5 problems

In a periodic energy surface, let the level-set trajectories induced by a magnetic field include strongly chaotic trajectories, meaning trajectories whose lifts to the plane have n…

Let a ball contain two conducting materials with conductivities $\beta$ and $\beta$ and with a fixed prescribed proportion. Consider the first eigenvalue of the associated Dirichle…

Let $C^{\mathrm{+}}_{v}(\xi(p))\subset\overline{\xi(p)^c}$, for $v\in(0,u^{\mathrm{+}}_{0})$, and let $C^{\mathrm{-}}_{u}(\xi(p))\subset\xi(p)$, for $u\in(0,u^{\mathrm{-}}_{0})$, b…

Isotropic realizability conjecture. Then $\nabla u$ is isotropically realizable in the torus with a positive conductivity $\sigma$ such that…

Let $\Xi_0$ be the set of chemical potentials for which the zero-temperature conductivity measure and in-phase linear-response current have been constructed, and let…