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

流形是局部类似于欧几里德空间的形状。这个项目关注的是那些闭合的流形，因为它们没有边界边，也不会延伸到无穷远。一个封闭的一维流形等价于圆，而一个封闭的(可定向的)二维流形等价于球体、圆环的表面或具有两个或多个孔的“圆环”的表面。封闭的三维流形不能如此容易地可视化，而封闭的四维流形可能具有非常复杂的结构，并且不容易被理解。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。四维流形也在规范理论中发挥着核心作用，规范理论的发展是为了统一四种已知基本力中的三种(电磁、弱相互作用和强相互作用)。该项目的第一个目标是完成超对称量子场理论预测的数学证明，超对称量子场理论将两种不同的规范理论联系起来，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可能结构的理解，这是近一个世纪来数学家和物理学家的魅力和灵感来源。近几十年来，对三维流形可能结构的分类有了很大的进步，但四维流形仍然是个谜，尽管数学家们付出了巨大的努力来分析它们。该项目的第三个目标是开发方法，将理解三维流形结构的不同方法联系起来。该项目邀请研究生参与这项研究。为了帮助培养下一代数学家，校长们还将继续他们的传统，组织研讨会和会议，提供说明性文章，帮助更多的观众对了解数学职业和研究感兴趣，指导本科生和研究生以及博士后研究人员，并通过暑期计划和国家数学博物馆的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完成Witten公式的证明，该公式涉及具有可接受的拓扑和简单类型的封闭、定向、光滑的四维流形的Donaldson和Seiberg-Witten不变量，采用基于非阿贝尔单极的模空间的数学严格方法。这项工作将应用一种新的方法来粘合几何分析中出现的非线性偏微分方程解，以建立对非阿贝尔单极子的预期粘合定理的证明。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简单类型的单连通四维流形猜想的Bogomolov-Miyaoka-Yau不等式的证明。该方法将奇异解析空间的Morse理论应用于非阿贝尔单极子的奇异模空间，证明了四维流形上具有给定拓扑的二阶厄米特向量丛上的另一类非线性偏微分方程解的存在性.该项目的第三个目标是推导封闭三维流形的瞬子和Seiberg-Witten Floer同调之间的关系，潜在地关联基本群和联系结构。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.