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, nondestructive 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 nonlinear 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 nonEuclidean 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: Longawaited mathematics proof could help scan Earth's innards. The linearization of this problem is the following integral geometry problem: determine a 2tensor (actually, determine only its solenoidal part) from its Xray 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 nonlinear boundary rigidity/lens rigidity problem
 Partial data problems (with local information)
 Those two problems for manifolds with conjugate points
 Possible generalizations for nonRiemannian families of curves motivated by inverse problems for hyperbolic systems and relativity
 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.