The aim of this project is to investigate some instances of the theory of derived categories of coherent sheaves. In particular, the P.I. has three main objectives: (1) To study the geometry of moduli spaces of complexes for the projective plane, K3 surfaces and cubic 4-folds. (2) To realize ``classical'' questions and conjectures in enumerative geometry as wall-crossing phenomena in derived category, the starting example being the total space of the canonical bundle over the projective plane. (3) To give a complete description of equivalences between the derived categories of K3 surfaces and of their deformations.
The broader context of this project is the area of algebraic geometry. Starting as a pure mathematical subject, in the recent years algebraic geometry has seen many interactions with other areas of science. The present project is motivated and inspired precisely by connections, envisioned by the Fields Medal winner M. Kontsevich almost twenty years ago, with theoretical physics and string theory. In particular, by bringing intuition and techniques from physics to tackle classical problems in pure mathematics and, vice versa, to provide mathematical rigorous foundations to physics constructions.
