12 problems

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

Let $A_1,\ldots,A_n\in\mathbb R^{n\times n}$ be symmetric matrices satisfying $\|A_i\|_{\mathrm{op}}\le 1$. A coloring is a vector $x\in\{\pm1\}^n$. Variance-sensitive Matrix Spenc…

Let $G$ be a finite group of size $n$, and let $A_1,\dots,A_n$ be the $n\times n$ matrices corresponding to the regular representation of $G$. Group Spencer conjecture. The Matrix…

GOE asymptotic discrepancy conjecture. With high probability, the asymptotic discrepancy satisfies

Kunisky–Zhang conjecture. Asymptotically, with high probability,

Small-density second-moment conjecture. When $\kappa$ approaches $2$, the range in which the second-moment ratio remains bounded by a constant may include values of $\tau$ arbitrar…

Let $\bm A_1,\dots,\bm A_T\in\mathbb{R}^{n\times n}_{\operatorname{sym}}$ be i.i.d. random symmetric matrices, where $T=T(n)$, $\|\bm A_i\|\leq1$ almost surely, the distribution is…

Let $A_1,\ldots,A_m$ be positive matrices, and suppose their traces satisfy … The discrepancy of these matrices is the least possible norm of a signed sum obtained by assigning sig…

Let $\xi_1,\ldots,\xi_d$ be independent Rademacher variables, meaning random variables taking values $\pm1$ with equal probability. Let…

The rank-sensitive matrix discrepancy conjecture. One has

Let $A_1,\ldots,A_m$ be Hermitian, positive semi-definite matrices of the same size satisfying … and $\operatorname{tr} A_i \leq \delta$ for all $i$. The matrix paving conjecture.…

Let $V$ be an $n \times m$ real matrix with $|V_{i,j}| \leq 1$, $\|v^i\|_1 \leq R$, and $\|v_j\|_1 \leq \Delta$ for all $i \in [n]$ and $j \in [m]$. Assume $R \geq \Delta$. The bou…