12 problems

Norm formula conjecture. The norm has the expansion

Subsingular-vector conjecture. Outside the submodules built on the level-zero singular vectors, there are generically only subsingular vectors, which are singular only for some dis…

Let $r$ and $s$ index a positive-parity singular vector at level $\frac{rs}{2}$, and let $\Delta_{r,s}$ be the corresponding highest weight. Write this vector as … Here $a$ and $b$…

Charge-restriction conjecture. Singular vectors in chiral Neveu–Schwarz Verma modules ${\cal V}_{h/2,h}^{NS,ch}$ should exist only for

The pivot containment conjecture. For every $w\in W_{2n}$,

Let $w$ be a word, let $H(w)$ be its space of $\ell$-singular vectors of weight $0$, and let $C$ be the standard configuration of $w$. Let $v_C(u)$ denote the associated generic si…

Let $w$ be a word, let $H(w)$ be its space of $\ell$-singular vectors of weight $0$, and let $\mathrm{Conf}(w)$ be the set of arc configurations of $w$. For each…

Let $Z$ be the noise matrix, let ${\bm v}$ and ${\bm u}$ be the signal directions, let $\phi$ denote the aspect-ratio parameter, and let $\hat{\bm v}$ be the left singular vector a…

Let $\nu_1,\ldots,\nu_5$ be the five parameters of the Grassmannian family, and let $N_1,\ldots,N_5$ be parameters such that the stated linear relation among the $\nu_j$ holds. Pen…

Let $\mathfrak g$ be a basic Lie superalgebra, and let the identity referred to as … holds for basic Lie superalgebras in general. This would extend the explicit odd singular-vecto…

Let $X$ be an $m\times n$ noise matrix with $q=\min(m,n)$ and singular values $\sigma_i(X)$, and let $b$ be the right edge of its limiting singular-value distribution with D-transf…

Let $X$ be an $m\times n$ noise matrix with singular values $\sigma_i(X)$, and let $b$ be the right spectral edge of its limiting singular-value distribution with D-transform…