The Ruelle Zeta Function of a chaotic (Anosov) flow is a meromorphic function in the complex plane defined as an infinite product over closed orbits. Its behaviour at zero is expected to carry interesting topological and dynamical information, and is encoded in certain resonant spaces of differential forms for the action of the Lie derivative on suitable spaces with anisotropic regularity. In this talk, I will introduce a new notion of helicity (average self-linking), and explain how this can be used to compute the resonant spaces for any Anosov flow in 3D, with particular emphasis in the dissipative (non volume-preserving) case. A prototype example is given by the geodesic flow of an affine connection with torsion and we shall see that for such a flow the order of vanishing drops by 1 in relation to the usual geodesic flow due to the Sinai-Ruelle-Bowen measure having non-zero winding cycle.

Many different ray transform problems arise from seismology. My examples are periodic ray transform problems in the presence of interfaces, linearized travel time tomography in strong anisotropy, and a partial data problem originating from shear wave splitting. I will discuss the underlying inverse problems and the arising integral geometry problems. This talk is based on joint work with de Hoop, Katsnelson, and Mönkkönen.

We discuss recent results regarding $C^\infty$-isomorphism properties of weighted normal operators of the X-ray transform on manifolds with boundary, in joint work [1] with Francois Monard and Rohit Kumar Mishra. The crux of the result depends on understanding the Singular Value Decomposition of weighted X-ray transforms/backprojection operators, which itself can be obtained via intertwining with certain degenerately elliptic differential operators. We also discuss recent work [2] with Francois Monard on developing tools to study such degenerately elliptic operators even further. Such tools include a scale of Sobolev spaces which take into account behavior up to the boundary, as well as generalizations of Dirichlet and Neumann traces called boundary triplets associated to degenerately elliptic operators which pick out the first and second most singular terms of a function near the boundary.

[1] R. Mishra, F. Monard, Y. Zou. The $C^\infty$-isomorphism property for a class of singularly-weighted X-ray transforms. Inverse Problems 39: 024001, 2023. https://doi.org/10.1088/1361-6420/aca8cb

[2] F. Monard, Y. Zou. Boundary triples for a family of degenerate elliptic operators of Keldysh type, arXiv: 2302.08133, 2023.

We discuss a nonlinear analogue of the Pestov-Uhlmann range characterisation for geodesic X-ray transforms on simple surfaces. The transform under consideration takes as input matrix-valued and possibly direction dependent functions (which may encode magnetic fields or connections on a vector bundle) and outputs their 'scattering data' at the boundary. The range of this transform can be completely described in terms of boundary objects, and this description is reminiscent of the Ward correspondence for anti-self-dual Yang-Mills fields, but without solitonic degrees of freedom. The talk is based on joint work with Gabriel Paternain.