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2000 Mathematics Subject Classification: 53C24, 53C65, 53C21. This is a survey of the recent results by the author and Gunther Uhlmann on the boundary rigidity problem and on the associated tensor tomography problem. Author partly supported by NSF Grant DMS-0400869.

We study semiclassical resonances in a box $\Omega \left(h\right)$ of height $h^N$, $N\gg 1$. We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set $𝒯$ of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator $P^{\#}\left(h\right)$ with discrete spectrum the number of resonances in $\Omega \left(h\right)$ is bounded by the number of eigenvalues of $P^{\#}\left(h\right)$ in an interval a bit larger than the projection of $\Omega \left(h\right)$ on the real line. As an application, we prove a...

We prove an estimate of the kind $\parallel q_1-q_2{\parallel }_{L^{\infty }}\le C\varphi (\parallel A_{q_1}-A_{q_2}{\parallel }_{R,3/2-1/2})$, where $A_{q_i}(\omega ,\theta )$, $i=1,2$ is the scattering amplitude related to the compactly supported potential $q_i\left(x\right)$ at a fixed energy level $k=$ const., $\varphi \left(t\right)=(-lnt{)}^{-\delta }$, $0\<\delta \<1$ and $\parallel ·{\parallel }_{R,3/2-1/2}$ is a suitably defined norm.