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

穿墙：从经典代数几何到微分几何、镜面对称和派生代数几何

• 批准号: EP/X032779/1
• 负责人: Fatemeh Rezaee
• 金额: $ 35.28万
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX032779%2F1
• 关键词: Wall; crossing; classical; algebraic; geometry

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.

该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/微分几何和编程。我利用我在布里奇兰稳定性条件和穿墙方面的专业知识，并使用一些现代工具来解决这个项目，该项目与当今纯数学中非常活跃的研究课题相关。该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墙。由于穿墙方法在更高维度上很容易变得不可控，因此我建议结合现代语言“派生代数几何”和“二次整数规划”中的技术，使过程更容易处理。然后我将用它来研究一些称为希尔伯特方案的模空间。该项目的另一半是通过 Bridgeland 稳定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 Bridgeland 稳定条件的一步，这对微分几何学家来说非常有趣）。另一方面，我们更进一步，通过找到一些相应的“镜像不等式”将它们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（这对当今的辛几何学家非常重要）的可解性，并且还可以帮助找到一些无法直接求解的偏微分方程的镜像方程。布里奇兰稳定性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，例如教学、监督、面试等，这对我的下一个职业很重要，希望能成为一份好的学术工作。

项目成果

Geometry of canonical genus 4 curves

规范4曲线的几何

• DOI: 10.1112/plms.12577
• 发表时间: 2024
• 期刊: Proceedings of the London Mathematical Society
• 影响因子: 1.8
• 作者: Rezaee F