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Let $X$ be a CR manifold with transversal, proper CR $G$-action. We show that $X/G$ is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold…

We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the…

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…

We prove a uniqueness result for the broken ray transform acting on the sums of functions and $1$-forms on surfaces in the presence of an external force and a reflecting obstacle. We assume that the considered twisted geodesic flows have…

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.

Let $(M,g)$ be a compact manifold without conjugate points and with visibility universal covering. We show that its geodesic flow has a time-preserving expansive factor which is topologically mixing and has a local product structure. As an…

Einstein's equations of gravitation are not invariant under geodesic mappings, i. e. under a certain class of mappings of the Christoffel symbols and the metric tensor which leave the geodesic equations in a given coordinate system…

We first give a constructive answer to the attenuated tensor tomography problem on simple surfaces. We then use this result to propose two approaches to produce vector-valued integral transforms which are fully injective over tensor fields.…

In this paper we consider the local X-ray transform for general flows. We extend the results on the local and global invertibility of the geodesic ray transform proved by Uhlmann and Vasy \cite{UV} to the X-ray transform for a general flow.…

Let $\FF$ be a codimension one foliation on a closed manifold $M$ which admits a transverse dimension one Riemannian foliation. Then any continuous leafwise harmonic functions are shown to be constant on leaves.

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.…

We consider the nonlinear problem of determining a connection and a Higgs field from the corresponding parallel transport along geodesics on a Riemannian manifold with boundary, in any dimension. The problem can be reduced to an integral…

In this article, we study the microlocal properties of the geodesic ray transform of symmetric $m$-tensor fields on 2-dimensional Riemannian manifolds with boundary allowing the possibility of conjugate points. As is known from an earlier…

The geometric quantization of the geodesic flow on a compact Riemannian manifold via the BKS "dragging projection" yields the Laplacian plus a scalar curvature term. To avoid convergence issues, the standard construction involves somewhat…

We study a form of cyclic pursuit on Riemannian manifolds with positive injectivity radius. We conjecture that on a compact manifold, the piecewise geodesic loop formed by connecting consecutive pursuit agents either collapses in finite…

We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the…

Starting from well-known absolute instruments for perfect imaging, we introduce a type of rotational-symmetrical compact closed manifolds, namely geodesic lenses. We demonstrate that light rays confined on geodesic lenses are closed…

We derive explicit reconstruction formulas for the attenuated geodesic X-ray transform over functions and, in the case of non-vanishing attenuation, vector fields, on a class of simple Riemannian surfaces with boundary. These formulas…

Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we…

Weighted V-line transforms map a symmetric tensor field of order $m\ge0$ to a linear combination of certain integrals of those fields along two rays emanating from the same vertex. A significant focus of current research in integral…