Algebraic shifting is an operation on simplicial complexes introduced by Kalai. It associates with any given complex a simpler complex (combinatorially speaking), while preserving and encoding important information about the original complex.
This operation is quite heavy to compute *deterministically* in the general case. In my work for my thesis with Eran Nevo, we came up with a polynomial-time algorithm for calculating the exterior shifting of triangulations of the torus, the projective plane, and the Klein bottle.
I will explain this result and the relevant background in the talk.