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：
Mohammad Farajzadeh Tehrani
金额：
$ 17.84万
依托单位：
University of Iowa
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：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2003340&HistoricalAwards=false
关键词：
Logarithmic Moduli Spaces Symplectic Geometry

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.

辛流形是概括经典力学中相空间概念的几何对象。在过去的四十年中，辛几何领域发展迅速，与代数几何、低维拓扑和高能物理等其他重要研究领域建立了新的联系。该奖项支持对称为全纯曲线及其相应不变量和代数结构的基本对象的研究。特别是，研究人员将解决基本问题，例如在存在称为除数的对象的情况下构建表现良好的全纯曲线族。这项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及其在其他领域的应用。他还将在公共图书馆发起一个针对高中生的数学俱乐部。该提案的主要目标是为任意对辛流形和法线交叉辛除数构造全纯曲线模空间，满足特定性质。构建此类模空间需要紧致化、变形理论的分析框架、解决横向问题并证明粘合定理。这些模空间可直接应用于枚举几何、镜像对称、深谷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。 PI 与 M. McLean 和 A. Zinger 合作，引入了正态交叉辛除数和簇的拓扑概念。他们构建了正则化和对数正切束等工具来处理此类对象。最近，PI 开发了一种基于对数切丛的新颖的紧致化和变形理论。他将使用此设置来完成构建的其余步骤。主要项目是定义相对于任意法线交叉除数的 Gromov-Witten 不变量。其他项目包括证明 Gromov-Witten 不变量的退化公式、查找与代数方法的关系以及开发镜像对称的应用。该项目由 Geometric Analysis 和刺激竞争研究既定计划 (EPSCoR) 共同资助。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力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：
Geometry & Topology
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：
ITU Journal on Future and Evolving Technologies
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