5 problems

Consider a percolation model with a non-uniqueness phase, meaning that there is a parameter range in which infinitely many infinite clusters exist. Mean-field critical exponents ar…

Let $d$ be the dimension, let $\text{BI}$ denote the set of bi-infinite branches, and consider their asymptotic directions toward the past, represented by limits…

Threshold-one conjecture. If $p\geq 1+\frac{4}{n}$, then

Let $n\geq 2$, let $\lambda<\frac{(n-1)^2}{4}$, and let $u_0\in H^1(\mathbb H^n)$ be initial data for the focusing nonlinear Schrödinger equation on $\mathbb H^n$. Let $E_\lambda$…

Finite-energy conjecture. The solutions corresponding to $b1_k$ have finite energy.