Inverse Problems in Integral Geometry and Parabolic Type Partial Differential Equations

Abstract: In this talk we will discuss the following three inverse problems.

a. Momentum ray transforms (MRT) in the Schwartz space: Momentum
ray transform integrates symmetric tensor fields along straight lines with a power
tk. Here we will discuss the injectivity question, kernel description, inversion al-
gorithm and the range characterization of the momentum ray transforms. In this
case we will show that given sufficient momentum ray transforms of a symmetric
tensor eld, it is possible to recover the entire tensor field.

b.  Restricted transverse ray transform (TRT): Next we study the transverse
ray transform of symmetric m-tensor fields in R3 restricted to all lines passing
through a curve from microlocal viewpoint. More precisely, we are interested in
an approximate inversion of a TRT acting on symmetric tensor fields restricted to
all lines passing through a fixed curve in R3.

c.  Parabolic inverse problems: Finally we discuss the inverse problem of de-
termining the convection coefficient and the time-dependent density coefficient
appearing in a convection-diffusion equation from the boundary measurements.