My main research directions are on the intersection of pure analysis, mathematical physics, applied mathematics, and numerical analysis.

My current interests are mainly in foundations, applications and numerical methods of spectral theory and spectral geometry, and include the following ongoing projects.

Spectral geometry and Quantum Chaos

Some recent results (joint with I Polterovich (Montreal), D Jakobson (McGill), N Nadirashvili (Marseille) and L Parnovski (UCL)) opened the study of domains isospectral with respect to the Dirichlet-Neumann boundary condition swap (we suggested and studied the following modification of the famous Kac's question, Can one hear which part of a drum's boundary is free and which is fixed?). This leads to unexpected links with problems on manifolds of higher genus, as well as the deeper study of the Dirichlet-to-Neumann maps and properties of wavefunctions.

Universal and isopermetric estimates, and null varieties

Recent results include a method of deducing new universal trace identities in an abstract setting, which allow one to obtain in an easy and uniform manner most of the known and a number of new estimates on eigenvalues of differential operators. The current work involves the famous (and very long standing) Polya Conjecture as well as the study of the relation between eigenvalues and zeroes of Fourier transforms, which relates to another long-standing conjecture, the so-called Pompeiu problem. This is a joint project with L Parnovski (UCL) and R Benguria (Saniago), which was also supported by the Royal Society International Collaboration grant with two Chilean universities.The work on isoperimetric inequalities for low eigenvalues is conducted jointly with colleagues in Montreal.

Non-self-adjoint operators and operator pencils

Some very recent results on spectra of the pencils of the type A+λB, in case when both operators A, B are self-adjoint but non-sign-definite, have been obtained jointly with E B Davies (King's College London) in a descrete case, and with D M Elton (Lancaster) and I Polterovich (Montreal) for 1d Dirac problem arising from graphene studies. Both problems give rise to unexpected connections with subjects as diverse as asymptotics of orthogonal polynomials and number theory, and will lead to many interesting generalisations.

Theory and numerics for eigenvalues and resonances on manifolds with regular ends

This includes spectral problems for PDEs in infinite domains,in particular in the presence of the essential spectrum, for example on perturbed hyperbolic manifolds (joint with A Strohmaier (Loughborough)) and, as an application, a localization of shelf ocean waves (a long standing problem in oceanography).The computational approach is pursued together with M Marletta (Cardiff).

Foundations of numerical analysis

I am in particular interested in the question of spectral pollution in projection methods for operators with essential spectrum, and methods of its recognition and avoidance. This is a joint project with E Shargorodsky (King's College London) and L Boulton (Heriot-Watt). The main result includes the demonstration of the usefulness of the so-called second order spectrum. Current research concentrates on further work on foundations of this method, as well as applications in elasticity and quantum mechanics.