Some proposals for open ended problems.
Implement a numerical scaling algorithm that tests semistability of projective hy- persurfaces. The input is a homogeneous polynomial. The output is either destabilizing one-parameter subgroup, or –ideally– an invariant that does not vanish for the hypersurface.
The Grassmannian Gr(1,P^4) of lines in 4-space is a 6-dimensional variety in P^9. Write its Chow form as the determinant of a 5 × 5-matrix whose entries are linear forms in the 10 choose 6 = 210 brackets, ready to be evaluated in a computer algebra system. Do the same with the Hurwitz form, and pass to higher Grassmannians. Determine the Chow polytopes.
