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

这个项目涉及代数几何领域以及与其他领域的联系，包括代数拓扑、辛/Diffen几何和编程。我利用我在布里奇兰稳定性条件和穿越墙方面的专业知识，也使用了一些现代工具来解决这个项目，这个项目与当今纯数学中非常活跃的研究课题有关。这个项目的一半研究所谓的模(或参数)空间，它是代数几何中的基础。研究模空间的一种有效方法是跨越墙。由于越墙法在更高的维度上很容易失控，我建议将现代语言中的“派生代数几何”和“二次整数规划”的技术结合起来，使这一过程更容易处理。然后，我将用它来研究一些称为Hilbert格式的模空间。项目的另一半是通过Bridgeland稳定性条件将代数几何与微分几何和镜像对称性联系起来，从而产生一些新的Bogomolov-Gieseker型不等式(这将是将著名的Hitchin-Kobayashi类型对应扩展到Bridgeland稳定性条件的一步，这对微分几何来说是非常有趣的)。另一方面，我们进一步将它们与镜像对称性联系起来，找到了相应的“镜像不等式”，这将有助于揭示特殊的拉格朗日型方程的可解性(这在当今的辛几何中是非常重要的)，也可以帮助寻找一些不能直接求解的偏微分方程组的镜像方程。桥梁稳定理论在欧洲已经很成熟，上述应用被认为是该领域的前沿。在这次奖学金期间，我还将学习其他技能，如教学、监督、面试等，这些技能对我的下一份职业生涯将是重要的，希望这将是一份好的学术工作。

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.