40 problems

Consider a finite transitive graph with $n$ vertices, coalescing random walks, coalescence time $\tau_{\mathrm{coal}}$, relaxation time $t_{\mathrm{rel}}$, and hitting time scale…

Fix an integer $N$. For a periodic box of size $L$, let $E_L^N(\varphi)$ be the Hartree–Fock minimum in the charge sector with trace $N$, let $E_L(0)$ be the corresponding vacuum e…

Mean-field TDHF conjecture. When $N$ is large, the single-body density operator $D_{N:1}(t)$ should nearly obey the time-dependent Hartree-Fock equation, with $F(0)=D_{N:1}(0)$. Th…

Propagation of quantum molecular chaos conjecture. The sequence of Heisenberg maps $\phi_{n,t}(A)=e^{iH_nt/\hbar}Ae^{-iH_nt/\hbar}$ propagates chaos: if $\{\rho_n\}$ is $\rho$-chao…

Let $V$ be the interaction matrix, and let $F$ be the eigenspace associated with the largest eigenvalue of $V$. Let $F^{\mathrm{abs}}$ be the eigenspace associated with an eigenval…

Let $B$ and $V$ be weight matrices satisfying Assumption, with $V\neq\mathrm{Id}$, and let $(\rho_t^\beta)_{t\geq 0}$ be the token-distribution evolution on ${\mathbb S}^{d-1}$. Le…

Let $k$ be the network width, let $\mathbf v$ be the readout vector, and let $\mathbf 1$ be the all-ones vector. The neuron-decoupling identity is established in the special case…

Dirac-mass conjecture. The mean-field minimizer is the Dirac mass at $x_\star$:

Kuramoto's conjecture. There exists a sharp critical coupling

Let $\mathcal{L}_n(\boldsymbol{\theta})$ be the loss function for the single-index model, let $\delta$ denote the aspect ratio, and let $\mathcal{T}_s(\cdot)$ be the preprocessing…

Let $p_\mathfrak{s}:=\lambda/(1+\lambda)$ be the sleep probability, let $\rho_\mathrm{c}$ denote the critical density for activated random walk on $\mathbb{Z}^d$, and let $d$ be th…

Let a graph have $N$ vertices, and construct a configuration-model graph by assigning each vertex a prescribed number of stubs and pairing the stubs uniformly at random, allowing s…

Partial homogenization conjecture. If the corresponding solutions of are denoted by $(\bm{X}^{(k)}_i(t))_{i\in[N_k]}$ and $(\bm{w}^{(k)}_{i,j}(t))_{i,j\in[N_k]}$, then

Let the assumptions of Theorem hold. For $\epsilon>0$, let $(\bm{X}^{(\epsilon)}(t),\bm{w}^{(\epsilon)}(t))$ solve, and let $(\bm{X}^{(0)}(t))$ solve the limiting system with the s…

Nam's radial-measure conjecture. The infimum defining $\beta_2$ is achieved by radially symmetric probability measures $\mu$; equivalently, $\beta_2$ can be calculated using only r…

Hirsch's conjecture. The gap $\Delta$ vanishes like

Let $\rho_\infty(\cdot,t)$ be the limiting parameter distribution at time $t$, and suppose that the origin $0_{\mathrm{dim}\theta+\mathrm{dim}w}$ is an interior point of…

The pair-approximation approach describes the stochastic epidemic exactly on locally tree-like graphs. Pair-approximation exactness conjecture. The pair approximation is conjecture…

Let $G$ be a nonamenable transitive graph, and consider critical Bernoulli percolation on $G$. Schonmann's conjecture. Critical percolation on every nonamenable transitive graph ex…

Consider a conservative long-range interacting particle system at thermal equilibrium with a tagged particle whose mean-field evolution is trivial. Let the linearized mean-field op…

Nam's coefficient conjecture. The main part of the conjecture is that

Let $E(N,lambda v)$ and $E(N-1,lambda v)$ denote the ground-state energies of the trapped Bose gas with interaction scaling $lambda$, and let $T$ and $v$ satisfy appropriate condit…

Critical-regime conjecture. Under the remaining choice of initialization scales corresponding to the solid blue line in the phase diagram, the training dynamics of the two-layer ne…

Incremental learning conjecture. Incremental learning also occurs for higher-order polynomials.

Let $N$ be the number of neurons and let $\mathfrak{Dim}$ denote the dimension parameter governing the model's technical difficulty. Assume that the quantity $L$ from the source re…