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.

符号歧管是几何对象，它概括了经典力学中的相位概念。在过去的四十年中，符号几何形状领域迅速发展，从而与其他重要的研究领域建立了新的联系，例如代数几何，低维拓扑和高能量物理学。该奖项支持对称为Holomorphic曲线的基本对象的研究及其相应的不变性和代数结构。特别是，研究人员将解决基本问题，例如在存在称为分裂的物体的情况下，举止良好的全体形态曲线家庭的建设。这项研究包含特定的项目，可以由研究生和博士后进行。研究人员将组织年度迷你群岛，以引入本科生来研究几何学和拓扑的机会，以及它们在其他领域的应用。他还将在针对高中生的公共图书馆开设一个数学俱乐部。该提案的主要目的是为任意的象征性歧管和正常的交叉符号分歧构建模态曲线的模量空间，以满足特定的特定属性。这种模量空间的构建需要一个紧凑的变形理论的分析框架，解决了横向问题并证明了胶合定理。这些模量空间在枚举的几何形状，镜像对称性，福卡亚类别的构建以及其他活跃的研究领域中有直接应用，在符号几何，代数几何学和弦理论中。与M. McLean和A. Zinger合作，PI引入了正常交叉互合式除数和多样性的拓扑概念。他们构建了用于使用此类对象的正规化和对数切线捆绑包等工具。最近，PI基于对数切线束开发了一种新颖的压缩和变形理论。他将使用此设置来处理施工的其余步骤。主要项目是定义Gromov-witten的不变性，相对于任意正常的越过除数。其他项目包括证明对格罗莫夫（Gromov）的不变性人士的退化公式，找到与代数方法的关系，并利用应用程序来镜像对称性。该项目由几何分析和启发竞争性研究的既定计划共同资助（EPSCOR）。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来提供支持。

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