13 problems

Scaling-critical regularity conjecture. The equation is well-posed in $B^s$ for $s>s_\text{crit}$ and ill-posed in $B^s$ for $s<s_\text{crit}$.

Universal amplitude-combination conjecture. All products

Spectral scaling conjecture. The spectra satisfy

Let ${\cal W}_{k}(L)$ be the nested-path correlation function for a two-dimensional critical percolation model of linear size $L$, and let $X_{\textsc{mnp}}(k)$ denote the monochro…

Fisher's scaling conjecture. The annealed density has the scaling form

Let $J>0$ be the intrinsic length of a moving polymer taking values in $\mathbf{R}^2$. Because the path $x\mapsto\mathbf{u}(t,x)$ behaves like a two-dimensional Brownian motion, it…

In a renormalization-group description, distinguish relevant eigenvalues, singular-irrelevant eigenvalues associated with genuinely new universal corrections, and gauge-irrelevant…

Consider gauge eigenvalues arising from coordinate changes in a renormalization-group description, and universal eigenvalues associated with the renormalization group. Gauge-eigenv…

Let a critical point serve as a fixed point of a renormalization-group transformation, and distinguish gauge-irrelevant corrections arising from redundant coordinate changes from s…

Consider an $n\times n$ input-queued switch with load factor $\rho$, and let the total queue size denote the sum of the queue sizes over all input-output pairs. The optimal scaling…

Let $O^\sigma$, $O^I$, and $O^\varepsilon$ denote the operators appearing in the susceptibility expansion, and let the leading-order contribution of $O^\sigma$ be defined by its lo…

Critical-dimension conjecture. The critical dimension for cluster-cluster aggregation with Lévy diffusion should be

Let $\rho(\theta,\theta',t)$ denote the interfacial fluctuation field for a radially growing cluster, and let $\rho$ evolve with dilution in one- and two-dimensional settings. In t…