55 problems

Raghunathan's conjecture. For any initial point $x\in X$, the orbit closure $\overline{U.x}$ is a periodic orbit $L.x$ of some subgroup $U\leq L\leq G$.

Pattern-avoidance conjecture for lci-ness. Non-lci-ness in general can be characterized by pattern avoidance.

Completeness conjecture. The finite lists of exceptions in the classifications of exceptional triangles and exceptional quadrilaterals are complete.

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 an affine invariant manifold in a stratum of Abelian or quadratic differentials have rank at least two. Mirzakhani's conjecture. Its orbit closure should cover a stratum of Abe…

Let $G/H$ be a spherical homogeneous space, and let $\mathfrak B(G/H)$ denote the set of $B$-orbits on $G/H$. A partial order on $\mathfrak B(G/H)$ is required to satisfy propertie…

Margulis's generalized conjecture. Every finite $W$-invariant $W$-ergodic measure on $G/\Gamma$ should be homogeneous, and, when $G/\Gamma$ has finite volume, every $W$-orbit closu…

Let $(f,g)=(\phi(h),g)$ be the pair associated with the arithmetic, nonuniform $P$ versus $NP$ problem, where $n$ is the input-size parameter and $m$ is the circuit-size parameter.…

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 $Y$ be an $m\times m$ variable matrix, let $X$ be an $n\times n$ submatrix with $n<m$, let $g=\operatorname{det}(Y)$, and let $f=\phi(h)$ with $h=\operatorname{perm}(X)$. The a…

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…

Let ${\mathbf{V}}$ denote the enhanced module and let $\mathcal{O}\subseteq {\mathbf{V}}$ be an enhanced nilpotent orbit. Its orbit closure is $\overline{\mathcal{O}}$. Achar–Hende…

Let $W_{\lambda}$ be the object whose stabilizer under the relevant $\operatorname{PGL}(3)$ action is denoted by $\operatorname{stab}(W_{\lambda})$, let $B$ and $A_g$ be the matric…

Symplectic juggling-pattern order conjecture. The inherited combinatorial order on symplectic juggling patterns corresponds to the $G^{sp}$-orbit closure inclusion order.

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 $A$ and $B$ be tuples of $m$ matrices, and let $f$ range over all noncommutative polynomials in $m$ variables. Write $\operatorname{rk} f(A)$ for the rank of the matrix obtaine…

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

The cubic-orbit conjecture. There exists a homogeneous polynomial $Q_d\in\mathbb{C}[x_1,x_2,\ldots,x_{d^2+4d+3}]$ of degree $3$, independent of $\lambda$, which vanishes on…

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…