# Research Interests

I consider myself an analyst, both pure and applied. I started mostly on the pure side but some of my most recent works are either applied or somewhere in between the two, and there are no clear distinctions between those two fields anyway. If I have to list areas formally, I would say that my mathematical interests include Partial Differential Equations, Applied Mathematics, and applications of Microlocal Analysis. Currently, I am working mostly on various *Inverse Problems,* *Integral Geometry*, and *Nonlinear Wave Propagation* but I am still interested in *Scattering Theory and Resonances* (scattering poles). My work is often motivated by problems coming from physics/geophysics, engineering, and medical imaging.

**Microlocal Analysis **is the study of functions, or more generally, distributions, in the phase space. We are interested not just in the local behavior near a fixed point (which is what classical analysis does); we want to understand the behavior near a point and a (co)direction. The main tools are the theory of pseudo-differential operators and the Fourier Integral Operators (FIOs). Microlocal Analysis is a powerful technique for analyzing PDEs with variable coefficients and for geometric analysis on manifolds. Semiclassical (microlocal) analysis studies the behavior of systems when a small parameter tends to zero. It gets its name from its application to Quantum Mechanics, where we take the Planck constant to tend to zero.

**Inverse Problems** is an area that is both quite challenging and is of great applied interest (aren't they all?). Medical Imaging, Geophysics, and 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
- Nonlinear wave propagation and inverse problems for nonlinear wave type of equations.

The **boundary rigidity (lens rigidity)** problem for compact Riemannian manifolds with boundaries 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: to recover the inner structure of Earth from the 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. It received one of the Frontiers of Science Awards at the International Congress of Basic Science 2023. 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, and its properties as an FIO
- Uniqueness and stability for the non-linear boundary rigidity/lens rigidity problem
- Scattering rigidity of Lorentzian manifolds
- 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 and relativity
- Other Integral Geometry problems
- Applications to elasticity and ultimately, to seismology

Another direction is **discretizations of inverse problems** and the related question of **sampling**. It turns out that there is a natural link between sampling and semiclassical (microlocal) analysis. In a recent work with Tindel, we study the effect of added noise on the data from a semiclassical point of view.

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.