I work on Riemannian Geometry in its broadest sense, especially where it meets other areas like: topology of manifolds and singular spaces, dynamics, global analysis, metric geometry and geometric group theory. 

Over the years, I've gotten into :

• Special structures: Einstein manifolds, symmetric spaces, asymptotically harmonic manifolds etc. 

• Low-dimensional geometry: 3D geometric structures (hyperbolic, Heisenberg and Sol geometries, plus non-geometric manifolds) 

• Invariants: related to systolic geometry and minimizing problems: simplicial volume and bounded cohomology, minimal volume and minimal entropy, Seiberg-Witten invariants, and more

• Rigidity: moduli spaces of special Riemannian structures, convergence theory in Riemannian geometry, pinching, collapsing and finiteness problems

• Negatively curvature: counting geodesics, entropy, quasiconformal measures, dynamics

• Metric geometry: geometry of CAT(0) and Gromov-hyperbolic spaces 

• Geometric group theory:  growth, splittings and related questions

I am also interested in math oureach: I've been involved with math competitions for years, translated J.P. Petit's Topologicon, and love taking in science events like Caffè Scienza, Caffè Matematici, Scienza 3, La Scienza a Scuola and more.