Research
Research
Geometry of Nilpotent Lie Groups
I study curvature properties of nilpotent Lie groups with a left invariant metric. The geometry of nilpotent Lie groups has a close connection with the geometry of solvable Lie groups. Any homogeneous Riemannian manifold of negative curvature is isometric to a solvable Lie group with a left invariant metric, so I am also interested in such a class of homogeneous Riemannian manifolds.
Abstract Group Automorphisms of Lie Groups
Given a Lie group, I am interested in the difference between its abstract (not necessarily continuous) group automorphisms and its Lie group automorphisms.
For simple, connected, and compact Lie groups, they coincide due to Cartan (1930) and van der Waerden (1933). This kind of phenomenon is called automatic continuity and is studied a lot in the context of simple groups. Motivated by this, I study abstract group automorphisms of nilpotent Lie groups. For a brief history, see the introduction of my paper.