Research topics

My research focuses on several aspects of the mathematical analysis of Partial Differential Equations (PDEs) and Stochastic Differential Equations (SDEs) arising in Physics, Biology, and Fluid Mechanics:

• Hydrodynamic and mean-field limits: Hyperbolic and parabolic hydrodynamic limits from kinetic PDEs to macroscopic PDEs, and mean-field limits from multi-agent systems to kinetic PDEs. Particular attention is given to models of collective dynamics with singular interactions arising in Biology and Neuroscience, including the Cucker–Smale and Kuramoto models.

• Propagation of chaos for non-exchangeable multi-agent systems: Derivation of non-exchangeable Vlasov PDEs and McKean–Vlasov SDEs from underlying stochastic multi-agent systems using probabilistic methods. Particular attention is given to non-exchangeable stochastic systems arising in Neuroscience, such as Kuramoto-type models with heterogeneous interaction networks described by graphon-type structures.

• Long-time behavior: Quantitative analysis of the long-time dynamics of multi-agent systems and kinetic PDEs arising in collective and evolutionary dynamics. Particular emphasis is placed on emergent patterns in models from Neuroscience, including Kuramoto and Stuart–Landau systems, as well as on the dynamics of phenotypic traits and other models of biological propagation phenomena at the mesoscale, such as the Fisher infinitesimal model.

• Optimal transport: Fibered Wasserstein spaces and the associated optimal transport framework, including fibered gradient-flow structures, variational formulations, and JKO schemes for nonlinear and nonlocal evolution equations.

• Incompressible and compressible fluids: Linked and knotted streamlines and vortex tubes in 3D steady solutions of the homogeneous incompressible Euler equations, and stability of generalized Beltrami fields. Well-posedness of inhomogeneous compressible barotropic Euler and Navier–Stokes equations.

• Modeling in developmental biology: Mathematical modeling and analysis of cell-to-cell communication through cytonemes in Drosophila, with a focus on morphogen transport and its role in tissue patterning and development.