Quantum geometry of orbifolds and their resolutions

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In the physics of string theory, the universe is a ten dimensional space. While four of the dimensions comprise the usual notions of space and time, the remaining six are curled up into tiny complicated geometric spaces called Calabi-Yau threefolds. Understanding the geometry of Calabi-Yau threefolds is central to both algebraic geometry and physics. In string theory, geometric properties of a Calabi-Yau threefold determine the physical properties of the particles (i.e. strings) that move through it. In algebraic geometry, Calabi-Yau threefolds play a special role in a variety of fields of study, for example Gromov-Witten theory, Donaldson-Thomas theory, and birational geometry.      The Calabi-Yau threefolds of string theory are usually taken to be smooth spaces, but can also be mildly singular spaces called orbifolds. In algebraic geometry, a singular space can be replaced by a smooth space by a so-called resolution of singularities. A resolution of singularities that is compatible with the Calabi-Yau condition is called a crepant resolution. A guiding principle in physics asserts that string theory on an orbifold should be equivalent to string theory on any crepant resolution.     There are a variety of invariants associated to Calabi-Yau threefolds --- their Gromov-Witten invariants, their Donaldson-Thomas invariants, and their Gopakumar-Vafa invariants. These invariants all have physical interpretations in string theory. The long term objective of this project is to make mathematical sense of the principle that string theory on an orbifold is equivalent to string theory on any of its crepant resolutions. In particular, I want to formulate, and prove in some important cases, the various Crepant Resolution Conjectures, namely the equivalence between the various invariants associated to a Calabi-Yau orbifold and its crepant resolution(s).