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

这个NSF奖项资助研究三维和四维空间与打结的曲线和曲面嵌入其中。科学家对这些物体特别感兴趣，因为它们与我们的物理世界和时空连续体有联系。这些研究将加深我们对宇宙的理解。此外，纽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，一些新的技术已经被引入到研究这些低维空间。该项目将致力于进一步发展技术，提供令人惊讶的见解，并证明现有理论之间意想不到的关系。在合作研究中，PI将探索新的不变量，以帮助我们区分不同的空间。该项目致力于增强规范理论和Floer同调的力量，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，与Tirasan Khandhawit和Hirofumi Sasahira联合，PI将进一步发展一般三流形的展开Seiberg-Witten Floer谱不变量理论，并使用它得出关于四流形的新结论。 在这个项目的第二部分，联合丹尼尔鲁伯曼和尼古拉Saveliev，PI将进行研究的不变量的四个流形的同调相同的空间作为一个产品的一个圆与三个领域，与令人惊讶的应用程序的研究三维同调配边群。在这个项目的第三部分，PI将从规范理论中寻求不变量的层理论描述，目标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.