New Perspectives in Heegaard Floer Homology

A fundamental question in topology is whether we can deform one shape into another while preserving certain intrinsic properties. By adding a time dimension, it is natural to think of the deformation as being a 4-dimensional object which has one object at one end, and the other shape at the other end. A fundamental question in low-dimensional topology is whether we can build a 4-dimensional space which connects two given 3-dimensional objects.

Heegaard Floer homology is an important tool for studying such questions. It gives topologists a way of knowing that two 3-dimensional spaces cannot be related by a 4-dimensional space. Heegaard Floer homology involves the counts of complicated solutions to differential equations and has deep connections to Seiberg-Witten theory, Yang-Mills theory, and mathematical physics.

This project aims to develop new tools for computing Heegaard Floer homology. A main focus of this project is to develop a new minus version of bordered Heegaard Floer homology for 3-dimensional spaces with torus boundary components. This theory is based on the link surgery formula of Manolescu and Ozsváth.

The project aims to use this theory to study the lattice homology conjecture of Némethi. Additionally, the project aims to study symmetries in this theory and in the link surgery formula. With these tools, the PI hopes to give computable invariants of homology cobordism.

In addition, the PI endeavors to mentor graduate students and undergraduates, as well as organize events to disperse knowledge.

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.