${\rm ch}_k(L_i^{[n]})$…

Let $E$ be an elliptic curve, let $\overline{M}_{g,n}$ be the moduli space of stable curves, and let $\mathcal C_g(\gamma_1,\ldots,\gamma_n)$ denote the class defined in the paper…

Let $\Delta$ be a root system of ADE type, let $\Delta_+$ be its set of positive roots, and let $r=|\Delta_+|$. For integers $k_\alpha\geq 1$, write … for the associated generalize…

For fixed $g$, $h$, and $m_2$, let $\overline{\mathcal{H}}_{g/h,d,m_2}$ denote the $d$-elliptic Hurwitz cycle with $m_2$ marked pairs of points having equal image. Lian's quasi-mod…

Götsche's conjecture. Certain quasi-modular forms are the generating functions for the number of curves in the fixed linear system $|C|$.

Let $K$ be the base field, let $\Omega$ be the domain of the Drinfeld modular forms, and let $t(z)$ be the usual parameter at infinity for $z\in\Omega$. The functions $E$, $g$, and…

Let ${\mathcal{H}}(2)$ be the stratum of abelian differentials on genus-$2$ surfaces with one double zero. For each invariant of the primitive square-tiled surfaces in this stratum…

For the moduli spaces $M_G(A)$ considered in the paper, form the generating series in $q$ of their stringy Hodge polynomials and specialize $(x,y)$ to obtain the stringy signature…

Bounded-degree and quasi-modularity conjecture. For any genus, the tropical refined invariants have bounded degree, and the coefficients of these polynomials are given by quasi-mod…

Let $\mathfrak{bm}_0$ be the space of all $\Phi\in\mathbb{Q}\langle\mathcal{B}\rangle$ satisfying $(\Phi\mid b_0)=0$, $(\Phi\mid b_k)=0$ for $k=2,4,6$, primitiveness for the balanc…

Let $g$ and $d$ be positive integers, and consider the locus of curves $[C]\in \overline{\mathcal{M}}_g$ admitting an admissible cover $f:C\to E$ of degree $d$ to some elliptic cur…

Let $X$ be a nonsingular surface, let $\beta$ be a curve class, let $g$ be a genus, let $g_{\max}$ denote the maximal genus associated with $\beta$, and let $p(x)$ and $\chi$ be th…

Let $S$ be a K3 surface and let $\beta\in H_2(S,\mathbb Z)$ be primitive. For cohomology classes $\gamma_0,\gamma_1,\ldots\in H^*(S)$, define the partition function…

Let $n\geq 1$, let $k_{[i,j]}\geq 1$ and $s_{[i,j]}\geq 0$ be integers for $1\leq i\leq j\leq n$, and let $\zeta_{\mathfrak{sl}(n+1),q}[\cdots]$ denote the associated type-A bi-bra…

Let $\mathfrak{g}=\mathfrak{sl}(n+1)$ with $n\geq 1$, let $k\geq 3$, and let $\mathcal{QM}=\mathbb{C}[E_2(\tau),E_4(\tau),E_6(\tau)]$ and…

Let $Z_2$ and $Z_3$ be expanded in the normalized Kähler parameters $Q_2$ and $Q_3$, and let $P^{(0)}$ be the genus-zero quantity appearing in the expansion. Write $Z_{2,3}$ for ei…

Let $d_B$ and $d_E$ denote the base and fiber degrees, respectively, and let $\mathcal{F}^{(n,g)}$ be the base-degree-zero amplitude of genus data satisfying $n+g\geq 2$. Let $q$ b…

Let $f_{l,w}$ be the normalised extremal quasi-modular form of weight $w$ and depth at most $l$, and let … be the set of weights for which its Fourier coefficients are integral. In…

Let $B$ be a smooth variety, let $X$ be the elliptic fibration under consideration, and let $\beta\in H_2(B,\mathbb Z)$ and $g$ satisfy the setup preceding the claim, including Ass…

The modularity conjecture. The relative potential has the form

Let $E_1$ and $E_2$ be nonsingular elliptic curves, let $\mathsf{A}=E_1\times E_2$ be fibered over $E_1$, and let ${\mathcal A}_{g,n}(\gamma_1,\ldots,\gamma_n)$ denote the reduced…

The quantities $c_{y}$, $c_{yz2}$, $c_{yz4}$, and $c_{z}$ arise in the generalized SYZ map for the elliptic orbifold under consideration. A function is quasi-modular if it belongs…

Calabi–Yau 1-fold quasi-modularity conjecture. The ancestor Gromov–Witten correlation functions defined above are quasi-modular forms. This is the one-dimensional formulation of th…

Let $\mathcal{X}$ be a compact Calabi–Yau orbifold, and let its Gromov–Witten invariants be assembled into generating functions in the relevant degree and marking variables. Folklo…

Let $\operatorname{\mathsf{qMZV}}$ denote the algebra of $q$-multiple zeta values, containing the ring of quasi-modular forms $\widetilde{M}_{\mathbb{Q}}=\mathbb{Q}[G_2,G_4,G_6]$,…