105 problems

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 $Q_1,Q_2,Q_3\subseteq\mathbb{R}^2$ 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…

Let $\lambda=(2^{e_2},\ldots,m^{e_m})$ be a partition without parts equal to $1$, set $N:=|\lambda|$ and $M:=|\widetilde\lambda|$, and write … For each admissible pair…

Shifted-binomial log-concavity conjecture. For every partition $\lambda$ without parts equal to $1$, and every admissible index $i$, the sequence…

Strong binomiality conjecture. For each fixed $k$ and each partition $\lambda\vdash k$, the coefficient $f_\lambda(n,d)$ is a polynomial in finitely many binomial coefficients of t…

Separate polynomiality conjecture. For each fixed $k$ and each partition $\lambda\vdash k$, the coefficient $f_\lambda(n,d)$ is polynomial in $d$ for fixed $n$, and polynomial in…

Do–He–Robertson's coefficient-gap conjecture. The coefficients satisfy

Stable-pair rationality and symmetry conjecture. The series $Z_\beta^\mathsf{PT}(q\mid D)$ is the Laurent expansion of a rational function, more precisely of $q^{d_\beta/2}f(q)$ fo…

Let $1\leq i<j\leq n$ with $j<n$, and let $\mathcal{S}_{ij}^2$ be the square image of the Schubert variety $\mathcal{S}_{ij}$. Degree recursion conjecture. The degrees satisfy … an…

The converse to the surplus condition criterion. One has

Maulik–Ranganathan's conjecture. The generating functions satisfy

Let $\vec{\lambda}_{A,+}$ be $n$-tuples of plane partitions and $\vec{\lambda}_{A,-}$ be $m$-tuples of plane partitions for the six surfaces. Let $v_\alpha=e^{\mathfrak{a}_\alpha}$…

Let $\vec{\pi}_{a,+}$ be $n$-tuples of plane partitions and $\vec{\pi}_{a,-}$ be $m$-tuples of plane partitions for each of the four legs. Let $v_\alpha=e^{\mathfrak{a}_\alpha}$,…

Let $n,m$ be nonnegative integers indexing D8/D8$'$ and opposite-brane pairs, and let $v_\alpha=e^{\mathfrak{a}_\alpha}$, $w_\alpha=e^{\mathfrak{b}_\alpha}$,…