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

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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.