The PI studies mathematical problems motivated by quantum mechanics and wave propagation. His central interest is the study of scattering resonances. These appear in many guises: as poles of Green functions, zeros of zeta functions or modes of decay and oscillation of waves, and the setting of their study ranges from geometry and automorphic forms, to microelectromechanical systems in engineering. The PI searches unifying themes and investigates specific cases. Special recent focus has been chaotic scattering and the relation between objects in thermodynamic formalism and the distribution of quantum resonances.
Waves encountered in experimental and theoretical studies have rates of oscillations and rates of decay. These two properties can be described by a single complex number with the real part corresponding to the rate of oscillations, and the imaginary part, to the rate of decay. In turn, these numbers appear as zeros of natural mathematical objects. In quantum mechanics, particles are described by wave functions, and waves with decay correspond to unstable particles. Classical/quantum correspondence suggests some subtle interplay between "classical" properties of the system and properties of waves. The PI investigates this in many settings, in particular when chaotic behaviour is present on the classical level.
