Logarithmic Moduli Spaces for Symplectic Geometry: Construction, Applications, and Beyond
辛几何的对数模空间:构造、应用及其他
基本信息
- 批准号:2003340
- 负责人:
- 金额:$ 17.84万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2020
- 资助国家:美国
- 起止时间:2020-09-01 至 2024-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Symplectic manifolds are geometric objects that generalize the concept of phase-space in classical mechanics. During the past four decades, the field of symplectic geometry has evolved rapidly, leading to new connections with other significant areas of research, such as algebraic geometry, low dimensional topology, and high energy physics. This award supports research on fundamental objects known as holomorphic curves and their corresponding invariants and algebraic structures. In particular, the investigator will address foundational questions, such as the construction of well-behaved families of holomorphic curves in the presence of objects known as divisors. This research contains specific projects that can be carried out by graduate students and postdocs. The investigator will organize annual mini-symposia for introducing undergraduate students to research opportunities in geometry and topology, and their applications to other fields. He will also initiate a math club at the public library aimed at high school students.The main objective of this proposal is to construct moduli spaces of holomorphic curves for arbitrary pairs of symplectic manifolds and normal crossing symplectic divisors, satisfying particular properties. Construction of such moduli spaces requires a compactification, an analytical framework for deformation theory, addressing the transversality issue, and proving a gluing theorem. These moduli spaces have immediate applications in enumerative geometry, Mirror Symmetry, construction of Fukaya categories, and other active areas of research in symplectic geometry, algebraic geometry, and string theory. In collaboration with M. McLean and A. Zinger, the PI introduced topological notions of normal crossing symplectic divisor and variety. They have constructed tools such as regularizations and logarithmic tangent bundle for working with such objects. Recently, the PI developed a novel compactification and a deformation theory based on the logarithmic tangent bundle. He will use this setup to work on the remaining steps of the construction. The main project is to define Gromov-Witten invariants relative to an arbitrary normal crossing divisor. Other projects include proving a degeneration formula for Gromov-Witten invariants, finding the relations with the algebraic approach, and exploiting the applications to Mirror Symmetry. This project is jointly funded by Geometric Analysis and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
辛流形是将经典力学中的相空间概念推广的几何对象。在过去的四十年中,辛几何领域发展迅速,导致与其他重要研究领域的新联系,如代数几何、低维拓扑和高能物理。该奖项支持研究被称为全纯曲线的基本对象及其相应的不变量和代数结构。特别是,研究者将解决基础问题,例如在被称为除数的物体存在下,构造全纯曲线的良好行为族。本研究包含可由研究生和博士后进行的具体项目。研究者将组织年度小型研讨会,向本科生介绍几何和拓扑的研究机会,以及它们在其他领域的应用。他还将在公共图书馆发起一个针对高中生的数学俱乐部。本文的主要目的是构造满足特定性质的辛流形和正规交叉辛因子的任意对的全纯曲线的模空间。构造这样的模空间需要一个紧化,一个变形理论的分析框架,解决横向性问题,并证明粘接定理。这些模空间在枚举几何、镜像对称、Fukaya范畴的构造以及辛几何、代数几何和弦理论的其他活跃研究领域中有直接的应用。在与M. McLean和A. Zinger的合作中,PI引入了正交辛因子和变异的拓扑概念。他们已经构建了诸如正则化和对数切线包之类的工具来处理这些对象。最近,PI发展了一种新的基于对数切线束的紧化和变形理论。他将使用此设置来处理构建的其余步骤。主要项目是定义相对于任意法向交叉除数的Gromov-Witten不变量。其他项目包括证明Gromov-Witten不变量的退化公式,找到与代数方法的关系,以及开发镜像对称的应用。该项目由几何分析和建立计划,以刺激竞争研究(EPSCoR)共同资助。该奖项反映了美国国家科学基金会的法定使命,并通过使用基金会的知识价值和更广泛的影响审查标准进行评估,被认为值得支持。
项目成果
期刊论文数量(4)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Pseudoholomorphic curves relative to a normal crossings symplectic divisor: compactification
相对于法向交辛除数的伪全纯曲线:紧致化
- DOI:10.2140/gt.2022.26.989
- 发表时间:2022
- 期刊:
- 影响因子:2
- 作者:Farajzadeh-Tehrani, Mohammad
- 通讯作者:Farajzadeh-Tehrani, Mohammad
RIS-aided mmWave beam-forming for two-way communications of multiple pairs
用于多对双向通信的 RIS 辅助毫米波波束成形
- DOI:10.52953/vbex2484
- 发表时间:2023
- 期刊:
- 影响因子:0
- 作者:Torkzaban, Nariman;Amir), Mohammad A.;Farajzadeh-Tehrani, Mohammad;Baras, John S.
- 通讯作者:Baras, John S.
Limits of stable maps in a semi-stable degeneration
半稳定退化中稳定图的极限
- DOI:10.1007/s10711-022-00731-5
- 发表时间:2022
- 期刊:
- 影响因子:0.5
- 作者:Farajzadeh-Tehrani, Mohammad
- 通讯作者:Farajzadeh-Tehrani, Mohammad
Deformation Theory of Log Pseudo-holomorphic Curves and Logarithmic Ruan–Tian Perturbations
对数伪全纯曲线的变形理论与对数阮田摄动
- DOI:10.1007/s42543-023-00069-1
- 发表时间:2023
- 期刊:
- 影响因子:0
- 作者:Farajzadeh-Tehrani, Mohammad
- 通讯作者:Farajzadeh-Tehrani, Mohammad
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Mohammad Farajzadeh Tehrani其他文献
Normal Crossings Degenerations of Symplectic Manifolds
辛流形的正态交叉简并
- DOI:
10.1007/s42543-019-00017-y - 发表时间:
2016 - 期刊:
- 影响因子:0
- 作者:
Mohammad Farajzadeh Tehrani;A. Zinger - 通讯作者:
A. Zinger
Mohammad Farajzadeh Tehrani的其他文献
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{{ truncateString('Mohammad Farajzadeh Tehrani', 18)}}的其他基金
Conference: Frontiers of Geometric Analysis
会议:几何分析前沿
- 批准号:
2347894 - 财政年份:2024
- 资助金额:
$ 17.84万 - 项目类别:
Standard Grant
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