8 problems

Let $\Omega$ be a domain, let $A$ be the prescribed parameter, and let $D=D_{\alpha,A}$ be an optimal configuration. Say that $D$ has the same symmetries as $\Omega$ when every sym…

Consider an annulus of thickness $\tau$, and write $\delta=A/|\Omega|$ for the prescribed area fraction. Let $D=D_{\alpha,A}$ be an optimal configuration, and let symmetry breaking…

Let $\Omega_h$ be a dumbbell domain, and let $D=D_{\alpha,A}$ be an optimal configuration. Dumbbell symmetry conjecture. For every $\alpha>0$, there is $\rho_1(\alpha,h)>0$ such th…

Let $\Omega_h$ be a dumbbell domain, let $D=D_{\alpha,A}$ be an optimal configuration, and let $\psi_{\Omega}$ be the function whose maxima determine the relevant localization regi…

Pemantle–Steif conjecture. For any Heisenberg model on any graph, SB is equivalent to SB+.

Let $\mu_0\in\mathfrak g$ be nonzero, let $\mu:[0,\delta]\to\mathfrak g$ be a regular deformation of $\mu_0$, meaning that $\mu(t)$ is regular for all $t>0$, and set…

Let $q_{\mathrm{ic}}=\delta_{-\zeta}+\delta_{\zeta}$ for $\zeta>0$, set $\mathfrak{m}=1$ and $\xi_1=0$, and consider the variational problem in which $T$ and these data are fixed.…

Let $(\lambda(\alpha),U(x;\alpha))$ and the constants $\alpha_1$, $\alpha_2$, and $\alpha_3$ be as in Theorem, where $\alpha_*<\alpha_1\leq\alpha_2\leq\alpha_3$ describe the degene…