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

流形是局部类似于欧几里得空间的形状。这个项目的重点是流形是封闭的，因为它们没有边界，也不会延伸到无限远。一个封闭的一维流形相当于圆，而一个封闭的（可定向的）二维流形相当于球体、一个甜甜圈的表面，或者一个有两个或更多洞的“甜甜圈”的表面。封闭的三维流形不容易可视化，而封闭的四维流形可能具有非常复杂的结构，并且不容易理解。具有三个空间方向和一个时间方向的四维流形在广义相对论中被用作宇宙的模型。四维流形在规范理论中也起着核心作用，规范理论的发展统一了四种已知基本力中的三种（电磁力、弱相互作用和强相互作用）。该项目的第一个目标是完成一个来自超对称量子场论的预测的数学证明，该理论涉及两种不同的规范理论，用于帮助理解四维流形。该项目的第二个目标是推进对四维流形可能结构的理解，近一个世纪以来，四维流形一直是数学家和物理学家的魅力和灵感来源。近几十年来，三维流形可能结构的分类取得了巨大的进展，但尽管数学家们付出了巨大的努力来分析四维流形，但它们仍然是神秘的。该项目的第三个目标是开发方法来联系不同的方法来理解三维流形的结构。这个项目有研究生参与研究。为了帮助培养下一代数学家，校长们还将继续他们组织研讨会和会议的传统，撰写说说性文章，以帮助吸引对数学职业和研究感兴趣的更广泛的受众，指导本科生、研究生和博士后研究人员。并通过国家数学博物馆的暑期课程和推广活动，鼓励高中生对数学的兴趣。该项目的第一个目标是利用基于非阿贝尔单极子模空间的数学严格方法，完成关于具有可接受拓扑和简单类型的封闭、定向、光滑四维流形的Donaldson和Seiberg-Witten不变量的Witten公式的证明。这项工作将应用一种新的方法来粘接几何分析中出现的非线性偏微分方程的解，以建立非阿贝尔单极子预期粘接定理的证明。他们项目的第二个目标是完成对具有非零Seiberg-Witten不变量的Seiberg-Witten简单型单连通四维流形的猜想Bogomolov-Miyaoka-Yau不等式的证明。该方法将奇异解析空间的莫尔斯理论的一个新版本应用于非阿贝尔单极子的奇异模空间，证明了另一个非线性偏微分方程——四维流形上具有规定拓扑的秩二厄米向量束上的反自对偶Yang-Mills方程的解的存在性。该项目的第三个目标是推导出封闭三维流形的瞬子和Seiberg-Witten Floer同调之间的关系，可能与基本群和接触结构有关。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。

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.