# Geometry and Inverse Problems (13w5036)

Arriving in Banff, Alberta Sunday, September 15 and departing Friday September 20, 2013

## Organizers

Gabriel Paternain (University of Cambridge)

Mikko Salo (University of Jyväskylä)

Gunther Uhlmann (University of Washington)

## Objectives

Inverse problems appear naturally in many areas of mathematics. This workshop will focus on several remarkable inverse problems which occur in Geometry broadly understood. Geometric thinking may also have impact in the resolution of inverse problems in PDEs, geophysics or medical imaging. There has been great activity in recent years around this circle of ideas, and the time seems ripe to bring together specialists with expertise in different aspects of these interconnected problems with the hope of making substantial further progress.

Below is a selection of geometric inverse problems that will be addressed at the workshop. Other closely related topics coming from conformal geometry, geometric scattering theory, and spectral geometry will also be considered.

Boundary and lens rigidity

An important geometric inverse problem consists in determining a Riemannian metric on a bounded set of Euclidean space, or on a manifold with boundary, from the boundary distance function, which is defined as the lengths of geodesics joining points of the boundary. Physically, these are the first arrival times of geodesics (rays) going through the domain. This is known in the geophysics literature as the inverse kinematic problem and in differential geometry as the boundary rigidity problem. There are known obstructions to determining a Riemannian metric up to isometry from the boundary distance function, so one needs some restriction on the metric. One such restriction is that the manifold is simple: any two points can be joined by a unique geodesic depending smoothly on the endpoints and the boundary is strictly convex. The conjecture proposed by Michel (1981) is that simple Riemannian manifolds with boundary can be determined uniquely up to an isometry from the boundary distance function.

The conjecture was proven in dimension two by Pestov and Uhlmann (2005) making a connection between the boundary distance function and the Dirichlet to Neumann map for a Riemannian metric discussed below. In higher dimensions this conjecture is unproven and this is one of the topics that will be discussed in the workshop. It was shown by Stefanov and Uhlmann (2005) that the conjecture holds locally and generically near an open and dense set of simple metrics. Burago and Ivanov (2010) extended the semiglobal result of Lassas, Sharafutdinov and Uhlmann (2003) showing that any metric close to Euclidean is boundary rigid.

For non-simple manifolds one can consider the behavior of all the geodesics going through the domain, not just the minimizing ones. This information is encoded in the scattering relation, which maps the point and direction of entrance of a geodesic to the point and direction of exit. The scattering relation was defined by Guillemin (1976) in the context of scattering theory. The natural lens rigidity conjecture is that for non-trapping manifolds the scattering relation determines the metric up to isometry. The lens rigidity and boundary rigidity problems are equivalent for simple manifolds. There are partial results about this conjecture on non-simple manifolds, due to Croke, Herreros, Stefanov-Uhlmann, and Vargo. Lens rigidity will also be one of the major topics that we will consider in the workshop.

One can discuss analogous problems to boundary rigidity and lens rigidity for more general flows. For instance, many of the results mentioned have been generalized by Dairbekov, Paternain, Stefanov and Uhlmann to magnetic geodesics (2007).

The Calderón and Gel'fand problems

Geometric inverse problems arise already in connection with the famous inverse conductivity problem due to Calderón. This problem is the model for Electrical Impedance Tomography, an imaging modality proposed for use in medical and seismic imaging. In the case of anisotropic media in dimensions three and higher, the problem reduces to the question of determining a smooth Riemannian manifold from the elliptic Dirichlet to Neumann map (or Cauchy data of harmonic functions) on its boundary. This is one of the major open questions related to the Calderón problem. Recently, Dos Santos Ferreira, Kenig, Salo and Uhlmann (2009) opened a new direction of research in the problem by showing that certain smooth manifolds in a fixed conformal class are determined by the Dirichlet to Neumann map. This result connects with the integral geometry questions described below, since the attenuated geodesic ray transform appears as a main ingredient in the proofs. The corresponding two-dimensional question for Schrödinger operators on Riemann surfaces was taken up by Guillarmou and Tzou (2011) based on methods introduced by Bukhgeim. This work raises the interesting possibility of extensions to complex manifolds in higher dimensions.

The geometric Calderón problem is also connected to another major question in geometric inverse problems, the Gel'fand problem which reduces to determining a Riemannian manifold from the hyperbolic Dirichlet to Neumann map. This problem has been thoroughly studied in a number of situations by Kurylev, Lassas and others using the Boundary Control method introduced by Belishev, and recent work of Kurylev, Lassas and Yamaguchi (2010) extends uniqueness results for this problem to the case of partly collapsed manifolds. The relation to the Calderón problem arises since the special form of manifolds treated in the elliptic case formally corresponds to the Lorentz metrics considered in the Boundary Control method. There is ongoing work by Dos Santos Ferreira, Kurylev, Lassas and Salo which exploits this relation. We mention that geodesic ray transforms are naturally encountered also in hyperbolic inverse problems on manifolds.

Tensor tomography

The geodesic ray transform, where one integrates a function or a tensor field along geodesics of a Riemannian metric, is closely related to the boundary rigidity problem. The integration of a function along geodesics is the linearization of the boundary rigidity problem in a fixed conformal class. The standard X-ray transform, where one integrates a function along straight lines, corresponds to the case of the Euclidean metric and is the basis of medical imaging techniques such as CT and PET. The case of integration along more general geodesics arises in geophysical imaging in determining the inner structure of the Earth since the speed of elastic waves generally increases with depth, thus curving the rays back to the Earth surface. It also arises in ultrasound imaging, where the Riemannian metric models the anisotropic index of refraction. In tensor tomography problems one would like to determine a symmetric tensor field up to natural obstruction from its integrals over geodesics.

The case of integrating tensors of order one corresponds to the geodesic Doppler transform in which one integrates a vector field along geodesics. This transform appears in ultrasound tomography and non-invasive industrial measurements. The integration of tensors of order two along geodesics, also known as deformation boundary rigidity, arises as the linearization of the boundary rigidity problem. The case of tensor fields of rank four describes the perturbation of travel times of compressional waves propagating in slightly anisotropic elastic media.

There are recent advances on tensor tomography on manifolds due to Dairbekov (2006), Sharafutdinov (2007) and Stefanov and Uhlmann (2009). Recently Paternain, Salo and Uhlmann (2011) settled the tensor tomography problem for simple surfaces by proving that the ray transform is injective on symmetric tensors of any order up to the natural obstruction. The proof introduces new methods and makes a connection to the attenuated ray transforms described below and also to methods in Complex Geometry such as the Kodaira Vanishing Theorem. This problem remains open on higher dimensional simple Riemannian manifolds, and there are related open problems on closed manifolds.

Attenuated ray transforms

There has been considerable activity in the study of ray transforms involving exponential attenuation factors. Besides their importance in imaging technology, they arise naturally in various inverse problems in geometry as explained above. In the case of Euclidean space with the Euclidean metric the attenuated ray transform is the basis of the medical imaging technology of SPECT and has been extensively studied. There are two natural extensions where this transform appears: one is to replace Euclidean space by a Riemannian manifold, and the other is to consider the case of systems where the attenuation is given for example by a unitary connection.

Injectivity of the attenuated ray transform in the Euclidean case was proved by Arbuzov, Bukhgeim and Kazantsev (1998) and an inversion formula was provided by Novikov (2002). Recently, Salo and Uhlmann (2010) proved that the attenuated ray transform is injective for simple two dimensional manifolds. In the case of systems, instead of scalar attenuation functions one can consider attenuations given by pairs of connections and Higgs fields on the trivial bundle on the manifold. These pairs often appear in the so-called Yang-Mills-Higgs theories. A good example of this is the Bogomolny equation in Minkowski (2+1)-space which appears as a reduction of the self-dual Yang-Mills equation in (2+2)-space and has been object of intense study in the literature of Solitons and Integrable Systems. There is a remarkable connection between the Bogomolny equation and scattering data which deserves further attention.

In recent work, Paternain, Salo and Uhlmann (2011) proved that the attenuated ray transform is injective for unitary pairs and simple surfaces. Injectivity for simple Riemannian manifolds in higher dimensions, and removing the condition that the pair is unitary, i.e. considering other structure groups like GL(n,C), are open questions. Injectivity properties of attenuated ray transforms have several applications. One of them implemented by Paternain, Salo and Uhlmann (2011) for arbitrary simple surfaces is to recover a unitary connection from the scattering data given by parallel transport along geodesics. Previous results in this direction for domains in Euclidean space were obtained by Finch and Uhlmann (2001), Novikov (2002) and Eskin (2004), and for simple Riemannian manifolds but for small connections by Sharafutdinov (2000). The problem of determining a unitary connection from its parallel transport on a simple Riemannian manifold of any dimension remains a major challenge.

Below is a selection of geometric inverse problems that will be addressed at the workshop. Other closely related topics coming from conformal geometry, geometric scattering theory, and spectral geometry will also be considered.

Boundary and lens rigidity

An important geometric inverse problem consists in determining a Riemannian metric on a bounded set of Euclidean space, or on a manifold with boundary, from the boundary distance function, which is defined as the lengths of geodesics joining points of the boundary. Physically, these are the first arrival times of geodesics (rays) going through the domain. This is known in the geophysics literature as the inverse kinematic problem and in differential geometry as the boundary rigidity problem. There are known obstructions to determining a Riemannian metric up to isometry from the boundary distance function, so one needs some restriction on the metric. One such restriction is that the manifold is simple: any two points can be joined by a unique geodesic depending smoothly on the endpoints and the boundary is strictly convex. The conjecture proposed by Michel (1981) is that simple Riemannian manifolds with boundary can be determined uniquely up to an isometry from the boundary distance function.

The conjecture was proven in dimension two by Pestov and Uhlmann (2005) making a connection between the boundary distance function and the Dirichlet to Neumann map for a Riemannian metric discussed below. In higher dimensions this conjecture is unproven and this is one of the topics that will be discussed in the workshop. It was shown by Stefanov and Uhlmann (2005) that the conjecture holds locally and generically near an open and dense set of simple metrics. Burago and Ivanov (2010) extended the semiglobal result of Lassas, Sharafutdinov and Uhlmann (2003) showing that any metric close to Euclidean is boundary rigid.

For non-simple manifolds one can consider the behavior of all the geodesics going through the domain, not just the minimizing ones. This information is encoded in the scattering relation, which maps the point and direction of entrance of a geodesic to the point and direction of exit. The scattering relation was defined by Guillemin (1976) in the context of scattering theory. The natural lens rigidity conjecture is that for non-trapping manifolds the scattering relation determines the metric up to isometry. The lens rigidity and boundary rigidity problems are equivalent for simple manifolds. There are partial results about this conjecture on non-simple manifolds, due to Croke, Herreros, Stefanov-Uhlmann, and Vargo. Lens rigidity will also be one of the major topics that we will consider in the workshop.

One can discuss analogous problems to boundary rigidity and lens rigidity for more general flows. For instance, many of the results mentioned have been generalized by Dairbekov, Paternain, Stefanov and Uhlmann to magnetic geodesics (2007).

The Calderón and Gel'fand problems

Geometric inverse problems arise already in connection with the famous inverse conductivity problem due to Calderón. This problem is the model for Electrical Impedance Tomography, an imaging modality proposed for use in medical and seismic imaging. In the case of anisotropic media in dimensions three and higher, the problem reduces to the question of determining a smooth Riemannian manifold from the elliptic Dirichlet to Neumann map (or Cauchy data of harmonic functions) on its boundary. This is one of the major open questions related to the Calderón problem. Recently, Dos Santos Ferreira, Kenig, Salo and Uhlmann (2009) opened a new direction of research in the problem by showing that certain smooth manifolds in a fixed conformal class are determined by the Dirichlet to Neumann map. This result connects with the integral geometry questions described below, since the attenuated geodesic ray transform appears as a main ingredient in the proofs. The corresponding two-dimensional question for Schrödinger operators on Riemann surfaces was taken up by Guillarmou and Tzou (2011) based on methods introduced by Bukhgeim. This work raises the interesting possibility of extensions to complex manifolds in higher dimensions.

The geometric Calderón problem is also connected to another major question in geometric inverse problems, the Gel'fand problem which reduces to determining a Riemannian manifold from the hyperbolic Dirichlet to Neumann map. This problem has been thoroughly studied in a number of situations by Kurylev, Lassas and others using the Boundary Control method introduced by Belishev, and recent work of Kurylev, Lassas and Yamaguchi (2010) extends uniqueness results for this problem to the case of partly collapsed manifolds. The relation to the Calderón problem arises since the special form of manifolds treated in the elliptic case formally corresponds to the Lorentz metrics considered in the Boundary Control method. There is ongoing work by Dos Santos Ferreira, Kurylev, Lassas and Salo which exploits this relation. We mention that geodesic ray transforms are naturally encountered also in hyperbolic inverse problems on manifolds.

Tensor tomography

The geodesic ray transform, where one integrates a function or a tensor field along geodesics of a Riemannian metric, is closely related to the boundary rigidity problem. The integration of a function along geodesics is the linearization of the boundary rigidity problem in a fixed conformal class. The standard X-ray transform, where one integrates a function along straight lines, corresponds to the case of the Euclidean metric and is the basis of medical imaging techniques such as CT and PET. The case of integration along more general geodesics arises in geophysical imaging in determining the inner structure of the Earth since the speed of elastic waves generally increases with depth, thus curving the rays back to the Earth surface. It also arises in ultrasound imaging, where the Riemannian metric models the anisotropic index of refraction. In tensor tomography problems one would like to determine a symmetric tensor field up to natural obstruction from its integrals over geodesics.

The case of integrating tensors of order one corresponds to the geodesic Doppler transform in which one integrates a vector field along geodesics. This transform appears in ultrasound tomography and non-invasive industrial measurements. The integration of tensors of order two along geodesics, also known as deformation boundary rigidity, arises as the linearization of the boundary rigidity problem. The case of tensor fields of rank four describes the perturbation of travel times of compressional waves propagating in slightly anisotropic elastic media.

There are recent advances on tensor tomography on manifolds due to Dairbekov (2006), Sharafutdinov (2007) and Stefanov and Uhlmann (2009). Recently Paternain, Salo and Uhlmann (2011) settled the tensor tomography problem for simple surfaces by proving that the ray transform is injective on symmetric tensors of any order up to the natural obstruction. The proof introduces new methods and makes a connection to the attenuated ray transforms described below and also to methods in Complex Geometry such as the Kodaira Vanishing Theorem. This problem remains open on higher dimensional simple Riemannian manifolds, and there are related open problems on closed manifolds.

Attenuated ray transforms

There has been considerable activity in the study of ray transforms involving exponential attenuation factors. Besides their importance in imaging technology, they arise naturally in various inverse problems in geometry as explained above. In the case of Euclidean space with the Euclidean metric the attenuated ray transform is the basis of the medical imaging technology of SPECT and has been extensively studied. There are two natural extensions where this transform appears: one is to replace Euclidean space by a Riemannian manifold, and the other is to consider the case of systems where the attenuation is given for example by a unitary connection.

Injectivity of the attenuated ray transform in the Euclidean case was proved by Arbuzov, Bukhgeim and Kazantsev (1998) and an inversion formula was provided by Novikov (2002). Recently, Salo and Uhlmann (2010) proved that the attenuated ray transform is injective for simple two dimensional manifolds. In the case of systems, instead of scalar attenuation functions one can consider attenuations given by pairs of connections and Higgs fields on the trivial bundle on the manifold. These pairs often appear in the so-called Yang-Mills-Higgs theories. A good example of this is the Bogomolny equation in Minkowski (2+1)-space which appears as a reduction of the self-dual Yang-Mills equation in (2+2)-space and has been object of intense study in the literature of Solitons and Integrable Systems. There is a remarkable connection between the Bogomolny equation and scattering data which deserves further attention.

In recent work, Paternain, Salo and Uhlmann (2011) proved that the attenuated ray transform is injective for unitary pairs and simple surfaces. Injectivity for simple Riemannian manifolds in higher dimensions, and removing the condition that the pair is unitary, i.e. considering other structure groups like GL(n,C), are open questions. Injectivity properties of attenuated ray transforms have several applications. One of them implemented by Paternain, Salo and Uhlmann (2011) for arbitrary simple surfaces is to recover a unitary connection from the scattering data given by parallel transport along geodesics. Previous results in this direction for domains in Euclidean space were obtained by Finch and Uhlmann (2001), Novikov (2002) and Eskin (2004), and for simple Riemannian manifolds but for small connections by Sharafutdinov (2000). The problem of determining a unitary connection from its parallel transport on a simple Riemannian manifold of any dimension remains a major challenge.