10 problems

Let $n=\sum_{j=1}^k n_j$ and consider a $(k+1)$-tensor space $V$ of format $(n_1+1,\dots,n_k+1,n+2)$, with $T\in V$ a general tensor. Higher-factor defective-span conjecture. If…

Let $T$ be a general tensor of format $(a+1,b+1,n+2)$, where $n=a+b$ and $a,b\geq2$. Three-factor defective-span conjecture. … except in the cases $(3,3,6)$, $(3,4,7)$, and…

Let $V=\mathbb{C}^{n_1+1}\otimes\dots\otimes\mathbb{C}^{n_k+1}\otimes\mathbb{C}^{n+2}$, where $n=\sum_{j=1}^k n_j$, and let $T\in V$ be a general tensor. Let $\alpha_T$ be the coho…

Let $G$ be a forest on vertex set $V=\\{1,\ldots,d\\}$, with each connected component a tree having at least three nodes, and let $A$ be its adjacency matrix. Let $\mathcal{V}_G$ b…

Let $G=(V,E)$ be a graph with vertex set $V=\\{1,\ldots,d\\}$ and adjacency matrix $A$. Let $S^r(\mathbb{R}^d)$ denote the space of symmetric order-$r$ tensors, and let…

Let $V_i$ be vector spaces with $\dim V_i=n_i+1$, and let $\mathscr{S}_{n_1,\dots,n_d}$ be the affine cone over the Segre variety of rank-one tensors in…

Let $V$ be the tensor space of a fixed format, let $T\in V$ be a generic tensor, and let $Z_T$ be the set of singular tuples of $T$. Write $\langle Z_T\rangle$ for their linear spa…

Let $T$ be a generic tensor of format $(n_1,\ldots,n_k)$, and let $\langle Z_T\rangle$ denote the linear span of its singular $k$-tuples. For fixed $n_1,\ldots,n_{k-1}$, consider t…

Young-symmetrizer constructions give explicit descriptions of Killing prolongations for some tensor symmetry types. The question concerns whether analogous constructions exist for…

Gram locus conjecture. The Gram locus $\mathcal{G}(\mathcal{B})$ is given by