191 problems

Let $A_1,B_1,C_1,A_2,B_2,C_2$ be finite-dimensional vector spaces over a field $\mathbb{K}$, and let $T_1\in A_1\otimes B_1\otimes C_1$ and $T_2\in A_2\otimes B_2\otimes C_2$. Thei…

Let $A$ be a nonsingular $d\times d$ matrix over a finite field $k$ of cardinality $q\geq4$. Alon–Jaeger–Tarsi conjecture. There exists a vector $v\in k^n$ such that both $v$ and…

Let $A,A^*$ be a tridiagonal pair on a finite-dimensional vector space $V$ over an algebraically closed field $\mathbb K$. Let $d$ be its diameter, and let…

Let $n\geq 3$. A linear subspace of $\operatorname{M}_n(\mathbb{F}_2)$ is $\overline{2}$-spec when every matrix in it has characteristic polynomial of one of the allowed forms over…

Let $n\geq 2$. A linear subspace of $\operatorname{M}_n(\mathbb{F}_2)$ is $\overline{1}^\star$-spec when every matrix in it has minimal polynomial of the form $t^a(t-1)^b$. Dimensi…

Akbari–Cameron–Khosrovshahi conjecture. There exists a nonzero ${0,1}$-vector in the row space of $A_G$ that is not one of the rows of $A_G$.

Continuous orthonormal trivialization conjecture. The bases can be chosen so that the map $T:G\to GL_d(\mathbb{K})$ defined by

Let $\mathcal{A}$ be a matrix. A set of initial minors is a set of minors of $\mathcal{A}$ whose cardinality is bounded by a polynomial or a sub-exponential function in the size of…

Let $M\in\mathbb{F}_p^{n\times n}$ be a matrix. Say that $M$ is $(a,b)$-choosable if, for all subsets $X_1,\ldots,X_n,Y_1,\ldots,Y_n\subset\mathbb{F}_p$ with $|X_i|=a$ and…

Let $M$ be an $n\times n$ nonsingular matrix. The permanent rank of a matrix is the size of its largest square submatrix with nonzero permanent. Kahn's Permanent conjecture. The pe…

For a positive integer $n$, write $[n]=\{1,\ldots,n\}$. For subsets $I_1,\ldots,I_t\subseteq[n]$, define their weight by … and write…

Tasaka's isomorphism conjecture. The map $\widetilde{\xi}$ is an isomorphism. The conjecture concerns the depth-three case and is attributed to Tasaka. The paper does not report a…

Kontsevich's Hadamard-inverse rank conjecture. The rank of $Hinv(A)$ is never two, although every other rank might be possible. This is a concrete problem about rank constraints fo…

Optimality conjecture for Krylov information. For the class $\Lambda$ of matrix-vector multiplications and every orthogonally invariant class $F$, Krylov information is optimal.

Let $L$ be an $(n-k)$-dimensional linear subspace of $K^n$ whose tropical Plücker coordinates are all finite, and let $M\in K^{m\times n}$, with $m\geq k$, have nonzero rows in the…

Let $G_n(\mathbb{C})$ denote the set of pairs $(A,B)$ of $n\times n$ complex matrices that generate the matrix ring, and let $\operatorname{msl}_{\mathbb{C}}(A,B)$ be their matrix…

Let $M_{d,d}$ be the generic $d\times d$ matrix over a field $k$, and let $I$ be the ideal of $d\times d$ permanents of the $d\times2d$ matrix $(M_{d,d}\mid M_{d,d})$. De Loera's r…

Let $V=\bigoplus_i V_i$ be a finite-dimensional vector space, with each $V_i$ equipped with a non-degenerate symmetric bilinear pairing, and let $K\subset V$ be a subspace such tha…

Assume $K$ is algebraically closed, $q\in\mathbb K$ is not a root of unity, $V$ is a finite-dimensional nonzero $K$-vector space, and $R,L:V\to V$ are nilpotent linear maps satisfy…

Let $n$ be a positive integer. A plane configuration is a finite set of vectors in $\mathbb{R}^2$; it is balanced if, for every vector $v_i$ in the configuration, the multiset of d…

Let $r\geq 2$ and let $A\in \operatorname{Mat}_r(\mathbf Z)$ be nonsingular. Define $\gcd(A-I)$ as the greatest common divisor of the entries of $A-I$, and call $A$ primitive when…

Finite affine-generation conjecture. For any matrix problem there exists an integer $n$ such that all indecomposable representations of fixed dimension are generated by a finite nu…

Harish-Chandra diagram-category conjecture. For any semisimple group, the category of Harish-Chandra modules is equivalent to a certain category of diagrams in the category of fini…

Let $C$ and $C'$ be arbitrary complex antisymmetric matrices, and let $W$ be a matrix giving their simultaneous canonical forms. In the fully diagonalizable case, the simple forms…

Kontsevich's conjecture. The rank of $B$ is never equal to three.