Math 548A
Geometric Structures: Integral Geometry
Vladimir Sharafutdinov
Winter 2004, Monday-Wednesday-Friday 1:30-2:20
The course is mostly devoted to geometric analysis of inverse problems.
We demonstrate how different geometric structures; like Riemannian
metrics, vector bundles, symplectic structures, geodesic flows, stable
and unstable foliations of an Anosov flow; can be used in studying
inverse problems of mathematical physics.
We start with discussing the boundary rigidity problem: how far is a
Riemannian metric on a compact manifold with boundary determined by
distances between boundary points? After linearization, this gives us
the integral geometry problem: to which extent is a symmetric tensor
field on a Riemannian manifold determined by its integrals over
geodesics joining boundary points? The latter question is equivalent to
an inverse problem for the kinetic equation that is a differential
equation on the unit tangent bundle. To treat the kinetic equation, we
develop some tensor analysis machinery that is based on the structure
of the tangent bundle. We also discuss corresponding periodic integral
geometry problems for closed Riemannian manifolds which are closely
related to the spectral rigidity problem. In the periodic setting, we
have to use the machinery of Anosov flows.
Tentative Syllabus
- Inverse kinematic problem of seismics in two dimensions
- Some questions of tensor analysis
- The ray transform
- Inversion of the ray transform
- Local boundary rigidity
- The modified horizontal derivative
- Inverse problem for the transport equation
- Integral geometry on Anosov manifolds