1 Introduction
1.1 Inverse problems
1.2 Background
1.3 Outline of this work

2 Geometric Description
2.1 Basic notations
2.1.1 Admissible polyhedron
2.2 Admissible Riemannian polyhedron
2.2.1 Metric structure
2.2.2 Coordinates
2.3 Distance structure on (M4,g)
2.3.1 Intrinsic distance dx
2.4 Laplace-Beltrami operator
2.5 Spectral problem

3 Gaussian Beams near the Interface
3.1 Gaussian beams (��quasiphotons�� )
3.1.1 Solution form
3.1.2 Main results
3.1.3 Formal series
3.2 Phase functions
3.2.1 Main equations
3.2.2 Required preparations
3.2.3 Impulses pref ptr construction
3.2.4 Qua~tratic forms
3.2.5 Phase functions ON (t,q,��), Ntr(t,q,��)
3.3 Reflection and transmission laws
3.4 Amplitudes
3.4.1 Amplitude values on the interface
3.4.2 Amplitude equations
3.4.3 Initial values
3.5 Exact and Approximate Solutions Estimates
3.5.1 Convergence
3.6 Conclusion

4 Reconstruction of a Smooth Manifold
4.1 BSD on the entire boundary
4.1.1 Formulation of the inverse problem for a smooth Riemannian manifold
4.1.2 Reconstruction of the Fourier coefficients of thewaves
4.1.3 Domains of influence. Tataru's theorems. Wave basis
4.1.4 On the role of Ganssian beams and boundary distance functions
4.2 IP with data given on a part of the boundary
4.2.1 First submanifold reconstruction
4.2.2 Recalculation of the boandary spectral data of (��p,D)
4.2.3 Reconstruction of M

4.2.4 Iterating procedure; Mm = Mint
5 Uniqueness Problem for the Polyhedron
5.1 Formulation of the uniqueness problem
5.2 The Holmgren-John uniqueness theorem
5.3 Uniqueness inverse problem
5.4 Meeting the interface
5.5 Crossing the interface
5.6 Polyhedra isometry

6 Conclusions and Outlook
References
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