Wall-crossing: from classical algebraic geometry to differential geometry, mirror symmetry and derived algebraic Geometry

该项目涉及代数几何领域以及与代数拓扑、辛/异几何和编程等其他领域的联系。我利用我在桥地稳定条件和过墙方面的专业知识，并使用一些现代工具来解决这个项目，这是当今纯数学中非常活跃的研究课题。这个项目的一半研究所谓的模（或参数）空间，这是代数几何的基础。研究模空间的一种有效方法是穿越墙。由于过墙法在高维情况下容易变得不可控，我建议将现代语言“派生代数几何”中的技术和“二次整数规划”中的技术结合起来，使这个过程更容易处理。然后我会用它来研究一些叫做希尔伯特格式的模空间。项目的另一部分是通过桥地稳定性条件将代数几何与微分几何和镜像对称联系起来，以产生一些新的Bogomolov-Gieseker型不等式（这将是将著名的Hitchin-Kobayashi型对应扩展到桥地稳定性条件的一步，这对微分几何学者来说非常有趣）。另一方面，我们更进一步，通过找到一些相应的“镜像不等式”，将这些与镜像对称联系起来，这反过来又会揭示特殊拉格朗日型方程的可解性（这对当今的辛几何学者非常重要），也有助于找到一些不能直接解决的偏微分方程的镜像方程。桥地稳定性理论在欧洲已经很好地建立起来，上述应用被认为是该地区的前沿。在此期间，我还将学习其他技能，如教学、监督、面试等，这对我的下一份职业很重要，希望能成为一份好的学术工作。

This project involves the field of algebraic geometry and connections to other areas including algebraic topology, symplectic/diffen geometry, and programming. I use my expertise in Bridgeland stability conditions and wall-crossing, and also use some modern tools to tackle this project which is related to very active research subjects in today's pure mathematics. Half of this project studies so-called moduli (or parameter) spaces which are fundamental in algebraic geometry. One effective way to study moduli spaces is via wall-crossing. As the wall-crossing method can get easily un-controllable in higher dimensions, I propose to incorporate the technology from the modern language of "Derived Algebraic Geometry" and also "Quadratic integer Programming" to make the process more tractable. Then I will use this in studying some moduli spaces which are called Hilbert schemes. Another half of the project is connecting algebraic geometry to differential geometry and mirror symmetry via Bridgeland stability conditions to produce some new Bogomolov-Gieseker type inequalities (which will be a step towards extending the celebrated Hitchin-Kobayashi type correspondence to Bridgeland stability conditions, which is very interesting to differential geometers). On the other hand, we take a further step and relate these to mirror symmetry via finding some corresponding "mirror inequalities", which in turn would shed light on the solvability of special Lagrangian type equations (which are very important to symplectic geometers these days), and also could help to find mirror equations to some Partial Differential Equations which cannot be tackled directly. The theory of Bridgeland stability is well established in Europe and the applications as above are considered as cutting edge in the area. During this fellowship, I will also learn other skills e.g. teaching, supervision, interviewing, etc, which will be important for my next career which will be hopefully a good academic job.