23 problems

Oblak's conjecture. The size-preserving function $P\mapsto\operatorname{Obk}(f(P))$ is precisely $\mathfrak{D}$.

Box Conjecture. The elements of $\mathcal{D}^{-1}(Q)$ can be arranged in this box so that the partition in position $(i_1,i_2,\cdots,i_k)$ has exactly

Let $\mathsf k$ be an infinite field, let $B$ be a nilpotent matrix, and let $\mathfrak D(P)$ denote the maximum Jordan partition arising from a nilpotent matrix commuting with one…

Let $A$ be an $n$-by-$n$ nilpotent matrix, and let $W(A)$ denote its numerical range. A flat portion is a line segment contained in the boundary of $W(A)$. Gau–Wu conjecture. The b…

Let $n\in\mathbb{N}-\{0\}$ and let $K$ be a field. Let $S$ be an affine subspace of $M(n\times n,K)$ such that every element of $S$ is nilpotent, and assume that each element of…

Let $Q$ be a stable partition of length $\ell$, let $J_Q$ be a Jordan matrix of type $Q$, and let $\mathcal N_{J_Q}$ be the variety of nilpotent matrices commuting with $J_Q$. For…

Let ${\mathbb F}={\mathbb F}_q$ be a finite field, and let $\mathcal{N}_n^m$ denote the set of $m$-tuples of nilpotent $n\times n$ matrices over ${\mathbb F}$. The group…

Let ${\mathbb F}={\mathbb F}_q$ be a finite field, and let $\mathcal{N}_n^m$ denote the set of $m$-tuples of nilpotent $n\times n$ matrices over ${\mathbb F}$. The group…

Let $Q=(u,u-r)$ be a stable partition, with $r\geq 2$ and $u-r\geq 1$, and let $\mathcal{Q}^{-1}(Q)$ denote the set of partitions having generic commuting partition $Q$. Oblak's co…

Let $T$ be a nilpotent matrix in Jordan canonical form, let ${\rm Gr}(\ell,n)^T$ be the variety of $\ell$-dimensional $T$-fixed subspaces, and let the shuffle equations be the line…

Let $m,n\geq 2$ be natural numbers and let $p$ be a prime. Consider the matrix ring $\mathbb{M}_n(\mathbb{Z}_{p^m})$, where $\mathbb{Z}_{p^m}$ is the ring of integers modulo $p^m$.…

For integers $g\ge 2$ and $n\ge 1$, let $A_g(n,q)$ be the polynomial associated in the paper with the counting of tuples of nilpotent matrices under simultaneous conjugation. Write…

Let $B$ be an $n\times n$ matrix, let $\mathcal N_B$ be its nilpotent commutator, and let $\mathfrak Z_P$ denote the locus in $\mathcal N_B$ of matrices having Jordan type $P$. Sup…

Kirillov–Melnikov conjecture. The same sequence of polynomials arises in a number of different ways.

Simple-pole conjecture. Generic local submodule zeta functions associated with nilpotent matrix algebras have a simple pole at zero.

Let $F:K^d\rightarrow K^d$ be a polynomial map with $F_i=X_i+H_i$, where … and … for $1\leq i\leq d$. Write $H=(H_1,\dots,H_d)$ and let $JH$ be its Jacobian matrix. Define polynomi…

Fix $Q=(u,u-r)$ with $u>r\ge2$, put $B=J_Q$, and let $E(\mu,\nu)$ and $X_s(k_1,k_2)$ be the coordinate expressions defined in the source. All-table loci equations conjecture. For…

Fix $Q=(u,u-r)$ with $u>r\ge2$, put $B=J_Q$, and use coordinates $a_1,\ldots,a_{u-1},b_1,\ldots,b_{u-r-1}$ on $\mathcal U_B$. For $\mu\in\{0,1,\ldots,u-1\}$ and…

Fix $Q=(u,u-r)$ with $u>r\ge 2$, put $B=J_Q$, and let $\mathcal U_B$ be the maximal nilpotent subalgebra of the centralizer of $B$. For a partition $P$, let…

Nilpotence conjecture. One has

Let $P$ be a partition, let $\mathcal U_B$ be the relevant space of nilpotent matrices commuting with a fixed nilpotent matrix of Jordan type $P$, and let $A$ be an adequate matrix…

Let $M$ be a square matrix of size $m$ whose entries are functions from a set $S$ to a reduced noncommutative ring $R$. For $v\in S$, write $M|_v$ for the matrix obtained by evalua…

Spectral arbitrariness conjecture. The patterns $\mathcal{T}_n$ are spectrally arbitrary for all $n\geq 2$, and hence are potentially nilpotent.