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, 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; 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 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

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 self-adjoint. 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 Lax-Phillips 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.