10 problems

Let $n$ bosonic particles have relativistic kinetic energy, and let $\mathcal{H}_{V\equiv 0}$ denote the pseudo-relativistic Hartree energy functional with vanishing external poten…

Let $\ell$ be the angular-momentum label of a nonrelativistic $\ell$-boson star, namely a spherically symmetric stationary solution of the multi-field Schrödinger–Poisson system, a…

Let a nonlinear sigma model be a scalar field theory describing maps from spacetime to a target manifold $\mathcal{T}$, minimally coupled to gravity. A toric action is an action of…

Let an $\ell$-boson star have $\ell>1$, and consider non-spherical linearized perturbations, meaning perturbations with nonzero total angular momentum sector. Non-spherical instabi…

Let an excited nonrelativistic $\ell$-boson star be a background configuration with $n>0$ radial nodes, and let $J=0$ denote radial perturbations. Radial instability conjecture for…

Let a nonrelativistic $\ell$-boson star have $\ell\geq3$, and let $\lambda=\lambda_R+i\lambda_I$ be an eigenvalue of the linearized perturbation system, where $\lambda_I$ is its im…

Let $\sigma_1^{(0)}$ denote the radial scalar-field profile of a nonrelativistic ground-state $(\ell=1)$-boson star, and let $J=0$ denote radial perturbations. Let $\mathcal{E}$ be…

Let $\ell$ be a nonnegative integer, and consider the stationary zero modes of the linearized equations with total angular momentum $J\in\{0,1,\ldots,2\ell\}$ and $|M|\leq J$. For…

Let a nonrelativistic $\ell$-boson star be a ground-state configuration, and let stability refer to linear perturbations, including radial and non-spherical modes. Ground-state ℓ-b…

Lieb–Yau conjecture. For each $N<N^*$, there exists a unique ground state (minimizer) of $e(N)$.