Topology, Geometry and Physics

这个项目将研究某些偏微分方程系统的解的行为，这些方程被认为是四维空间的基本结构的信息。感兴趣的方程来自高能物理学，在那里它们被引入作为基本粒子相互作用的非常对称的模型。 关于解决方案要解决的基本问题是：对于特定的应用程序，是有无穷多个还是无穷多个解决方案？ 在这两种情况下，解决方案是什么样的？ 它们对四维空间有什么影响？ 作为背景：这些方程太复杂了，计算机无法求解（更不用说手工求解了）。 因此，几乎所有关于解的信息都必须通过对方程本身的详细分析来获得。 关于四维空间的问题，这里的中心问题是理解可能的四维空间的列表。 最后一个问题也与物理学有关，因为可观测的宇宙有四维--三个空间维度，然后是时间方向。 由于感兴趣的方程来自高能物理理论，并且与那里和凝聚态物理中使用的其他方程非常相似，因此该项目中开发的分析工具也可以应用于这些其他领域的问题。 该项目还有一个重要的教育支持方面，即旨在使研究生加快相关的几何，拓扑和分析。该项目的研究有三个主要重点。 第一个问题是三维流形和半直线的乘积上的Kapustin-Witten方程的解的行为，其中半直线的一端具有由三维流形中的纽结给出的奇异边界条件。 这个重点的目标是看看非收敛序列的解决方案是否有病理，防止解决方案计数定义的结不变量的结在三维流形。 可能出现两个麻烦的来源，一个是边界条件的新病理，另一个是由于Z/2调和1-形式的存在。 第二个焦点是关于这些相同的Kapustin-Witten方程的解空间的结构的代数含义，以及关于三流形与直线的乘积的Kapustin-Witten方程的解空间的结构的代数含义：给定非紧性问题和形式上零维解空间的新奇，可以从这些空间定义代数不变量吗？ 第三个焦点涉及上述Z/2谐波1-形式的意义，更一般地说，自旋束的Z/2谐波部分。 这样的Z/2调和截面表现为四维流形上的Vafa-Witten方程和三维和四维流形上的多旋量Seiberg-Witten方程的解序列的重整化极限。 尽管有特定的主题和问题，该项目的长期目标是了解光滑的四维流形的结构。该奖项反映了NSF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。

This project will study the behavior of solutions to certain systems of partial differential equations that are thought to give information about the underlying structures of four dimensional spaces. The equations of interest came from high energy physics where they were introduced as very symmetric models for elementary particle interactions. Basic question to address with regards to solutions are these: Are there finitely many or infinitely many solutions for the specific applications? In either case, what do the solutions look like? And what are their implications with regards to four dimensional spaces? By way of background: The equations are far too complicated for a computer to solve (let alone to solve by hand). Because of this, almost all information about the solutions must be obtained via a detailed analysis of the equations themselves. With regards to questions about four dimensional spaces, the central issue here is to understand the list of possible four dimensional spaces. This last question is also relevant to physics by virtue of the fact that the observable universe has four dimensions--three spatial dimensions and then the time direction. Since the equations of interest come from high energy physics theory and are close cousins to others used there and in condensed matter physics, the analysis tools developed in this project may also find applications to questions in these other fields. There is also an important educational support aspect of this project that is meant to bring graduate students up to speed on the relevant geometry, topology and analysis.The research in this prject has three primary focus points. The first concerns the behavior of the solutions to the Kapustin-Witten equations on the product of a three-dimensional manifold and the half-line with singular boundary conditions at one end given by a knot in the three-manifold. The goal for this focus point is to see whether non-convergent sequences of solutions have pathologies that prevent solution counting definitions of knot invariants for knots in the three dimensional manifold. Two sources of trouble likely arise, one is a novel pathology from the boundary conditions, and the other due to the presence of Z/2 harmonic 1-forms. The second focus point concerns the algebraic implications of the structure of the solution space for these same Kapustin-Witten equations and for the Kapustin-Witten equations on the product of the three-manifold with the line: Can an algebraic invariant be defined from these spaces given the non-compactness issues and with the novelty of formally zero dimensional solution spaces? The third focus point concerns the significance of the afore-mentioned Z/2 harmonic 1-forms and, more generally, Z/2 harmonic sections of a spin bundle. Such Z/2 harmonic sections appear as renormalized limits of sequences of solutions to the Vafa-Witten equations on four dimensional manifolds and the multi-spinor Seiberg-Witten equations on manifolds of dimensions three and four. Specific topics and questions not withstanding, the long term goal of this project is to understand the structure of smooth, four dimensional manifolds.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.