Inverse problems

I currently work under the supervision of Colin Guillarmou on inverse problems in Riemannian geometry. My work is based on recent advances made in microlocal analysis and in the analytic study of flows (by Dyatlov-Guillarmou, Dyatlov-Zworski, Faure-Sjöstrand, …). A poster explaining my research (in french).

  • The marked length spectrum of Anosov manifolds, with Colin Guillarmou, submitted. Burns and Katok conjectured in 1985 that the marked length spectrum (the sequence of lengths of closed geodesics, differentiated by the homotopy) of a closed manifold with negative curvature should determine the metric in the sense that any two such metrics with same marked length spectrum should be isometric. Croke and Otal independently proved the conjecture in 1990 in dimension 2 but the question remains open in dimension greater or equal to 3. We prove a local non-linear version of Burns-Katok’s conjecture which extends to surfaces with Anosov geodesic flow and non-positively-curved Anosov manifolds of dimension greater or equal to 3.
  • Local marked boundary rigidity under hyperbolic trapping assumptions, submitted. The marked boundary distance of a manifold with boundary maps two points and a homotopy class of curves joining them to the distance between the points computed within the class. We prove that for a manifold (M,g) with strictly convex boundary, hyperbolic trapped set and under the assumption that the X-ray transform over solenoidal 2-tensors is injective, the marked boundary distance locally determines the metric in the following sense: if g’ is another metric in a neighborhood of g with same marked boundary distance, then there exists an isometry i: M -> M fixing the boundary such that i^*g’=g.
  • Boundary rigidity of negatively-curved asymptotically hyperbolic surfaces, submitted. For an asymptotically hyperbolic manifold, it is possible to define a notion of renormalized length between two points on the boundary at infinity. We show that two asymptotically hyperbolic surfaces (M,g) and (M,g’) with negative curvature are determined by their marked renormalized lengths in the following sense: if the lengths agree, then there exists an isometry i defined on the compactification of M which fixes the boundary and such that i^*g’=g.
  • On the s-injectivity of the X-ray transform for manifolds with hyperbolic trapped set, submitted. In this article, we study the X-ray transform (XRT) on smooth compact manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set. A prototype for these is given by surfaces with convex boundary and negative curvature. We prove an equivalence principle between the s-injectivity of the XRT and the surjectivity of a certain operator over the set of solenoidal tensors. We then deduce the s-injectivity of the XRT on tensors of any order in the case of such manifolds in dimension 2.

Systèmes dynamiques

  • On the genericity of the shadowing property for conservative homeomorphisms, with Pierre-Antoine Guihéneuf, published in Proceedings of the AMS 146 (2018), n°10, 4225—4237. We show that the shadowing property is generic among conservative homeomorphisms of a compact manifold. The shadowing property is satisfied as soon as any pseudo-orbit of a homeomorphism (an orbit for which the iterates are not exact but may exhibit an error of approximation) is shadowed by an exact orbit of the same homeomorphism.