359 problems

A cluster algebra is an algebra generated by cluster variables grouped into clusters; a Laurent polynomial with positive coefficients is a Laurent polynomial whose coefficients are…

Let $X$ be a projective manifold, meaning a smooth projective variety, whose tangent bundle $T_X$ is nef. Campana–Peternell conjecture. The variety $X$ should be homogeneous. This…

Let $E$ be an ample vector bundle over a compact complex manifold $M$. A Hermitian metric on $E$ is Griffiths-positive when its curvature is positive in the Griffiths sense. Griffi…

Beltrametti–Sommese conjecture. If $D-K_X$ is nef, then $D$ is ample.

Let $P_M$ be the base polytope of a matroid $M$. A lattice polytope is Ehrhart positive when all coefficients of its Ehrhart polynomial are nonnegative. De Loera–Haws–Köppe conject…

Let $\mathbb{I}^{\rm br}$ be the conjectural ${A}_2$-bracelets basis map for $\mathscr{O}(\mathscr{X}_{{\rm PGL}_3,\frak{S}})$, and let $\mathbb{I}$ denote the paper's correspondin…

Hankel total positivity conjecture. The sequences of polynomials

Atom-positivity conjecture. The row operator applied to a key polynomial is atom-positive:

Boucksom–Demailly–Păun–Peternell duality conjecture. The full cones $\mathcal{E}$ and $\mathcal{M}$ should be dual with respect to the Poincaré pairing for any compact Kähler manif…

Let $\mathscr{A}_q$ be a quantum cluster algebra generated by cluster variables $\{Z_i\}_{i\in\mathsf{K}}$, contained in the non-commutative Laurent polynomial ring…

Let $J_{\lambda}(q,t)$ be the integral form Macdonald symmetric function and let $s_{\mu}$ be the Schur function, with $\lambda$ and $\mu$ partitions. For a non-negative integer…

Let $\lambda$ be a partition and let $\theta^{(\alpha)}_\mu(\lambda)$ be the Jack-character coefficient defined by the paper, with $b:=\alpha-1$. Let $m_1(\mu)$ denote the number o…

Let $c^rho{tau, u}(alpha)$ be the coefficients defined by … Here $b:=\alpha-1$ is the shifted Jack parameter. Goulden–Jackson's conjecture. The coefficients satisfy…

Let $m\in\mathbb Z_{>0}$, let $\mathcal C\subseteq\mathsf{RPD}_{m,n}$ be a $\sqrt{\mathfrak{gl}_n}$-crystal, and let $w\in S_n$. Write … A polynomial is Lascoux positive if it is a…

Thurston's conjecture. The bracelets bases of ordinary skein algebras are positive bases; equivalently, their structure constants lie in

Let $\mathcal{A}(X,\mathbf{h},\Lambda,\widetilde{B})$ be a generalized quantum cluster algebra with initial extended cluster $\{X_1,\ldots,X_m\}$. A Laurent polynomial in this clus…

Let $X$ be a projective manifold and let $L$ be an ample line bundle on $X$. For a very general point $x\in X$, write … the maximal value of the Seshadri constant of $L$ on $X$. La…

Let $\mathcal{A}$ be a cluster algebra, and let a cluster monomial mean a monomial in cluster variables all belonging to the same cluster. Strong positivity conjecture. Any cluster…

A Schubitope is a polytope in the class introduced by Monical, Tokcan, and Yong. A polytope is Ehrhart positive when all coefficients of its Ehrhart polynomial are nonnegative. Mon…

Let $X_{w,2}(z)$ be the depth-two extremal quasimodular form of weight $w$, and define … A quasimodular form is completely positive when all coefficients in the relevant complete-p…

Let $\lambda$ be a partition, let $m_\lambda$ denote the corresponding monomial symmetric function, and let $\Phi_{\mathbf{x},\mathbf{y}}$ be the map defined in the source from sym…

Fix $n\geq 1$. For partitions $\pi,\mu,\nu\vdash n$, define $d^\pi_{\mu,\nu}(\alpha)$ by … where $m_\nu$ denotes the monomial symmetric function in the alphabet $\boldsymbol{q}$. M…

Stieltjes moment conjecture. The sequences of polynomials

Let $n$ be a positive integer, let $w\in S_n$ be a permutation, let $\zeta$ be a weak composition, and let $\boldsymbol{x}=(x_1,\ldots,x_n)$. Write…

Let $A_{Q,\alpha}$ and $B_{\mu_Q,\alpha}$ be the quiver-representation and moment-map counting polynomials whose normalised limits, when they exist, are rational fractions in $q$.…