Plamen D Stefanov
My mathematical interests include Partial Differential Equations, Mathematical Physics, 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. Its linearization 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

Other Integral Geometry problems
Scattering Theory studies properties of systems (usually governed by some PDE) at large distances from the "scatterer" and/or large times. One of the most important aspects of it are inverse scattering problems  what can we say about the system from such remote observations?
Resonances, also known as scattering poles, are "eigenvalues" of systems in unbounded domains, or more generally, on infinite volume manifolds. Their "eigenfunctions", called resonant states, however are required to be outgoing, instead of having finite energy. Resonances are complex numbers, even though the Hamiltonian is typically selfadjoint. They can also be defined as the poles of the scattering matrix, which makes resonances part of Scattering Theory. My interests in this area include

Relationship between the underlying classical mechanical system and the behavior of resonances, resonant states, the scattering operator, and the resolvent

In particular, the LaxPhillips Conjecture  resonances near the real axis and their relationship to trapped rays

Bounds and asymptotics on the counting function of the resonances in a disk in the complex plane (n odd)
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.