Inverse problems on Riemannian manifolds, spring 2010

Lecturer

Mikko Salo

Scope

5 cu.

Type

Advanced studies.

Lectures

IV period, Mon 12-14 B322 and Thu 14-16 C124 (Easter holiday 1.-7.4.). The first lecture is on 15th March.

Lecture notes

Lecture notes (more or less final version, September 2010)

Exercises

Exercises

Course description

This course is an introduction to Calderón's inverse conductivity problem on Riemannian manifolds. This problem arises as a model for electrical imaging in anisotropic media, and it is one of the most basic inverse problems in a geometric setting. The problem is still largely open, but we will discuss recent developments based on complex geometrical optics and the geodesic X-ray transform.

The course is an independent sequel to the class Inverse problems on Riemann surfaces given by Leo Tzou in Fall 2009. The main difference is that in this course we will consider manifolds of dimension three and higher, where one has to rely on real variable methods instead of using complex analysis. The course can be considered as an introduction to geometric inverse problems, but also as an introduction to the use of real analysis methods in the setting of Riemannian manifolds.

Prerequisites

Real analysis I, Introduction to differential geometry, basics of Riemannian geometry.

Bibliography

We will follow parts of the articles

D. Dos Santos Ferreira, C. Kenig, M. Salo, and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178 (2009), p. 119-171, arXiv:0803.3508.
C. Kenig, M. Salo, and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations. Preprint, arXiv:0905.3275.

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