We discuss generalizations of Ozsvath-Szabo's spectral sequence relating
Khovanov homology and Heegaard Floer homology, focusing attention on an
explicit relationship between natural Z (resp., 1/2 Z) gradings appearing in
the two theories. These two gradings have simple representation-theoretic
(resp., geometric) interpretations, which we also review.
We prove a conjecture of Khovanov which identifies the topological space
underlying the Springer variety of complete flags in C^2n stabilized by a fixed
nilpotent operator with two Jordan blocks of size n.