Elements of noncommutative geometry in inverse problems on manifolds

Abstract

We deal with two dynamical systems associated with a Riemannian manifold with boundary. The first one is a system governed by the scalar wave equation, the second is governed by the Maxwell equations. Both of the systems are controlled from the boundary. The inverse problem is to recover the manifold via the relevant measurements at the boundary (inverse data). We show that the inverse data determine a C*-algebras, whose (topologized) spectra are identical to the manifold. By this, to recover the manifold is to determine a proper algebra from the inverse data, find its spectrum, and provide the spectrum with a Riemannian structure. The paper develops an algebraic version of the boundary control method, which is an approach to inverse problems based on their relations to control theory

Noncommutative geometry and the tomography of manifolds

“…The key to using eikonals in the inverse problem is the following result [6]. We have the representation (3.5) ε σ y = τ σ y + Ky, in which τ σ is understood as an operator from C into H , multiplying fields pointwise by functions τ σ := dist(·, σ), and the operator K : C → H is compact.…”

“…As shown in [6],Ė is a commutative Banach algebra, isometrically isomorphic to the algebra of scalar eikonals T. The isometryĖ D −→ T is given by the connection of generatorsĖ As shown in [6],Ė is a commutative Banach algebra, isometrically isomorphic to the algebra of scalar eikonals T. The isometryĖ D −→ T is given by the connection of generatorsĖ…”

“…6 This structure defines a family of conformally equivalent metrics compatible with it. We find the spectrum A (Γ) =: Ω and define the algebra A ( Ω) = GA (Γ) ⊂ C c ( Ω).…”

“…We identify Γ with Γ according to (1.6). 6 Functions from A ( Ω) play the role of local complex coordinates. In addition, the metrics are connected by the relation g = ρj * g with an (unknown) smooth function ρ > 0.…”

Eikonal Algebra on a Graph of Simple Structure

“…These algebras are used for reconstruction of Riemannian manifolds via dynamical and/or spectral boundary inverse data. Namely, these data determine the relevant eikonal algebra, which is a commutative C*-algebra, whereas its spectrum (a set of irreducible representations) provides an isometric copy of the manifold under reconstruction and, thus, solves the problem [5,6,7].…”