In this project, the PI will study stability conditions on the derived categories of coherent sheaves on algebraic varieties. More specifically, it will consider the behavior of moduli spaces of stable objects under a change of the stability conditions (``wall-crossing''). The PI will study the significance of wall-crossing to birational geometry, and to enumerative questions related to Donaldson-Thomas and Gromov-Witten invariants.
The study of moduli spaces has been one of the driving forces of algebraic geometry in the last 50 years. It can give answers to fundamental questions about how ``objects''---such as vector bundles, curves, maps---behave in ``families''. On the other hand, the topic of derived categories has seen a remarkable development in the last 15 years with input from string theory, symplectic geometry and algebraic geometry. The projects of this proposal combine these two guiding principles. Since wall-crossing is a phenomenon of general interest to each of the above-mentioned disciplines, the investigator hopes to contribute to the existing fruitful interdisciplinary collaboration.
