# Gabriel P. Paternain's research while affiliated with University of Cambridge and other places

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## Publications (138)

Let $\Sigma$ be a smooth closed oriented surface of genus $\geq 2$. We prove that two metrics on $\Sigma$ with same marked length spectrum and Anosov geodesic flow are isometric (via an isometry isotopic to the identity). This generalizes to the Anosov setting the marked length spectrum rigidity result of Croke and Otal for negatively-curved surfac...

The purpose of this paper is to study transport equations on the unit tangent bundle of closed oriented Riemannian surfaces and to connect these to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow - which...

This up-to-date treatment of recent developments in geometric inverse problems introduces graduate students and researchers to an exciting area of research. With an emphasis on the two-dimensional case, topics covered include geodesic X-ray transforms, boundary rigidity, tensor tomography, attenuated X-ray transforms and the Calderón problem. The p...

This up-to-date treatment of recent developments in geometric inverse problems introduces graduate students and researchers to an exciting area of research. With an emphasis on the two-dimensional case, topics covered include geodesic X-ray transforms, boundary rigidity, tensor tomography, attenuated X-ray transforms and the Calderón problem. The p...

This up-to-date treatment of recent developments in geometric inverse problems introduces graduate students and researchers to an exciting area of research. With an emphasis on the two-dimensional case, topics covered include geodesic X-ray transforms, boundary rigidity, tensor tomography, attenuated X-ray transforms and the Calderón problem. The p...

We study resonant differential forms at zero for transitive Anosov flows on $3$-manifolds. We pay particular attention to the dissipative case, that is, Anosov flows that do not preserve an absolutely continuous measure. Such flows have two distinguished Sinai-Ruelle-Bowen $3$-forms, $\Omega_{\text{SRB}}^{\pm}$, and the cohomology classes $[\iota_{...

We show that we can retrieve a Yang--Mills potential and a Higgs field (up to gauge) from source-to-solution type data associated with the classical Yang--Mills--Higgs equations in Minkowski space $\mathbb{R}^{1+3}$. We impose natural non-degeneracy conditions on the representation for the Higgs field and on the Lie algebra of the structure group w...

We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $$\Sigma $$ Σ with Betti number $$b_1$$ b 1 , the order of vanishing of the Ruelle zeta function at zero equals $$4-b_1$$ 4 - b 1 , while in the hyperbolic case it is equal to $$4-2b_1$$ 4 - 2 b 1 . This is in contrast to the 2-dimensional case where the ord...

This article considers the attenuated transport equation on Riemannian surfaces in the light of a novel twistor correspondence under which matrix attenuations correspond to holomorphic vector bundles on a complex surface. The main result is a transport version of the classical Oka-Grauert principle and states that the twistor space of a simple surf...

We consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form L2↦HT1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepack...

Results by van der Vaart (1991) from semi-parametric statistics about the existence of a non-zero Fisher information are reviewed in an infinite-dimensional non-linear Gaussian regression setting. Information-theoretically optimal inference on aspects of the unknown parameter is possible if and only if the adjoint of the linearisation of the regres...

We show that a connection can be recovered up to gauge from source-to-solution type data associated with the Yang–Mills equations in Minkowski space $${\mathbb {R}}^{1+3}$$ R 1 + 3 . Our proof analyzes the principal symbols of waves generated by suitable nonlinear interactions and reduces the inversion to a broken non-abelian light ray transform. T...

We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials...

We show that for a generic conformal metric perturbation of a hyperbolic 3-manifold $\Sigma$, the order of vanishing of the Ruelle zeta function at zero equals $4-b_1(\Sigma)$, contrary to the hyperbolic case where it is equal to $4-2b_1(\Sigma)$. The result is proved by developing a suitable perturbation theory that exploits the natural pairing be...

For M a simple surface, the nonlinear statistical inverse problem of recovering a matrix field from discrete, noisy measurements of the SO(n)‐valued scattering data CΦ of a solution of a matrix ODE is considered (n ≥ 2). Injectivity of the map Φ ↦ CΦ was established by Paternain, Salo, and Uhlmann in 2012. A statistical algorithm for the solution o...

Bayesian inference and uncertainty quantification in a general class of non-linear inverse regression models is considered. Analytic conditions on the regression model $\{\mathscr G(\theta): \theta \in \Theta\}$ and on Gaussian process priors for $\theta$ are provided such that semi-parametrically efficient inference is possible for a large class o...

This paper settles the question of injectivity of the non-Abelian X-ray transform on simple surfaces for the general linear group of invertible complex matrices. The main idea is to use a factorization theorem for Loop Groups to reduce to the setting of the unitary group, where energy methods and scalar holomorphic integrating factors can be used....

We show that a connection can be recovered up to gauge from source-to-solution type data associated with the Yang-Mills equations in the four dimensional Minkowski space. Our proof analyzes the principal symbols of waves generated by suitable nonlinear interactions and reduces the inversion to a broken non-abelian light ray transform. The principal...

We associate a flow $\phi$ to a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi$ always admits a dominated splitting and identify special cases in which $\phi$ is Anosov. In particular, starting from holomorphic differentials of f...

We consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form $L^2\mapsto H^{1/2}_{T}$, where the $H^{1/2}_{T}$-space is defined using the natural parametrization of geodesics as initial bounda...

We consider Anosov flows on closed 3-manifolds preserving a volume form $\Omega$. Following \cite{DyZw17} we study spaces of invariant distributions with values in the bundle of exterior forms whose wavefront set is contained in the dual of the unstable bundle. Our first result computes the dimension of these spaces in terms of the first Betti numb...

For $M$ a simple surface, the non-linear and non-convex statistical inverse problem of recovering a matrix field $\Phi: M \to \mathfrak{so}(n)$ from discrete, noisy measurements of the $SO(n)$-valued scattering data $C_\Phi$ of a solution of a matrix ODE is considered ($n\geq 2$). Injectivity of the map $\Phi \mapsto C_\Phi$ was established by [Pat...

We consider the statistical inverse problem of recovering a function f : M → ℝ, where M is a smooth compact Riemannian manifold with boundary, from measurements of general X-ray transforms Ia (f) of f , corrupted by additive Gaussian noise. For M equal to the unit disk with “flat” geometry and a = 0 this reduces to the standard Radon transform, but...

We consider the motion of a classical colored spinless particle under the influence of an external Yang–Mills potential A on a compact manifold with boundary of dimension ≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...

We consider the geometric non-linear inverse problem of recovering a Hermitian connection $A$ from the source-to-solution map of the cubic wave equation $\Box_{A}\phi+\kappa |\phi|^{2}\phi=f$, where $\kappa\neq 0$ and $\Box_{A}$ is the connection wave operator in the Minkowski space $\mathbb{R}^{1+3}$. The equation arises naturally when considering...

We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian $2$-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We show that the family of flows...

We show that the existence of a function in $L^{1}$ with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball.

We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an ellipt...

We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and vanishes on the rest. This is done in a geometry with non-positive sectional curvature and a strictly convex o...

In this article we introduce an approach for studying the geodesic X-ray transform and related geometric inverse problems by using Carleman estimates. The main result states that on compact negatively curved manifolds (resp. nonpositively curved simple or Anosov manifolds), the geodesic vector field satisfies a Carleman estimate with logarithmic we...

We show that a properly convex projective structure $\mathfrak{p}$ on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if $\mathfrak{p}$ is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that $\mathfrak{p}$ admits a compatible Weyl connection if and only...

We consider the statistical inverse problem of recovering a function $f: M \to \mathbb R$, where $M$ is a smooth compact Riemannian manifold with boundary, from measurements of general $X$-ray transforms $I_a(f)$ of $f$, corrupted by additive Gaussian noise. For $M$ equal to the unit disk with `flat' geometry and $a=0$ this reduces to the standard...

We establish an equivalence principle between the solenoidal injectivity of the geodesic ray transform acting on symmetric m-tensors and the existence of invariant distributions or smooth first integrals with prescribed projection over the set of solenoidal m-tensors. We work with compact simple manifolds, but several of our results apply to nontra...

We derive reconstruction formulas for a family of geodesic ray transforms with connection, defined on simple Riemannian surfaces. Such formulas provide injectivity of such all transforms in a neighbourhood of constant curvature metrics and non-unitary connections with curvature close to zero. If certain Fredholm equations are injective in the absen...

Consider a compact Riemannian manifold of dimension $\geq 3$ with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-forms. We also show that the conn...

We consider integral geometry inverse problems for unitary connections and
skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The
results apply to manifolds in any dimension, with or without boundary, and also
in the presence of trapped geodesics. In the boundary case, we show injectivity
of the attenuated ray transform on...

We establish an equivalence principle between the solenoidal injectivity of
the geodesic ray transform acting on symmetric $m$-tensors and the existence of
invariant distributions or smooth first integrals with prescribed projection
over the set of solenoidal $m$-tensors. We work with compact simple manifolds,
but several of our results apply to no...

We reconstruct a Riemannian manifold and a Hermitian vector bundle with
compatible connection from the hyperbolic Dirichlet-to-Neumann operator
associated with the wave equation of the connection Laplacian. The boundary
data is local and the reconstruction is up to the natural gauge transformations
of the problem. As a corollary we derive an ellipt...

We consider an exact magnetic flow on the tangent bundle of a closed surface.
We prove that for almost every energy level $\kappa$ below the Ma\~n\'e
critical value of the universal covering there are infinitely many periodic
orbits with energy $\kappa$.

In the recent articles \cite{PSU1,PSU3}, a number of tensor tomography
results were proved on two-dimensional manifolds. The purpose of this paper is
to extend some of these methods to manifolds of any dimension. A central
concept is the surjectivity of the adjoint of the geodesic ray transform, or
equivalently the existence of certain distribution...

We discuss several symplectic aspects related to the Ma\~n\'e critical value
c_u of the universal cover of a Tonelli Hamiltonian. In particular we show that
the critical energy level is never of virtual contact type for manifolds of
dimension greater than or equal to three. We also show the symplectic
invariance of the finiteness of the Peierls bar...

We consider a closed orientable Riemannian 3-manifold $(M,g)$ and a vector
field $X$ with unit norm whose integral curves are geodesics of $g$. Any such
vector field determines naturally a 2-plane bundle contained in the kernel of
the contact form of the geodesic flow of $g$. We study when this 2-plane bundle
remains invariant under two natural alm...

Let L be a convex superlinear autonomous Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mañé in [23]. We define energy levels satisfying the Palais-Smale condition and we show that the critical value of the lift of L to any covering of N equals the infimum of the values of k such that the en...

In this article we define and compute the Novikov Floer homology associated
to a non-resonant magnetic field and a mechanical Hamiltonian on a flat torus
T^{2N}. As a result, we deduce that this Hamiltonian system always has 2N+1
contractible solutions, and generically even 2^{2N} contractible solutions.
Moreover if there exists a non-degenerate no...

We consider exact magnetic flows on closed orientable surfaces. We show that
for almost every energy $\kappa$ below Ma\~n\'e's critical value of the
universal covering there are always at least three distinct closed magnetic
geodesics with energy $\kappa$. If in addition the energy level is assumed to
be non-degenerate we prove existence of infinit...

We survey recent progress in the problem of recovering a tensor field
from its integrals along geodesics. We also propose several open
problems.

We describe the range of the attenuated ray transform of a unitary connection
on a simple surface acting on functions and 1-forms. We use this to determine
the range of the ray transform acting on symmetric tensor fields.

This article considers inverse problems on closed Riemannian surfaces whose
geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and
injectivity of the geodesic ray transform on solenoidal 2-tensors. We also
establish surjectivity results for the adjoint of the geodesic ray transform on
solenoidal tensors. The surjectivity res...

We show that the planar circular restricted three body problem is of
restricted contact type for all energies below the first critical value (action
of the first Lagrange point) and for energies slightly above it. This opens up
the possibility of using the technology of Contact Topology to understand this
particular dynamical system.

We show that the differential in positive equivariant symplectic homology or
linearized contact homology vanishes for non-degenerate Reeb flows with a
continuous invariant Lagrangian subbundle (e.g. Anosov Reeb flows). Several
applications are given, including obstructions to the existence of these flows
and abundance of periodic orbits for contact...

We show that on simple surfaces the geodesic ray transform acting on
solenoidal symmetric tensor fields of arbitrary order is injective. This solves
a long standing inverse problem in the two-dimensional case.

The Floer homology of a cotangent bundle is isomorphic to loop space homology
of the underlying manifold, as proved by Abbondandolo-Schwarz, Salamon-Weber,
and Viterbo. In this paper we show that in the presence of a Dirac magnetic
monopole which admits a primitive with sublinear growth on the universal cover,
the Floer homology in atoroidal free h...

We show that for a simple surface with boundary the attenuated ray transform
in the presence of a unitary connection and a skew-Hermitian Higgs field is
injective modulo the natural obstruction for functions and vector fields. We
also show that the connection and the Higgs field are uniquely determined by
the scattering relation modulo a gauge tran...

We define a new variant of Rabinowitz Floer homology that is particularly
well suited to studying the growth rate of leaf-wise intersections. We prove
that for closed manifolds $M$ whose loop space is "complicated", if $\Sigma$ is
a non-degenerate fibrewise starshaped hypersurface in $T^*M$ and $\phi$ is a
generic Hamiltonian diffeomorphism then th...

Let $M$ be a closed orientable Riemannian surface. Consider an
SO(3)-connection $A$ and a Higgs field $\Phi:M\to so(3)$. The pair $(A,\Phi)$
naturally induces a cocycle over the geodesic flow of $M$. We classify (up to
gauge transformations) cohomologically trivial pairs $(A,\Phi)$ with finite
Fourier series in terms of a suitable B\"acklund transf...

We construct examples of Tonelli Hamiltonians on
\mathbbTn{\mathbb{T}^n} (for any n≥ 2) such that the hypersurfaces corresponding to the Mañé critical value are stable (i.e. geodesible). We also provide a
criterion for instability in terms of closed orbits in free homotopy classes and we show that any stable energy level of a
Tonelli Hamiltonian...

In this note we study two index questions. In the first we establish the
relationship between the Morse indices of the free time action functional and
the fixed time action functional. The second is related to Rabinowitz Floer
homology. Our index computations are based on a correction term which is
defined as follows: around a non-degenerate Hamilt...

We study the dynamics and symplectic topology of energy hypersurfaces of mechanical Hamiltonians on twisted cotangent bundles. We pay particular attention to periodic orbits, displaceability, stability and the contact type property, and the changes that occur at the Mañé critical value c. Our main tool is Rabinowitz Floer homology. We show that it...

We construct examples of Tonelli Hamiltonians on $\T^n$ (for any $n\geq 2$) such that the hypersurfaces corresponding to the Ma\~n\'e critical value are stable (i.e. geodesible). We also provide a criterion for instability in terms of closed orbits in free homotopy classes and we show that any stable energy level of a Tonelli Hamiltonian must conta...

We give an example of a symplectic manifold with a stable hypersurface such that nearby hypersurfaces are typically unstable. Comment: 9 pages

Let $M$ be a closed orientable surface of negative curvature. A connection is said to be transparent if its parallel transport along closed geodesics is the identity. We describe all transparent SU(2)-connections and we show that they can be built up from suitable B\"acklund transformations. Comment: 10 pages

Consider the tangent bundle of a Riemannian manifold $(M,g)$ of dimension
$n\geq3$ admitting a metric of negative curvature (not necessarily equal to
$g$) endowed with a twisted symplectic structure defined by a closed 2-form on
$M$. We consider the Hamiltonian flow generated (with respect to that
symplectic structure) by the standard kinetic energ...

We establish a relationship between the helicity of a magnetic flow on a closed surface of genus ≥ 2 and the Mañé critical value. 1. Results Let N be a closed oriented 3-manifold with a volume form Ω. A vector field F on N that preserves Ω is said to be null-homologous or exact if the closed 2-form iFΩ is exact. Given a volume preserving null-homol...

Let $(M,g)$ be a closed oriented negatively curved surface. A unitary connection on a Hermitian vector bundle over $M$ is said to be transparent if its parallel transport along the closed geodesics of $g$ is the identity. We study the space of such connections modulo gauge and we prove a classification result in terms of the solutions of certain PD...

We consider a magnetic flow without conjugate points on a closed manifold $M$ with generating vector field $\G$. Let $h\in C^{\infty}(M)$ and let $\theta$ be a smooth 1-form on $M$. We show that the cohomological equation \[\G(u)=h\circ \pi+\theta\] has a solution $u\in C^{\infty}(SM)$ only if $h=0$ and $\theta$ is closed. This result was proved in...

We consider Anosov thermostats on a closed surface and the X-ray transform on functions which are up to degree two in the velocities. We show that the subspace where the X-ray transform fails to be s-injective is finite dimensional. Furthermore, if the surface is negatively curved and the thermostat is pure Gaussian (i.e. no magnetic field is prese...

We consider magnetic flows on compact quotients of the 3-dimensional solvable geometry Sol determined by the usual left-invariant metric and the distinguished monopole. We show that these flows have positive Liouville entropy and therefore are never completely integrable. This should be compared with the known fact that the underlying geodesic flow...

We explore the relationship between contact forms on $\mathbb S^3$ defined by Finsler metrics on $\mathbb S^2$ and the theory developed by H. Hofer, K. Wysocki and E. Zehnder in \cite{HWZ,HWZ1}. We show that a Finsler metric on $\mathbb S^2$ with curvature $K\geq 1$ and with all geodesic loops of length $>\pi$ is dynamically convex and hence it has...

We show that an arbitrary Anosov Gaussian thermostat on a surface is
dissipative unless the external field has a global potential.

For a compact Riemannian manifold with boundary, endowed with a magnetic potential α, we consider the problem of restoring the metric g and the magnetic potential α from the values of the Mañé action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this case, knowledge of the Mañé action...

We show that an arbitrary Anosov Gaussian thermostat close to equilibrium has positive entropy poduction unless the external field E has a global potential. The configuration space is allowed to have any dimension and magnetic forces are also allowed. We also show the following non-perturbative result. Suppose a Gaussian thermostat satisfies \(K_w(...

Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$. We consider the flow $\phi$ determined by the motion of a particle under the influence of a magnetic field $\Omega$ and a thermostat with external field ${\bf e}$. We show that if $\phi$ is Anosov, then it has weak stable and unstable foliations of class $C^{1,1}$ if and onl...

We consider an optical hypersurface $\Sigma$ in the cotangent bundle $\tau:T^*M\to M$ of a closed manifold $M$ endowed with a twisted symplectic structure. We show that if the characteristic foliation of $\Sigma$ is Anosov, then a smooth 1-form $\theta$ on $M$ is exact if and only $\tau^*\theta$ has zero integral over every closed characteristic of...

Let M be a closed oriented surface and let Ω be a non-exact 2-form. Suppose that the magnetic flow φ of the pair (g,Ω) is Anosov. We show that the longitudinal KAM-cocycle of φ is a coboundary if and only if the Gaussian curvature is constant and Ω is a constant multiple of the area form thus extending the results in [the second author, Math. Proc....

We construct F-structures on a Bott manifold and on some other manifolds obtained by Kummer-type constructions. We also prove that if M=E#X, where E is a fiber bundle with structure group G and a fiber admitting a G-invariant metric of non-negative sectional curvature and X admits an F-structure with one trivial covering, then one can construct on...

Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $\Omega$ be a 2-form. We show that the magnetic flow of the pair $(g,\Omega)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has constant Gaussian curvature, $\Omega$ is a constant multiple of the area form of $g$ and the magnetic flow is a...

We study the existence of Riemannian metrics with zero topological entropy on a closed manifold M with infinite fundamental group. We show that such a metric does not exist if there is a finite simply connected CW complex which maps to M in such a way that the rank of the map induced in the pointed loop space homology grows exponentially. This resu...

Let $M$ be a closed oriented surface of negative Gaussian curvature and let $\Omega$ be a non-exact 2-form. Let $\lambda$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la \Omega$ is a coboundary if and only if the Gaussian curvature is constant and $\Omega$ is a constant multiple of the a...

## Citations

... Concluding this introduction, we mention that the Nash-Moser implicit function theorem also finds application to related inverse problems on Riemannian surfaces, see [20,16]. ...

... In the case of a strictly convex obstacle, the uniqueness result for the broken ray transform of scalar functions on Riemannian surfaces of nonpositive curvature were obtained in [IS16]. This result was later generalized to higher dimensions and tensor fields of any order in [IP22]. In the case of rotational (or spherical) symmetry, one may sometimes solve these and related problems using local results and data avoiding the obstacle when the manifold satisfies the Herglotz condition [dHIK22,IM23,Sha97]. ...

... This has come up in a few places: in the elliptic inverse boundary value problems in which the CGO solutions are constructed by using the Gaussian Beams -they concentrate along geodesics in an asymptotic limit, yielding integrals of the quantities as in Step (4) [11,21,23]. Other applications include [66], where the authors determine a matrix X-ray transform from the scattering relation and thus reduce the problem. There are many other occasions where a geometric inverse problem is reduced to another one -one such important example is the proof of boundary rigidity for surfaces by Pestov and Uhlmann [67], the main part of which determines the DN map data for the metric from the boundary distances. ...

Reference: The Calderón problem for connections

... The nonlinear map A → C A is called the non-Abelian X-ray transform. The inverse problem of recovering an attenuation A from measurements of its scattering data C A has been subject of a number of recent papers [30,29,25] (with earlier in work [46,42,10,26,8]) and the question of injectivity is now well understood in the following setting: Let G ⊂ GL(n, C) be a Lie group with Lie algebra g and suppose that A is given in terms of a 1-form A ∈ Ω 1 (M, g) and a matrix field Φ ∈ C ∞ (M, g) as ...

... In higher dimensions, boundary rigidity is known for simple conformal metrics [36,17], generic simple manifolds including all analytic ones [44], metrics close to Euclidean or hyperbolic [25,18,33,12,11], and for manifolds foliated by strictly convex hypersurfaces [45,46]. See also the surveys [31,48,47,40]. ...

... In proofs of various cases of the Fried conjecture, proving meromorphic extension and regularity at zero is part of the problem. Separate proofs of meromorphic extension have also been given, such as the ones in [4,7,10,35]. In the case G = Z, a result related to convergence and meromorphic extension of the equivariant Ruelle ζ-function is proved for geodesic flows on sphere bundles of surfaces in [5]. ...

... The main focus of the literature so far has been the recovery of the underlying potential function f . Minimax posterior contraction rates using multi-scale analysis were derived for Gaussian priors in Nickl, van de Geer and Wang (2020); Monard, Nickl and Paternain (2021) and uniform sequence priors in Nickl (2018). Furthermore, semi-parametric Bernstein-von Mises results on linear functionals were derived in Nickl (2018); Monard, Nickl and Paternain (2021). ...

... For example, based on the scheme, the nonlinear interactions of distorted plane waves were analyzed to recover the metric of a Lorentzian space-time manifold and nonlinear coefficients using the measurements of solutions to nonlinear hyperbolic equations [28,40,48]. In contrast the underlying problems for linear hyperbolic equations are still open, see also [9,40] and the references therein. The method is also applied to study elliptic equations with power-type nonlinearities, including stationary nonlinear Schrödinger equations and magnetic Schrödinger equations, see [26,27,29,30,31,36,37,41]. A demonstration of the method can be found in [2,3] on nonlinear Maxwell's equations, in [32,33] on nonlinear kinetic equations, and in [38] on semilinear wave equations. ...

... Krieger et al. [15] discussed some key issues such as renormalization and blow up for the critical Yang-Mills problem. The result of a connection can be recovered up to gauge from source to solution type data associated with the Yang-Mills equations in Minkowski space R 1+3 has been proved in [5]. Earlier results include existence and uniqueness for higher-order Sobolev norms by Segal [19]. ...

... Having sorted out the resonant forms at zero, the only remaining obstacle to compute the order of vanishing at zero of the Ruelle zeta function using geometric multiplicities is semisimplicity for L X acting on Ω 1 0 . As explained in [CP20] this is a subtle issue that could in principle depend on the parametrisation of the flow. For example [CP20,Theorem 1.4] shows that there are time changes of geodesic flows of hyperbolic surfaces that are not semisimple. ...