107 problems

Let $p\colon C\to B$ be a smooth proper family of curves, let $L$ be a line bundle on $C$, and let $\pi_n\colon C^{[n]}\to B$ be the relative Hilbert scheme of length-$n$ subscheme…

The conjecture asserts that the Gross–Joyce–Tanaka wall-crossing formula holds for equivariant Donaldson–Thomas $4$ invariants of Calabi–Yau $4$ categories equipped with framing fu…

For a $3$-dimensional $\mathcal{N}=4$ gauge theory with Higgs branch $\mathcal{M}_H$, dual Coulomb branch $\mathcal{M}_C$, and quantized Coulomb-branch algebra $\mathcal{A}_C$, the…

Let $\mathcal{M}$ be an $SU(2)$ Hitchin moduli space with prescribed singularities, and let $\hat{\mathcal{M}}$ be its SYZ mirror, an $PGL(2)$ Hitchin moduli space with prescribed…

Let ${\mathsf r}\geq 2$, and let $\check{\mathsf{QM}}^{\mathsf a}_{{\mathsf d},0}$ and $\hat{\mathsf{QM}}^{\mathsf a}_{{\mathsf d},0}$ denote the zero-degree quasimap invariants in…

The converse to the surplus condition criterion. One has

Pandharipande–Thomas rationality conjecture. The partition function $\operatorname{Z_{PT}}\left(M;q\mid\prod_{i=1}^r\tau_{k_i}(\gamma_i)\right)_\beta$ is the Laurent expansion of a…

Klemm–Pandharipande integrality conjecture.

Let $\vec{\pi}_a$ be $n$-tuples of plane partitions for the four legs, and let $\vec{v}=(v_1,\ldots,v_n)$ and $\vec{w}=(w_1,\ldots,w_n)$ be the flavor fugacities of the $n$ D8 and…

Let $g\geq 2$ be the genus, let ${\mathsf{r}}=2$ be the rank, and let ${\mathsf{d}}$ be the degree. Define the generating series $\check{\mathsf{QM}}^i_{\mathsf{d}}(q)$ and…

Let $X$ be a nonsingular projective 3-fold, let $\beta\in H_2(X,\mathbb{Z})$, and let … be the stable pairs descendent series, where $\gamma_i\in H^*(X)$ and $k_i\geq 0$. Stable pa…

LMOV conjecture. For every $\mu\in\mathcal{P}_+$, the function $F_{\mu}^{(X,\mathcal{D})}$ has the integral expansion given by the right-hand side of the open multiple-covering for…

Let $(X,L)$ be a very general polarized $K3$ surface. A rational curve in $|L|$ is a curve of geometric genus $0$. Nodal rational curves conjecture. All rational curves in $|L|$ ar…

Let $Q_1,Q_2,Q_3\subseteq\mathbb{R}^2$ be three general real conics. A real circle is a circle defined over $\mathbb{R}$ that is tangent to each of $Q_1,Q_2,Q_3$. Maximality conjec…

Let $D^{s,n}_{n^2-m}$ denote the determinantal locus of general $n\times n$ matrices of the indicated rank parameter. For any fixed $m>0$, the general determinantal-locus conjectur…

The B-model of a Calabi–Yau threefold has a partition function associated with its monodromy action. B-model partition-function conjecture. The analogue of Theorem holds for the B-…

Labastida–Mariño–Vafa conjecture. For partitions $\lambda^1,\dots,\lambda^l$,

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…

Assume that $\mathbb{P}^2_k$ is obtained from $\mathbb{P}^2$ by blowing up $k$ generic real points and is equipped with its natural real structure. Let $D\subset \mathbb{P}^2_k$ be…

Let $X$ be a Calabi–Yau threefold, fix $g\ge0$, and suppose $I_{1-g}(X,\beta)$ parametrizes ideals of local complete intersection curves of arithmetic genus $g$. Let…

Positivity and monotonicity conjecture. The Welschinger invariants $W_{D,m}$ are positive if $m<[(c_1(\Sigma)\cdot D-1)/2]$, non-negative if $m=[(c_1(\Sigma)\cdot D-1)/2]$, and sat…

Hurwitz–quadrangulation correspondence conjecture. The constants satisfy

Let $z:=\phi^3$, let $\theta:=z\frac{d}{dz}$, and let $I_3$ be the logarithmic solution of the Picard–Fuchs equation described in the source. Define … where $I^{(0)}_2$ is the holo…

Let $G(t,-k)$ denote the generating function considered in the paper, with $t$ a formal variable and $k$ an integer. Generating-function conjecture. For every integer $k\geq 1$, ……

Let $X$ be either the Hirzebruch surface $\mathbb F_2$ or the surface $B'$ obtained by blowing up six points on a conic, and let $E$ be its $(-2)$-curve. Let $D$ be an effective di…