Wall-crossing, stability conditions and mirror symmetry

矢量束的稳定性一直是代数几何中的一个核心概念，因为Atiyah在50多年前开始了他们的研究。这个概念后来被扩展到相干束，它是向量束的自然推广，可以被认为是一种允许有奇点的向量束的方法。虽然相干轴的稳定性取决于选择，但直到最近才对可能的选择空间进行了研究。特别是，由于bridgeeland的开创性工作，我们知道，如果我们愿意将稳定性的概念从相干束扩展到相干束的复合体，即扩展到派生范畴，则存在多种稳定性条件。这种稳定性的新概念是由弦理论激发的，镜像对称与辛几何和深谷范畴的性质密切相关。这项提议的一个重点是研究弦理论提出的布里奇兰稳定条件问题，但目前还没有一个令人满意的数学答案。例如，PI将致力于构建一类缺失的稳定性条件的例子，其存在与弦理论中d膜的稳定性概念密切相关。代数几何是研究多项式方程组的解集所产生的形状的学科。虽然它们在数学中无处不在，但它们在数学物理中也变得很重要，更具体地说，在弦理论中。相反，弦理论家的见解，他们用不同的背景和直觉来解决类似的问题，在过去的20年里对代数几何产生了巨大的影响。本课题将进一步促进代数几何与弦理论之间的相互作用。

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.