My research is broadly centred around PDEs with links to variational problems and probability, as well as applications in kinetic theory, statistical physics, material sciences and quantum mechanics. Quite often, very few and simple basic mechanisms can lead to a rich class of complex phenomena, whose description involves interesting mathematics. The projects I am working on are related to a series of current and highly active lines of research: optimal transportation, singular stochastic PDEs, Boltzmann equation, pattern formation problems, solitons. These diverse projects, though seemingly unconnected, upon closer inspection often share some common features:
- Ideas from regularity theory play a crucial role,
- nonlocal interactions are driving mechanisms,
- the analysis of continuum models is combined with probabilistic tools and thinking.
Optimal Transportation
- ϵ-regularity for general cost functions
- Multi-marginal optimal transport: sparsity of generalized Wasserstein barycenters and regularity
Variational methods for singular SPDEs
- The Magnetisation Ripple
Calculus of variations
- Domain branching in micromagnetics and superconductivity
- Nonlocal Variational Problems related to Nonlinear Optics
Kinetic Theory
- Entropy decay for the Kac evolution
- Smoothing Properties of the Boltzmann Equation
Quantum Mechanics
- Harmonic Analysis and Semi-classical Bounds on the Number of Eigenvalues of Schrödinger Operators