index theory of elliptic operators;

secondary invariants of elliptic operators;

geometric quantization: noncompact context;

and noncommutative geometry.

Several mathematical problems that I currently work on originate
from physics, for example:

topological field theories,

Mathematics of Condensed matter physics:

1. (fractional) quantum Hall effect 1985 & 1998 Nobel Prizes in Physics
2. topological insulators, 2016 Nobel Prize in Physics
3. topological semimetals, 2016 Nobel Prize in Physics

Mathematics of string theory & duality in a background flux.