8 problems

Masoero–Raimondo string-equation conjecture. Under the nonvanishing assumption, $U(T,X)$ is a global solution of this equation.

Let $Z$ denote the zero set of the function $\tau(T,X)$ associated with the critical Benjamin–Ono profile. Nonvanishing conjecture. … i.e., $\tau(T,X)\neq 0$ for all…

Let $t>t_b$, fix $x_0\in(X^-(t),X^+(t))$, and let $0<r<q<1$. For $|x-x_0|\le\epsilon^r$ outside the exceptional set $X_o$, consider small eigenvalues $\alpha$ with…

Let $\psi_U(x,t)$, $\psi_L(x,t)$, and $\phi_L(x,t)$ be the modulation functions defined from the scaling functions of the upper- and lower-branch eigenvalues, and let…

Let $t>t_b$ and let $\nu_{L,k}(t)$ be the imaginary parts of the lower-branch eigenvalues. Lower-branch imaginary-part conjecture. There is an $\epsilon$-independent function…

Let $\nu_{U,k}(t)$ be the imaginary parts of the upper-branch eigenvalues and let $N_U(\epsilon)$ be their number. Upper-branch imaginary-part conjecture. There is a scale…

Let $S_U$ and $S_L$ denote the upper and lower branches of the bulk eigenvalues, with real parts indexed increasingly by $\mu_{U,k}(t)$ and $\mu_{L,k}(t)$, and let $t_b$ be the bre…

Let $S_o$ be the set of outlier eigenvalues and let $\epsilon>0$ be sufficiently small. Outlier sparsity conjecture. There is an exponent $0\le p<1$ and a constant $K>0$ independen…