Collaborative Research: Geometric Analysis, Monopoles, and Applications to Low-Dimensional Manifolds

合作研究：几何分析、单极子以及低维流形的应用

• 批准号: 2104871
• 负责人: Thomas Leness
• 金额: $ 18.97万
• 依托单位: Florida International University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104871&HistoricalAwards=false
• 关键词: Collaborative; Research; Geometric; Analysis; Monopoles

Manifolds are shapes that locally resemble Euclidean space. This project focuses on manifolds that are closed in the sense that they have no boundary edges and do not extend to infinity. A closed one-dimensional manifold is equivalent to the circle, while a closed (orientable) two-dimensional manifold is equivalent to the sphere, the surface of a donut, or the surface of a “donut” with two or more holes. Closed three-dimensional manifolds cannot be so easily visualized, while closed four-dimensional manifolds can have very complicated structures and are not well-understood. Four-dimensional manifolds, with three spatial directions and one temporal direction, are used in general relativity as models for the universe. Four-dimensional manifolds also play a central role in gauge theories developed to unify three of the four known fundamental forces (the electromagnetic, weak, and strong interactions). The first goal of the project is to complete a mathematical proof of a prediction from supersymmetric quantum field theory, one that relates two different gauge theories used to help understand four-dimensional manifolds. The second goal of the project is to advance understanding of the possible structures of four-dimensional manifolds, a source of fascination and inspiration for mathematicians and physicists for nearly a century. The classification of possible structures of three-dimensional manifolds advanced tremendously in recent decades, but four-dimensional manifolds remain mysterious, despite intense effort by mathematicians to analyze them. The third goal of the project is to develop methods to relate different approaches to understanding the structure of three-dimensional manifolds. The project involves graduate students in the research. To help train the next generation of mathematicians, the principals also will continue their tradition of organizing seminars and conferences, contributing expository articles to help engage a broader audience interested in learning about careers and research in mathematics, mentoring undergraduate and graduate students and postdoctoral researchers, and encouraging the interest of high-school students in mathematics through summer programs and outreach activities at the National Museum of Mathematics.The first goal of the project is to complete a proof of Witten's formula relating the Donaldson and Seiberg-Witten invariants of a closed, oriented, smooth four-dimensional manifold with admissible topology and simple type, employing a mathematically rigorous method based on moduli spaces of non-Abelian monopoles. The work will apply a new approach to gluing solutions to non-linear partial differential equations that arise in geometric analysis to establish a proof of an expected gluing theorem for non-Abelian monopoles. The second goal of their project is complete a proof of the conjectured Bogomolov-Miyaoka-Yau inequality for simply connected four-dimensional manifolds of Seiberg-Witten simple type and having non-zero Seiberg-Witten invariants. The approach uses a new version of Morse theory for singular analytic spaces applied to the singular moduli space of non-Abelian monopoles to prove existence of solutions to another non-linear partial differential equation – the anti-self-dual Yang-Mills equation on a rank-two Hermitian vector bundle with prescribed topology over a four-dimensional manifold. The third goal of the project is to derive relations between the instanton and Seiberg-Witten Floer homologies of closed three-dimensional manifolds, potentially relating fundamental groups and contact structures.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.

歧管是局部类似欧几里得空间的形状。该项目的重点是在没有边界边缘并且不会扩展到无限的意义上是封闭的。封闭的一维歧管等同于圆，而封闭的（可定向）二维歧管等效于球体，甜甜圈的表面或带有两个或更多孔的“甜甜圈”的表面。封闭的三维流形不能那么容易地可视化，而封闭的四维流形可以具有非常复杂的结构，并且不理解。具有三个空间方向和一个临时方向的四维流形被用作宇宙的模型。四维流形在衡量理论中也起着核心作用，以统一四个已知的基本力中的三个（电磁，弱和强相互作用）。该项目的第一个目标是完成超对称量子场理论的预测的数学证明，该预测与两个不同的量学理论有关，用于帮助理解四维流形。该项目的第二个目标是促进对四维流形的可能结构的理解，这是近一个世纪以来对数学家和物理学家的迷恋和灵感的来源。近几十年来，三维流形的可能结构的分类大大提高了，但是四维流形仍然是神秘的，数学家对它们进行分析非常有效。该项目的第三个目标是开发方法，以关联不同的方法来理解三维流形的结构。该项目涉及研究生。为了帮助培训下一代数学家，校长还将继续其组织半手和会议的传统，贡献说明性文章，以帮助吸引更广泛的受众，以吸引有兴趣学习的职业和研究的数学，心理学习和研究生和研究生，以及通过暑期研究人员对暑期学生的兴趣和鼓励的数学兴趣，并鼓励暑期学生的兴趣，并鼓励国有兴趣，并鼓励国有学位的兴趣。项目是为了完成Witten公式的证明，该公式将唐纳森和塞伯格（Seiberg）的封闭式，定向，光滑的四维歧管的不变式与可允许的拓扑和简单类型相关联，采用了基于数学上严格的方法，基于非阿贝尔单调摩托车的模量空间。这项工作将在几何分析中出现的非线性部分偏微分方程中使用一种新方法将溶液粘合解决方案，以建立预期的非亚伯利亚单托尔定理的证明。他们的项目的第二个目标是完整证明了猜想的Bogomolov-Miyaoka-yau不平等现象，简单地连接了Seiberg-Witten Simple类型的四维流形，并且具有非零Seiberg-witten不变性。该方法使用一种新版本的莫尔斯理论来用于奇异的分析空间，该空间应用于非阿布尔单孔的奇异模量空间，以证明存在于另一个非线性偏微分方程方程的解决方案 - 反对双向的YANG-MILLS方程，上面是在级别的Hermitian Vector Bundle上，具有与四位数的拓扑拓扑相比。该项目的第三个目标是在封闭的三维流形的Instanton和Seiberg-Witten的浮子同源物之间获得关系，这可能与基本群体和联系结构有关。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力和更广泛影响的评估来审查Criteria通过评估来通过评估来获得的支持。