33 problems

Let $\mathbf{M}$ be the random matrix from theorem $\mathrm{th:sym}$, with eigenvalues ordered so that $\lambda_1(\mathbf{M})$ is the dominant eigenvalue and…

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 $G=(V,E)$ be a finite graph with positive edge rates $r_{xy}$, and let $L$ be the generator of the symmetric simple exclusion process on the $n$-particle configuration space…

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

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…

Contingency-table norm conjecture. For arbitrary table dimensions $m$ and $n$, table sum $\Sigma$, and row and column sums $r_k$ and $c_p$, this multi-decomposition is non-degenera…

Exact spectral-gap conjecture. The inequality relating the $m$-urn and two-urn gaps is an equality:

Let $\Gamma$ be a hypergraph with non-negative weights. For every $k\in\mathbb{Z}_{\geq 1}$, let $\omega_k=\omega_k(\Gamma)$ denote the smallest new eigenvalue of the Laplacian…

Let $\Gamma=([n],w)$ be an arbitrary hypergraph with non-negative weights. Let $\mathcal{KMP}_k$ be the discrete KMP process with $k$ indistinguishable particles on $\Gamma$, and l…

Let $\Gamma=([n],w)$ be an arbitrary hypergraph with non-negative weights. For each irreducible representation $\rho$ of $\mathrm{U}(n)$, let $\lambda_{\min}(\Gamma,\rho)$ denote t…

Let $\Gamma=([n],w)$ be a weighted hypergraph on $[n]$, with non-negative weights, and let $\rho$ be a non-trivial irreducible representation of $\mathrm{Sym}(n)$. Write…

Let $\alpha$ be irrational and let $v\in C^\omega(\mathbb{T},\mathbb{R})$. Write $\Sigma_{v,\alpha}$ for the spectrum and $N_{v,\alpha}$ for the integrated density of states. An en…

Cesi's conjecture.

Let $p$ and $q$ be coprime integers with $q$ odd. For the periodic problem $\Sigma(p/q)$, let $\gamma_k(p/q)$ denote the length of the gap with label $k$. Gap-decay and extremal-fr…

Let $X$ be a simplicial complex on vertex set $V$ of size $n$, with $h(X)=d$. Write $\mu_k(X)$ for the parameter appearing in the equality below, and let $\Delta_m$ denote the comp…

Let $G$ be a finite weighted hypergraph with non-negative weights. At each step, choose a hyper-edge with probability proportional to its weight and then choose a uniformly random…

Let $G_{\star}$ be a star graph with $n_{\star}$ vertices, local dimension $q$, and moment $k$, and let $\Delta(G_{\star},n_{\star},k)$ denote its spectral gap. Higher-moment star-…

For a star graph $G_{\star}$ with $n_{\star}$ vertices, let $\Delta(G_{\star},n_{\star},k)$ denote the spectral gap of its Hamiltonian at moment $k$. Low-moment star graph conjectu…

Let … be the Hamiltonian associated with a one-dimensional local random circuit, where $T_k^{(i,i+1)}$ acts on neighboring sites. Let the circuit have open boundary conditions, and…

Let $k=\frac12$ or $k=-\frac12$, and let $u$ be a weight $k$ multiplier system on $\Gamma0(N)$. Let $S(m,n,c,nu)$ be the corresponding half-integral-weight Kloosterman sum, let…

Let $\mathbb{P}_g^{\mathrm{\scriptsize{WP}}}$ denote the Weil–Petersson probability measure on compact hyperbolic surfaces of genus $g$, and let $\lambda_1>0$ be the smallest non-z…

Quarter-gap conjecture. For every $\epsilon>0$,

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 $V$ be a single-well potential with centered transition point $\tau=0$ or a convex potential, and let $\Lambda(V,\alpha)$ denote the fundamental spectral gap of the one-dimensio…