WORKSHOP "GEOMETRIC DYNAMICS DAYS"

NOVEMBER 8-9, 2013 (BOCHUM)

**Invited speakers**:

1. **Gonzalo Contreras** (Cimat, Mexico) - *Homogenization (of Hamilton-Jacobi equations) in arbitrary manifolds *

2. **Gabriel P. Paternain** (University of Cambridge, U.K.) - *Geodesic ray transforms and geometric inverse problems. Spectral
rigidity and invariant distributions on Anosov surfaces.*

All the lectures will take place in the lecture room NA 5/99, on the 5th floor of the Mathematics Department (Building NA).

Coffee will be served in room NA 1/58, on the 1st floor of the same building.

Participation is open. Participants are kindly invited to send an e-mail to Frau Dzwigoll (Sekretariat Prof. Dr. Gerhard Knieper) by October 15, communicating whether they will attend the conference dinner. Frau Dzwigoll can also give advice about accommodation.

General directions can be found here. A campusmap, as well as other useful maps, can be found here.

The workshop is financially supported by

**SFB TR-12**and jointly organized with the Westfälische Wilhelms-Universität Münster and the Technische Universität Dortmund.

**Schedule:**

Friday

2-3 pm: Registration (NA 5/34)

3-4 pm: G.Paternain -

*Geodesic ray transforms and geometric inverse problems*(NA 5/99)

4-5 pm: Coffee break (NA 1/58)

5-6 pm: G.Contreras - Homogenization (of Hamilton-Jacobi equations) in arbitrary manifolds 1 (NA 5/99)

7:30 pm: Conference dinner

Saturday

10-11 am: G.Contreras - Homogenization (of Hamilton-Jacobi equations) in arbitrary manifolds 2 (NA 5/99)

11-11:30 am: Coffee break (NA 1/58)

11:30-12:30 am: G.Paternain -

*Spectral rigidity and invariant distributions on Anosov surfaces*(NA 5/99)

**Abstracts**

*G. Paternain - Geodesic ray transforms and geometric inverse problems.*

In this talk I will discuss recent progress in the analysis of geodesic ray transforms and their relevance for the solution of some geometric inverse problems on Riemannian manifolds. The standard X-ray transform, where one integrates a function along straight lines is a well-studied object (and the basis of several medical imaging techniques), but I will consider more general transforms in which we integrate tensor fields along geodesics of a certain Riemannian metric. These transforms arise naturally as linearizations of important geometric inverse problems like the boundary rigidity problem in which one tries to determine a Riemannian metric from the knowledge of its boundary distance function. The talk is based on joint work with Mikko Salo and Gunther Uhlmann.

*G. Paternain - Spectral rigidity and invariant distributions on Anosov surfaces.*

I will discuss inverse problems on a closed Riemannian surface whose geodesic flow is Anosov. More specifically, I will try to establish spectral rigidity (a problem that has been open for some time) and surjectivity results for the adjoint of the geodesic ray transform. These surjectivity results imply the existence of many geometric distributions invariant under the geodesic flow and play a key role to prove spectral rigidity. This is joint work with Mikko Salo and Gunther Uhlmann.