Overview: Homological invariants associated to 3-manifolds, knots, and links (notably Heegaard Floer homology and Khovanov homology) have been among the most important tools in low-dimensional topology in the 21st century. One major breakthrough in the past decade has been the reinterpretation of several of these algebraic invariants in terms of immersed curves on surfaces, whereby various topological gluing formulas can be interpreted as a combinatorial version of Lagrangian Floer homology. In this minicourse, we will first learn some of the basics of the Heegaard Floer invariants, with more an emphasis on their structure, basic properties, and applications than on all the details of their construction, and then dive into the immersed curve interpretation. If time permits, we will then explore the parallel story on the Khovanov side.
Prerequisites: Elementary algebraic topology, at the level of Duke's Math 611.
Time/location: Mondays and Wednesday, 10:05-11:20am, October 30 - December 4, Gross 324. You can also join by Zoom here (please contact me for the password).
References: I will be drawing on a number of different sources in this course, including:
Lecture videos: Links to videos will be posted here throughout the course.