**Title:** Boundary rigidity of Riemannian manifolds

**Speaker:** Professor Plamen Stefanov

**Speaker Info:** Purdue

**Brief Description:**

**Special Note**:

**Abstract:**

This talk is based on a series of joint works with Gunther Uhlmann. Let $(M,g)$ be a compact Riemannian manifold with boundary. Assume that for each pair of boundary points (x,y), we know the distance function $d(x,y)$. The manifold $M$ is called boundary rigid, if it is determined uniquely by $d(x,y)$ (known on the boundary). At present, boundary rigidity is known only for some classes of manifolds.We study this problem for simple manifolds, i.e., strictly convex ones with no caustics inside. The associated linearized problem is recovery of the so-called solenoidal part of a tensor from its X-ray transform $If$ along geodesics. We show that $N=I^*I$ is a pseudodifferential operator, analyze its principal symbol and invert it microlocally. We prove an a priori stability estimates for $N$. Using analytic pseudodifferential calculus, we show that for analytic metrics, the linearized problem is invertible. For the non-linear one, we prove generic local uniqueness and stability.

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