I am interested in the statistical structure of quantum mechanics: developing mathematical tools for quantum engineering and theoretical physics.
Currently, I am mainly studying quantum trajectories and interacting many-body systems. Quantum trajectories describe continuously measured quantum systems. From the mathematical point of view, it is a study of long products of random matrices with a specific form of long-range correlations. The critical question that I helped to answer is determining what information the measurement provides about the system, which on the mathematical side (quite astonishingly!) allows deriving ergodic properties of these systems.
In interacting systems, I am interested in the classification of states up to local automorphisms. To classify states, we are developing an index theory for ground states of gapped many-body systems. This is an ongoing project with S. Bachmann (UBC) and W. De Roeck (KU Leuven).