For more information about this meeting, contact Yakov Pesin, Anatole Katok, Svetlana Katok, Dmitri Burago.

Title: | Boundary rigidity, old and new (a geometric approach) |

Seminar: | Center for Dynamics and Geometry Seminars |

Speaker: | D. Burago, PSU |

Abstract: |

A Riemannian manifold with boundary is said to be boundary rigid if its metric
is uniquely determined by the boundary distance function, that is the restriction
of the distance function to the boundary. Loosely speaking, this means that
the Riemannian metric can be recovered from measuring distances between
boundary points only. The goal is to show that certain classes of metrics are
boundary rigid (and, ideally, to suggest a procedure for recovering the metric).
This problem has been extensively studied from PDE viewpoint: the distance
between boundary points can be interpreted as the "travel time" for a solution of
the wave equation. Hence this becomes a classic Inverse Problem when we
have some information about solutions of a certain PDE and want to recover its
coefficients. For instance such problems naturally arise in geophysics (when
we want to find out what is inside the Earth by sending sound waves), medical
imaging etc.
In a joint project with S. Ivanov we suggest an alternative geometric approach
to this problem. In our earlier work, using this approach we were able to show
boundary rigidity for metrics close to flat ones (in all dimensions), thus giving
the first class of boundary rigid metrics of nonâ€“constant curvature beyond two
dimensions. We were now able to extend this result to include metrics close to
a hyperbolic one.
The approach is based on a very transparent scheme, which however has a
few blocks where certain formulas have to be carefully selected, and this turns
out to be a rather non-trivial task. The purpose of the talk is to explain this
scheme and to explain what is needed from those blocks to make it work. The
talk assumes no background in inverse problems and is supposed to be
accessible to a general math audience including graduate students (in other
words, we will not get into technical details of the proofs). |

### Room Reservation Information

Room Number: | MB106 |

Date: | 11 / 03 / 2008 |

Time: | 03:30pm - 05:30pm |