Plamen D Stefanov
My mathematical interests include Partial Differential Equations, Applied Mathematics, and applications of Microlocal Analysis. Currently, I am working mostly on various Inverse Problems and Integral Geometry but I am still interested in Scattering Theory and Resonances (scattering poles).
Inverse Problems is an area that is both quite challenging and is of great applied interest (aren't they all?). Medical Imaging, Geophysics, non-destructive material testing rely heavily on Inverse Problems. A typical inverse problem is to recover the coefficients of a PDE from measurements on the boundary of the domain, or at infinity. Very often, those problems are highly non-linear and ill posed. My interests here include
Inverse Boundary Value Problems, including elliptic and hyperbolic ones, inverse problems for the transport equation (optical tomography)
- Mathematics of medical imaging
- Inverse Scattering Problems
- Integral Geometry, especially integral transforms on non-Euclidean spaces including Riemannian and Lorentzian manifolds; and tensor tomography
- Questions of uniqueness, stability, recovery algorithms, numerical recovery.
The boundary rigidity (lens rigidity) problem for compact Riemannian manifolds with boundary is to show that a manifold of a certain class is uniquely determined by its boundary distance function, respectively, by its scattering relation. It is an inverse problem but it is also of independent interest in geometry. One of the motivations comes from seismology: recover the inner structure of Earth from travel times of seismic waves. A recent work in this direction by me, G. Uhlmann and A. Vasy was featured in the News section of Nature: Long-awaited mathematics proof could help scan Earth's innards. The linearization of this problem is the following integral geometry problem: determine a 2-tensor (actually, determine only its solenoidal part) from its X-ray transform: integrals along geodesics connecting boundary points. This is called sometimes Tensor Tomography. I am interested in
Analyzing the linearized integral geometry problem, (s-)injectivity, stability estimates, its properties as an FIO
- Uniqueness and stability for the non-linear boundary rigidity/lens rigidity problem
- Partial data problems (with local information)
- Those two problems for manifolds with conjugate points
- Possible generalizations for non-Riemannian families of curves motivated by inverse problems for hyperbolic systems
- Other Integral Geometry problems
A new direction (for me) is discretizations of inverse problems and the related question of sampling. It turns out that there is a natural link between sampling and semiclassicsal (microlocal) analysis.
I am interested in working with students who are excited by some of those areas. Prospective or current graduate students should feel free to contact me.