16 problems

Let $\mathbf{u}_1,\ldots,\mathbf{u}_m\in\mathbb{C}^{d}$ satisfy $\|\mathbf{u}_i\|\leq 1$ for all $i$, and suppose there are universal constants $\eta\geq 2$ and $\theta>0$ such tha…

Kadison–Singer conjecture. There is a partition $X_1,\ldots,X_r$ of $[m]$ such that

For a set $S\subseteq\mathbb{N}$, let $P_S$ be the projection onto the coordinates indexed by $S$, and let $E(T)$ denote the diagonal operator of an operator…

Paving conjecture. There is a natural number $r$, independent of $n$ and $T$, such that $\forall n$ and every such $T$ there is a partition ${A_j}_{j=1}^r$ of…

Quantitative $R_\epsilon$-conjecture. There is a natural number $r=r(\epsilon,A,B)$ such that, for every $n\in\mathbb N$ and every such sequence, there is a partition…

$R_\epsilon$-conjecture. Every unit norm Riesz basic sequence is a finite union of $\epsilon$-Riesz basic sequences.

Let $M_n(\mathbb{C})^{+}_{1}$ denote the positive semidefinite $n\times n$ complex matrices bounded above by the identity, and let $\chi_r[A]$ be the $r$-characteristic polynomial…

Let $M_n(\mathbb{C})^{+}$ denote the positive semidefinite $n\times n$ complex matrices, and let a positive contraction be a positive semidefinite matrix bounded above by the ident…

Let $A_1,6,6,A_m$ be matrices satisfying … and $\operatorname{trace}(A_i)\leq\epsilon$ for every $i$. The mixed characteristic polynomial extremal-family conjecture. The largest ro…

Let $ell_2^n$ be the $n$-dimensional Hilbert space, let $T$ be a linear operator on $ell_2^n$ whose matrix has zero diagonal, and for $A\triangleleft\{1,\ldots,n\}$ let $Q_A$ denot…

Let $\mathcal{H}$ be a reproducing kernel Hilbert space, and for each point $z$ let $k_z^{\mathcal{H}}$ be its kernel function and $\widetilde{k}_z^{\mathcal{H}}$ the corresponding…

Wandering-point conjecture. If $\boxed{\textstyle\text{?}}$ is wandering for the $G$-action on $G^*$, then the state corresponding to evaluation at $\boxed{\textstyle\text{?}}$ ext…

Non-recurrent-point range conjecture. If $\boxed{\textstyle\text{?}}$ is non-recurrent, then

Non-recurrent-point conjecture. If $\boxed{\textstyle\text{?}}$ is non-recurrent, then $s_{\boxed{\textstyle\text{?}}}$ extends uniquely to $B(\ell^2(G))$.

Minimal-idempotent reduction conjecture. If a minimal idempotent has a unique state extension, then every point has a unique state extension. Thus, for any fixed minimal idempotent…

Paulsen's conjecture. Whether a given state has a unique extension should be related to dynamical properties of that state.