41 problems

Let $Y\to X$ and $Y^{\vee}\to X^{\vee}$ be a pair of dual symplectic resolutions with Hamiltonian torus actions. Fix a generic embedding of $\mathbb{C}^{\times}$ into the torus tha…

Let $X$ and $Y$ be toric Fano manifolds, and let $H^*(X)$ and $H^*(Y)$ denote their cohomology rings. A cohomology ring isomorphism preserves their first Chern classes when it maps…

Let $\ell\in\mathbb{R}^n$ be a positive length vector, and let $\mathcal{M}_2(\ell)$ denote the moduli space of closed linkages with length vector $\ell$ in the plane. Length vecto…

Multiplicative splitting conjecture. There exists such a splitting that is multiplicative under cup product:

Let $\mathcal{B}(X,h)$ be a regular nilpotent Hessenberg variety in Lie type $A$, and let the associated Ding variety be the Schubert variety whose Betti numbers agree with those o…

Let $\mathcal{A}$ be a unimodular arrangement and let $\mathfrak{M}(\mathcal{A})$ be the associated hypertoric variety over $\mathbb{C}$. Let…

Let $\mathcal{A}$ be a unimodular arrangement, let $\mathfrak{M}(\mathcal{A})_{\mathbb{C}}$ be its hypertoric variety, and let $R_0(\mathcal{A})$ be the combinatorially defined rin…

The $e_m$-overkiller conjecture. For $3\leq m\leq k$,

The exact $e_m$-killer conjecture. For $1\leq m\leq k$,

The filtration-saturation conjecture. For $1\leq m\leq k$,

The filtered Hilbert-series conjecture. For $m\geq 0$,

Wang's conjecture. The structure constants $B_{\rho\sigma}^{\nu}(n)$ are independent of $n$ for every smooth quasi-projective surface $X$. The statement extends the known result fo…

Let $X$ be a smooth quasi-projective surface, and let $X^{[n]}$ denote its Hilbert scheme of $n$ points. Let $[x]_c\in H_c^4(X)$ be the Poincaré dual of the homology class in…

Let $X$ be a quasi-projective surface. For each $n$, let $H^*(X^{[n]})$ be expressed in the Heisenberg monomial basis $(\mathfrak h_\mu(n))$. Constant Conjecture. The structure con…

Let $X$ be a quasi-projective surface, and let $X^{[n]}$ denote its Hilbert scheme of $n$ points. A Farahat-Higman-type ring associated to $X$ should govern the cohomology rings…

Cohomology-ring presentation conjecture. The de Rham cohomology ring is

Let $\mathcal{M}$ be the moduli space above, let $H_I^*(\mathcal{M})\subset H^*(\mathcal{M})^\Gamma$ be the subring generated by the universal classes $\alpha_{\mathcal{M}}$,…

Let $G_{k,n}$ be the complex Grassmann manifold, and let $H^{*}(G_{k,n};\mathbb{Q})$ be its rational cohomology algebra. An endomorphism is a homomorphism of this graded cohomology…

Let $M_{d,c7}$ be the moduli space of one-dimensional sheaves on $\mathbb{P}^2$ with degree $d$ and Euler characteristic $\chi$, and let $P_\bullet H^*(M_{d,\chi},\mathbb{Q})$ and…

Complete homogeneous basis conjecture. For $1\leq m\leq\min(\ell,k)$,

Reiner–Tudose's conjecture. For each $m=0,1,2,\ldots,\min(k,\ell)$, one has

Cohomology algebra conjecture. The graded commutative $\mathbf{k}$-algebra

Let $\mathcal{A}$ be a toric arrangement of hypertori, with the notation and presentation of Theorem 5.2. For a relation indexed by $A$, $B$, and $j$, let $m(A)$ and…

Let $h$ be a Hessenberg function and let $\operatorname{Hess}(N,h)$ be the associated regular nilpotent Hessenberg variety. For a Hessenberg function $h'\subset h$, write…

Let $X_0$ be an irreducible nodal curve with exactly one node, and let $U_{X_0}(2,\mathcal{L}_0)$ be the moduli space of rank-two semistable sheaves on $X_0$ with determinant…