  We discuss some mathematical conjectures which have come out of the Dirichlet
branes in superstring theory, focusing on the case of supersymmetric branes in
Calabi-Yau compactification. This has led to the formulation of a notion of
stability for objects in a derived category, contact with Kontsevich's
homological mirror symmetry conjecture, and "physics proofs" for many of the
subsequent conjectures based on it, such as the representation of Calabi-Yau
monodromy by autoequivalences of the derived category.
