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Inverse Problems and Flows

Objective

In this project, we propose to study mixing and rigidity properties of a large class of
flows in geometry and dynamical systems using novel microlocal and spectral methods.
Such methods, recently developed for flows with hyperbolicity, are based on the construction of
new functional spaces, adapted to the considered flow, on which the generating vector field is Fredholm with
discrete spectrum, known as the resonance spectrum. We want to apply these tools to the following categories of long-standing problems:

1) Geometric inverse problems: boundary and lens rigidity problems, analysis of the X-ray transforms for geodesic flows and its applications to Calderon and Gelfand inverse problems.
2) Rigidity for Anosov flows: determination of the minimal regularity of the stable/unstable foliations providing rigidity,
rigidity questions for regular conjugacies of flows.
3) Rate of mixing for flows of hyperbolic type: decay of correlations of Axiom A flows, Anosov flows, and
their compact Lie groups extensions, frame flows of negatively curved manifolds.
4) Resonances for locally symmetric spaces: relation between resonances of the flow and the spectrum of Laplacians on bundles.

To address these problems, we will develop a thorough analysis for the regularity and the long time properties of solutions to transport equations. We are convinced that the proposed approach will provide solutions to these problems, which have been out of reach for a long time.

Call for proposal

ERC-2016-COG
See other projects for this call

Funding Scheme

ERC-COG - Consolidator Grant

Host institution

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Address
Rue Michel Ange 3
75794 Paris
France
Activity type
Other
EU contribution
€ 1 133 236

Beneficiaries (1)

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France
EU contribution
€ 1 133 236
Address
Rue Michel Ange 3
75794 Paris
Activity type
Other