Mathematical Aspects of D-branes
  In these notes we review recent work on describing D-branes with nonzero Higgs vevs in terms of sheaves, which gives a physical on-shell D-brane interpretation to more sheaves than previously understood as describing D-branes. The mathematical ansatz for this encoding is checked by comparing open string spectra between D-branes with nonzero Higgs vevs to Ext groups between the corresponding sheaves. We illustrate the general methods with a few examples. 
E. Sharpe