25 problems

Let $V$ be a finite-dimensional irreducible $\boxtimes_q$-module of type $1$, and let $\omega$ be the antiautomorphism of $\boxtimes_q$ from Lemma. Invariant-form conjecture. There…

Let $\mathbb K$ be an infinite field, and let $B_1$ and $B_2$ be non-degenerate symmetric $3\times 3$ matrices over $\mathbb K$. The matrices are Lie-equivalent when the Lie algebr…

Let $Phi$ be a generic tridiagonal-pair parameter array with diameter $d=3$, acting through $A,A^*$ on $V$, and let $w_1,w_2,w_3$ be its weights. Invariant-form conjecture. The fol…

Let $\sigma \\: \Gamma \to 2^V$ be a vertex-induced triangulation of a $(d-1)$-dimensional simplex, and let $k$ be a field. Let $B$ be the symmetric bilinear form on the local face…

Let $V_1$ and $V_2$ be vector spaces, and let $b:V_1\times V_2\to\mathbb{K}$ be a bilinear form of rank $2$ that is degenerate, so that $V_1$ or $V_2$ has dimension greater than…

Consider the GKZ family of bilinear forms given by the family in the source. Clarke's conjecture. The signed Euler characteristic can take any value between $1$ and its generic val…

Continuous simplex bilinear-form measure conjecture. If the Hausdorff dimension of $E$ is at least

Finite-field triangle value conjecture. Then

Codimension-two conjecture. The subset ${\mathtt{G}}_{\ge 2}(\mathsf{B},m,n)$ is contained in a submanifold of codimension two in ${\mathtt{G}}(m,n)$.

Let $G=\operatorname{GL}_{mn}$, let $D$ be the subgroup used in the paper, and let $\sigma,\sigma'\in\operatorname{Irr}G$ be $m$-homogeneous. Consider bilinear forms on…

Let $G=\operatorname{GL}_{mn}$, let $D$ be the subgroup used in the paper, and let $\sigma\in\operatorname{Irr}G$ be $m$-homogeneous. Consider bilinear forms on…

Characteristic-two generalization conjecture. Theorem 2 (the equality theorem for alternating bilinear forms) can be generalized to all fields with characteristic $2$.

Alternating-form symmetric-operator equality conjecture. There exists a maximal partially complete $b$-singular flag $\mathcal{F}$ of $V$ such that

Let $K$ be a field, let $V$ be an $n$-dimensional vector space over $K$, and let $\operatorname{Bil}(V)$ be the vector space of bilinear forms on $V\times V$. A subspace…

Let $V$ be the ambient vector space, let $P\subset V\times V$ have density $\delta>0$, and let $c(\delta)$ be the constant in Theorem 1.1, which asserts that there are subspaces…

Let $F$ be the real or complex field, let $\omega$ be a non-degenerate symmetric or skew-symmetric bilinear form on $F^2$, and let $P\subset F^2\setminus\{0\}$ be an $N$-element po…

Bump conjecture. There is a constant $C>0$, independent of $w$ and $\sigma$, such that

Let $E$ and $F$ be real or complex vector spaces, and let $u:E\times F\rightarrow\mathbb C$ be a bilinear form. Suppose that $u$ is majorized by a sum of the absolute values of mor…

Characteristic-two height-two conjecture. There exist a $(d-1)$-fold Pfister form $\pi$ and a $4$-dimensional form $\mathfrak{c}$ of nontrivial determinant such that

Let $R$ be a Frobenius algebra with an ideal $\mathfrak m$ such that … and let $B$ and $B'$ be forms on $R$ related by $B'(r,s)=B(r,su)$ for some $u\in U(R)$. Suppose that $B$ has…

Let $Y$ be a subspace of $c_0$. Suppose that every bilinear form on $Y$ can be extended to a bilinear form on the whole space $c_0$. The $c_0$-conjecture. Then $Y$ is isomorphic to…

Let $(A,\circ_1,\circ_2,\cdots,\circ_{2^n})$ be an algebra whose operations arise in the setting described above, and consider the algebraic equation associated to the skew-symmetr…

Let $A$ be an $n\times n$ matrix of nonzero entries, and let $x,y$ be chosen independently and uniformly from $\{-1,1\}^n$. For a matrix decomposition $A=A_1+A_2+A_3$, say that…

Assume $\mathbb F$ is algebraically closed. Let $V$ be a vector space over $\mathbb F$ with finite positive dimension, and let $A,A^*$ be a tridiagonal pair on $V$. The self-adjoin…

Non-degeneracy conjecture. (i) If $n=2k+1$, then the bilinear form on the $B_n(\hat{z})$-module $S^{(1)}$ determined by the referenced construction is non-degenerate if and only if…