Wall-crossing, stability conditions and mirror symmetry

自从Atiyah在50多年前开始研究向量丛以来，向量丛的稳定性一直是代数几何中的一个中心概念。这一概念后来被扩展到相干集束，这是向量丛的自然推广，可以被认为是允许具有奇点的向量丛的一种方式。虽然相干滑轮的稳定性概念取决于选择，但直到最近才对可能的选择空间进行了研究。特别是，由于Bridgeland的开创性工作，如果我们愿意将我们的稳定性概念从相干层扩展到相干层的复合体，即派生范畴，我们知道存在多种稳定性条件。这个新的稳定性概念是由弦理论激发的，而镜像对称性与辛几何和Fukaya范畴的性质密切相关。这项建议的一个重点是研究弦理论提出的布里奇兰稳定性条件的问题，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实例，其存在与弦论中D-膜的稳定性概念密切相关。代数几何是研究作为多项式方程系统的解集出现的形状的学科。虽然它们在数学中普遍存在，但它们在数学物理中也变得重要，更具体地说，在弦理论中也是如此。相反，弦理论家的洞察力在过去20年里对代数几何产生了巨大的影响，他们以不同的背景和直觉处理类似的问题。这项提议的项目将进一步促进代数几何和弦理论之间的这种互动。

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.