30 problems

Let $g=\operatorname{det}(Y)$, $h=\operatorname{perm}(X)$, and $f=\phi(h)$. A strong obstruction is a representation-theoretic obstruction of the type defined in the paper for the…

Let $v$ be the determinant function and let $\Delta_V[v]$ be the class variety associated with $NC$, namely the orbit closure of the determinant. Let $X(Pi_v)$ be the variety and sc…

Closure-order conjecture. If

Let ${\cal K}$ and ${\cal H}_{\overline{y_e}}$ be the stabilizer-related groups associated with the orbit-closure data, and let $\overline{O(z)}\subseteq\overline{O(y)}$ be orbit c…

Let $\ell$ be a linear coordinate on $\mathbb C^1$, and define the projective orbit closures … … Let $n(m)$ be a polynomial. Mulmuley–Sohoni's orbit-closure conjecture. For all suf…

Type C induction conjecture. In type $C$, Borel orbit closures are induced by type $A$.

Irreducible-component conjecture. If $I$ is an irreducible component of $R(\mathcal{A},V)^{\langle-,-\rangle,\varepsilon}$ and $M,N\in I$, then

Let $\mathcal{M}$ be a geminal orbit closure, and let the optimal covering map associated with $\mathcal{M}$ be the covering map of minimal degree arising in the classification of…

Fix $n>1$. For each $i\in\{1,\ldots,n\}$, let $\mathcal{S}_i$ be a component of a stratum of Abelian or quadratic differentials, and let $\mathcal{M}\subset\prod_i\mathcal{S}_i$ be…

The Strong Diamond Conjecture. Under these assumptions, $\mathcal{M}$ is a full locus of covers. This would extend the full-locus conclusion beyond the cases established by the pre…

Let $d\geq 2$ be an integer. Let $A\subset\mathbb{T}^d$ be an orbit closure under the $\times 2,\times 3$ actions. Orbit-closure conjecture. Then $A$ is a union of finitely many (p…

Let an orbit closure be an orbit closure of the $SL(2,\bR)$ action on a stratum of translation surfaces, and let its rank be the rank associated with the corresponding affine invar…

Let $\gamma$ be any $(p,2)$-clan with $p\geq 2$, and let $Y_{\gamma}$ be its orbit closure. lci–Gorenstein equivalence conjecture. $Y_{\gamma}$ is a local complete intersection if…

Let $\gamma$ be a $(p,2)$-clan with $p\geq 2$, and let $Y_{\gamma}$ be its orbit closure. Gorenstein pattern-avoidance conjecture. $Y_{\gamma}$ is Gorenstein if and only if…

Let $Y_{\gamma}$ be an orbit closure, and let $\operatorname{Maxsing}(Y_{\gamma})$ be the set of Bruhat-maximal orbits along which $Y_{\gamma}$ is singular. Let…

Let $K$ act on the ambient flag variety, let $Y_{\gamma}$ be the closure of the orbit indexed by $\gamma$, and let $\mathcal{O}_{\alpha}$ be the orbit indexed by $\alpha$ with…

Let $Q$ be a quiver. For an extended Dynkin quiver, let $T_1,\dots,T_m$ be pairwise non-isomorphic indecomposables in $\operatorname{Rep}(Q)$ such that…

Nonarithmetic rank-one sparsity conjecture. Outside of genus two, a sequence of nonarithmetic rank-one orbit closures cannot equidistribute in a component of a stratum.

Let $G=GL(p+q,\mathbb{C})$, $K=GL(p,\mathbb{C})\times GL(q,\mathbb{C})$, and consider $K$-orbit closures in $G/B$, indexed by clans. A clan avoids a pattern when it contains no occ…

Let $Y_{\gamma}$ be a $K$-orbit closure, and suppose the maximal components of its singular locus are $\mathcal O_{\alpha_1},\ldots,\mathcal O_{\alpha_k}$. Non-Gorenstein-locus con…

Let $Y_{\gamma}$ be a $K$-orbit closure and let $\mathcal O_{\alpha}$ be a $K$-orbit appearing in the maximal singular locus of $Y_{\gamma}$. Maximal-singular-locus conjecture. All…

Let $p\in Y_{\gamma}$ be a point of the $K$-orbit closure, and let $\operatorname{gr}_{\mathfrak m_p}\mathcal O_{p,Y_{\gamma}}$ be the associated graded local ring. Reduced tangent…

Hilbert-polynomial conjecture. The following assertions hold: (i) $\operatorname{gr}_{\mathfrak m_p}\mathcal O_{p,Y_{\gamma}}$ is Cohen–Macaulay; (ii)…

Let $I_{\gamma,\beta}$ be the patch ideal defining a neighborhood of the $K$-orbit closure $Y_{\gamma}$ at the point indexed by $\beta$. Patch-ideal radicality conjecture. The idea…

Let $\gamma$ be a clan and let $\Upsilon_{\gamma}(X;Y)$ denote the polynomial associated to the $K$-orbit closure $Y_{\gamma}$. K-orbit multidegree conjecture. Theorem(III) holds f…