10 problems

Matrix Spencer conjecture. There exists a universal constant $C$ such that

Weak variance conjecture. The expected spectral norm satisfies

Let $T$involve Bernoulli random tensor entries, and let $d_{i_{1},\dots,i_{k-m}}$ denote the degree of a tuple $(i_{1},\dots,i_{k-m})$, namely … Assume…

Upper bound conjecture. The lower bound for the expected spectral norm should be reversible:

Fix $n\in\mathord{\mathbb N}$ with $2<n$, and let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb C}^n$ be nonsingular. Let $\mathbf H$ denote the polynomial map whose fixed points are us…

Let $n\geq 3$ be fixed, and let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb C}^n$ be a symmetric $d$-qudit. For most such tensors, consider the spectral norm $\|{\cal S}\|_{\sigma}$.…

Let $X$ be the $d\times d$ symmetric random matrix with entries $X_{ij}=b_{ij}g_{ij}$, where $\{g_{ij}:i\ge j\}$ are independent standard Gaussian random variables and…

For a Boolean function $f:\{0,1\}^n\to\{\pm1\}$, let $\|\widehat f\|_{1,\epsilon}$ be the smallest spectral norm of a function $g$ satisfying … The function $f$ is symmetric when i…

Let $F:\{0,1\}^n \times \{0,1\}^n \to \{-1,1\}$ be a Boolean communication problem, and let $\|\widehat{F}\|_1$ denote its spectral norm. Grolmusz's conjecture. There is a constant…

Let $(M,\omega)$ be a symplectic manifold, let $\operatorname{Symp}_0(M,\omega)$ be its identity component, and let $\operatorname{Ham}(M,\omega)$ be the Hamiltonian subgroup. Let…