I am interested in applied analysis, applied probability, PDEs, spectral theory, semigroup theory, functional inequalities, optimal transportation and interpolation theory. In particular, I am interested in the interactions between these fields, and how such interactions shed new light on a given problem.
To be more exact, my research focuses on:
- Many particle/agent models, and their Mean Field Limit. I have been heavily involved with the investigation of the so-called Kac’s model.
- Mean Field Limits in general, and how one can utilise the field of optimal transportation to understand them better.
- Non-linear PDEs, usually motivated from physical, biological and chemical phenomena. A few examples are: The Boltzmann equation, The Boltzmann-Nordheim equation (a quantum variant of the Boltzmann equation), Becker-Döring equations and Reaction-Diffusion systems.
- Long time behaviour in equations/models that have an Entropy structure. In particular, I am very interested in cases where one can use the so-called Entropy Method and connect the investigation of the rate of convergence to a state of equilibrium to an underlying geometric functional inequality.
- Degenerate and Defective PDEs, and the question of convergence to equilibrium in them.
- Spectral and Semigroup theory in the service of linear PDEs. In particular, I am interested in associated functional inequalities such as Poincaré, or weak Poincaré inequalities.
- Interpolation Theory, with focus on weighted functional spaces.