Stability of vector bundles has been a central concept in algebraic geometry since Atiyah initiated their study over 50 years ago. The concept was later extended to coherent sheaves, which are a natural generalization of vector bundles, and can be thought of as a way to allow for vector bundles with singularities. While the notion of stability for coherent sheaves depends on choices, only recently has the space of possible such choices been studied. In particular, due to the seminal work by Bridgeland, we know that there is a manifold of stability conditions, if we are willing to extend our notion of stability from coherent sheaves to complexes of coherent sheaves, i.e., to the derived category. This new concept of stability is motivated by string theory, and mirror symmetry relates it closely to symplectic geometry and properties of the Fukaya category. One focus of this proposal is to work on questions for Bridgeland stability conditions that are suggested by string theory, but don't have a satisfactory mathematical answer yet. For example, the PI will work on constructing a missing class of examples of stability conditions, whose existence is closely related to a notion of stability for D-branes in string theory.
Algebraic Geometry is the study of shapes arising as solution sets of systems of polynomial equations. While these are ubiquitous in mathematics, they have also become important in mathematical physics and, more specifically, in string theory. Conversely, insights by string theorists, who approach similar question with different background and intuition, have had an enormous influence on algebraic geometry over the last 20 years. The projects of this proposals will contribute further to this interaction between algebraic geometry and string theory.
