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和其他领域的连接，包括代数拓扑，辛/差分几何和编程。我用我的专业知识在Bridgeland稳定性条件和跨壁，并使用一些现代工具来解决这个项目，这是涉及到非常活跃的研究课题，在今天的纯数学。这个项目的一半研究所谓的模（或参数）空间，这是基本的代数几何。研究模空间的一个有效方法是通过跨壁。由于越墙方法在高维空间中很容易变得不可控，我建议将现代语言“导出代数几何”和“二次整数规划”中的技术结合起来，使过程更易于处理。然后，我将使用它在研究一些模空间，这是所谓的希尔伯特计划。该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稳定性条件的一步，这对微分几何学家来说非常有趣）。另一方面，我们进一步通过寻找相应的镜像不等式将这些方程与镜像对称联系起来，这将有助于揭示特殊拉格朗日型方程的可解性（这对辛几何学家来说非常重要），也有助于找到一些不能直接处理的偏微分方程的镜像方程。Bridgeland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。在此期间，我还将学习其他技能，如教学，监督，面试等，这将是我的下一个职业，希望是一个很好的学术工作的重要。

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