6 problems

Real equivariant almost-commuting sphere matrices conjecture. For every $\eta>0$ there is a $\delta>0$, independent of $n$, such that there is a triple $(X_2,Y_2,H_2)$ of commuting…

Localization dichotomy conjecture. The following statements are equivalent: (a) $P$ admits a generalized Wannier basis that is exponentially localized; (b) $P$ admits a generalized…

Let $V\in C^\infty(\mathbb{R}^3/\mathbb{Z}^3)$ be a smooth periodic potential, and consider the periodic Schrödinger operator $-\Delta_{\mathbb{R}^3}+V$. A degeneracy of its Bloch…

Let $H$ and $X$ be self-adjoint matrices in a matrix algebra, and suppose that $(X,H)$ belongs to an Altland–Zirnbauer symmetry class. Lin's theorem states that sufficiently almost…

Let $U_1$ and $U_2$ be approximately commuting unitary matrices, with $obreakbiglVert[U1,U2]bigrVertleq ecdelta$, and let $H1,H2,H3$ be the associated Hermitian matrices construct…

Let $H_r$, for $r=1,\ldots,d+1$, be matrices in one of the ten random-matrix-theory universality classes that give a soft representation of the sphere $S^d$. In the stable limit, c…