Research

Non-smooth differential geometry

• Low regularity Lorentzian geometry
• Synthetic Lorentzian geometry and Lorentzian length spaces
• Comparison geometry
• Optimal transport
• Distributional geometry

General relativity

• Low regularity spacetimes
• Singularity theorems
• Synthetic methods

Generalized functions and analysis

• Colombeau algebras
• Diffeomorphism invariant theories of nonlinear generalized functions
• Applications to differential geometry
• Applications to general relativity
• Generalized smooth functions

Symmetry groups of differential equations

• Group analysis for generalized functions
• Classical group analysis
• Classification problems

Partial differential equations

• Singular nonlinear problems
• Parabolic equations on Riemannian manifolds

Connections across areas

Many of these themes interact closely, especially in the study of geometric analysis on spaces with low regularity and in applications to mathematical relativity.