Riemannian and pseudo-Riemannian geometry

• Riemannian geometry, Ricci flows, topology of manifolds, hyperbolic geometry
• Lorentzian geometry, foliations, geodesic flows, transformation groups

Dynamical systems

• Teichmüller theory, flat surfaces, moduli spaces
• Differentiable dynamics

Algebraic and complex geometry

• Complex hyperbolic geometry, symmetric spaces, geometry of period domains
• Kählerian geometry, non-abelian Hodge theory, singular Kähler-Einstein metrics, hyperbolicity
• Algebraic and analytic singularities, o-minimal structures
• Birational geometry, Cremona groups