Gauge Theory, Floer Homology, and Invariants of Low-Dimensional Manifolds

这项NSF奖资助研究嵌入了打结曲线和曲面的三维和四维空间。科学家对这类天体特别感兴趣，因为它们与我们的物理世界和时空连续体有关。这些研究将加深对我们宇宙的理解。此外，打结曲线理论在其他科学领域也有广泛的应用，例如，在DNA和蛋白质分子结构的研究中。近年来，一些新的技术被引入来研究这些低维空间。这个项目将致力于进一步开发技术，提供令人惊讶的见解，并证明现有理论之间的意想不到的关系。在合作研究中，PI将探索新的不变量，帮助我们区分不同的空间。这个项目致力于增强规范理论和Floer同调的能力，并将这些工具应用于三维和四维流形的研究。在这个项目的第一部分，PI将与Tirasan Khandhawit和Hirofumi Sasahira一起，进一步发展一般三维流形的展开Seiberg-Witten Floer谱不变量理论，并利用它得出关于四维流形的新结论。在这个项目的第二部分，PI将与Daniel Ruberman和Nikolai Saveliev一起研究同调于一个圆与三个球面的乘积所得到的空间的四个流形的不变量，在三维同调余边群的研究中有着惊人的应用。在本项目的第三部分，PI将从规范理论中寻找不变量的束理论描述，目的是使这些不变量更加灵活和强大。

This NSF award funds research to study three- and four-dimensional spaces with knotted curves and surfaces embedded within them. Such objects are of particular interest to scientists because of their connection to our physical world and the space-time continuum. These studies will deepen the understanding of our universe. Moreover, the theory of knotted curves has seen broad applications in other fields of science, for example, in investigations of the structure of DNA and that of protein molecules. In recent years, several new techniques have been introduced to study these low dimensional spaces. This project will be devoted to further developing the techniques, providing surprising insights and proving unexpected relationships between existing theories. In collaborative research, the PI will explore new invariants that would help us distinguish between different spaces.This project is devoted to enhancing the power of gauge theory and Floer homology and applying these tools to the study of three- and four-dimensional manifolds. In the first part of this project, joint with Tirasan Khandhawit and Hirofumi Sasahira, the PI will further develop the theory of unfolded Seiberg-Witten Floer spectrum invariants for general three-manifolds and use it to draw new conclusions regarding four-manifolds. In the second part of this project, joint with Daniel Ruberman and Nikolai Saveliev, the PI will carry out research on invariants of four-manifolds with homology identical to the space obtained as a product of a circle with a three sphere, with surprising applications in the study of the three-dimensional homology cobordism group. In the third part of this project, the PI will seek for a sheaf theoretic description of invariants from gauge theory, with the goal to make these invariants more flexible and powerful.