12 problems

Let $(M,g_1)$ and $(M,g_2)$ be compact Riemannian manifolds with smooth boundary, with $\dim(M)\geq 3$. For each metric, let $C_{g_i}$ denote the Cauchy data set of harmonic functi…

Modified anisotropic Calderón conjecture at nonzero frequency. If

Anisotropic Calderón conjecture at zero frequency. If

Let $n \geq 3$ and let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain. Suppose that $\gamma_1,\gamma_2 \in W^{1,n}(\Omega)$ are uniformly elliptic, and let…

Let $d/2\leq p<\infty$, let $\Omega$ be a bounded domain with Lipschitz boundary, and let $H_c^{s,p}(\Omega)$ denote the compactly supported Sobolev space of potentials. For a pote…

Anisotropic partial Calderón conjecture. It should follow that $g=\tilde{g}$ up to the diffeomorphism gauge invariance if $\dim M\geq 3$, and up to the diffeomorphism and conformal…

Let $(M,g)$ be a compact Riemannian manifold with smooth boundary. Let $f \in C^{\infty}(M)$ vanish to infinite order at $\partial M$ and satisfy … for all $u_j \in C^{\infty}(M)$…

Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and assume that $q_1, q_2 \in C^{\infty}(M)$ so that $0$ is not a Dirichlet eigenvalue of $\Delta_g + q_j$ in $M$…

Uniqueness conjecture for the Schrödinger potential. The knowledge of $\Lambda_{g,q}$ uniquely determines $q$. This is the potential-recovery version of the Calderón problem in a k…

Let $(M,g)$ be a Riemannian manifold, let $E$ be a Hermitian vector bundle over $M$, and let $A$ and $B$ be unitary connections on $E$. Denote by $\Lambda_A$ and $\Lambda_B$ the Di…

Two-point determination conjecture. If both

Let $(M,g_1)$ and $(M,g_2)$ be two compact oriented Riemannian manifolds with smooth boundary and dimension $n\geq 3$. For a metric $g$, let … be the Cauchy data set of harmonic fu…