9 problems

Very efficient format conjecture. 1. The format $(n,R_c(n))$ is very efficient if $c$ is such that $|E_1|\geq |Y|$. 2. The format $(n,r)$ is efficient if there exists $c$ with…

Grammians conjecture. There exists a constant $\kappa_n>0$ and an increasing function $\gamma_n:(0,\kappa_n)\to(0,1)$, both depending only on $n$, such that $\gamma_n(\kappa)\to1$…

Defining-ideal conjecture. The Zariski closure of $SOT_{2,n}$ in $\mathbb{C}^{n^4}$ is cut out by the vanishing of $\mathcal P_n$, $\mathcal Q_n$, and $h_n$. The ideal defined by t…

Let $\{a_i\}_{i=1}^R$ be $R$ independent standard Gaussian vectors in $\mathbb{R}^L$, let $c_n<\frac{1}{n!}$ be a constant, and consider the power method iterations given by the pa…

Generalized Kruskal conjecture. Under these hypotheses, a nontrivial zero-sum subcollection must exist. This relaxes general position to lower bounds on the dimensions of the subsy…

Let $\{\mathbf{A}^k\}$ be the sequence generated by Algorithm, where $\mathbf{A}^k$ is the solution $(\mathbf{A}^{(1)},\mathbf{A}^{(2)},\ldots,\mathbf{A}^{(N)})$ in the $k$-th iter…

Let a tensor have format $(n_1,\dots,n_d)$, and let … Call the format perfect when $R(n_1,\dots,n_d)$ is an integer, and call a format generically identifiable when a general tenso…

Let $S^d(\mathbb C^n)$ be the space of symmetric tensors, and let the odeco variety be the Zariski closure of the orthogonally decomposable tensors. For a tensor…

Let $k\ge 3$ and let $n_i$ denote the dimensions of the factors, for $1\le i\le k$. Write $\mathbf{n}=(n_1,\ldots,n_k)$ and define … A generic tensor of order $k$ and rank $r$ is a…