108 problems

Consider the antiferromagnetic Heisenberg model in one spatial dimension with integer spins. Haldane's conjecture. The model is gapped. This conjecture predicts a fundamental disti…

Consider the AKLT models on the two-dimensional honeycomb lattice and the two-dimensional square lattice, defined by nearest-neighbor AKLT interactions with the corresponding latti…

Let $B>0$ and let $\ell$ be a positive integer. For $N$ particles, let … be the Laughlin lowest-Landau-level ground-state space, where $F$ is a correlation factor and…

Let $\gamma(S)$ denote the spectral gap of the higher-spin XXZ chain with spin magnitude $S$. Large-spin spectral-gap limit conjecture. The limit … exists and is strictly positive.…

Kong–Li–Liu's iSWAP optimality conjecture. For a general fixed connected graph, Kong, Li, and Liu conjectured that $\mathrm{iSWAP}$ maximises the Hermitian second-moment spectral g…

Let $n\geq 2$ and consider $n$-tuples in the special unitary group $\mathrm{SU}(2)$. The subgroup generated by an $n$-tuple acts naturally on $L^2(\mathrm{SU}(2))$. Gamburd–Jakobso…

Caputo's conjecture. The spectral gap $\psi_{S_n}(w)$ comes from the irreducible representation corresponding to the partition $[n-1,1]$.

A two-dimensional PEPS has a bulk Hamiltonian and a corresponding one-dimensional boundary Hamiltonian. The bulk Hamiltonian is gapped when it has a spectral gap above its ground-s…

Baudoin–Tardif conjecture. The optimal convergence rate in this estimate converges to $\eta_1$ as $\gamma\to\infty$. The rate supplied by the existing exponential-convergence resul…

Smits's conjecture. If $\Omega$ is a convex bounded domain, then $\Lambda^{\Omega}(\alpha)$ is an increasing function for $\alpha\geq0$. The conjecture concerns monotonicity of the…

Let $Sym(n)$ act on the set $X_{n,4}$ of $4$-tuples of $n$ elements. For a generating set $S$, write $Cay(Sym(n);S)$ for the corresponding Cayley graph and $Sch(X_{n,4};S)$ for the…

Let $X$ be the real hyperbolic $n$-space and let $\Gamma\subset SO(n,1)$ be a congruence arithmetic subgroup. For differential forms of degree $i$, write…

Let $D\subset\mathbb{R}^2$ be a bounded convex domain symmetric with respect to both coordinate axes. Let $R$ be the smallest rectangle containing $D$ whose sides are parallel to t…

Let $D\subset\mathbb{R}^d$ be a bounded domain, and let $\lambda_1(D)$ and $\lambda_2(D)$ be the first and second eigenvalues for the symmetric $\alpha$-stable process, with…

Let $D$ be a convex bounded domain of diameter $d_D$, and let $u1(D)< u2(D)$ denote the first two eigenvalues of its Dirichlet Laplacian. The spectral gap is $u2(D)- u1(D)$. van…

Let $f$ be a real Fourier integral with spectral gap $(-a,a)$, meaning that its spectrum avoids this interval. The limit average density of zeros is the limiting average number of…

Let $\Delta=\cosh\eta$ with $\Delta\geq 1$, and let $\gamma_{J,M}$ denote the spectral gap above the ground-state eigenvalue of the spin-$J$ ferromagnetic XXZ chain in the kink sec…

Let a periodic conformally flat manifold be given, with curvature understood as its scalar curvature, and consider the spectral gaps of the associated periodic operator. Green's cu…

Spin-1/2 gap conjecture. Assuming this ordering holds true, for spin $1/2$ and $B\le A$ the gap equals

Large-spin Bose-gas conjecture. There is a function $\gamma_\infty:(1,\infty)\times\mathbb{R}\to\mathbb{R}$ such that

Let $S$ be the spin of a one-dimensional ferromagnetic XXZ chain, let $\Delta$ denote its anisotropy parameter, and let the spectral gap be rescaled by its linear-in-$S$ growth. Cl…

Vafa's precompactness conjecture. The space of CFTs with fixed $c$ and a lower bound on the gap is precompact: there is a distance function on the space such that every infinite se…

Consider the local Iwatsuka model with a nonzero magnetic field $b$ satisfying minimal regularity assumptions such as those formulated in Section 2.1. Finiteness conjecture. The nu…

Let $\Gamma$ be a cocompact Fuchsian group and let $\sigma$ be a multiplier on $\Gamma$. Let $\theta$ denote the flux associated with $\sigma$, and let $H_{\sigma,V}$ be the corres…

Let $L$ denote the linear system size, $V$ the volume, $d$ the spatial dimension, and $V/L$ the system width. Assume that the width $V/L$ is of order $L^{d-1}$ and that $d>1$. Even…