Courses
The following courses are foundational for Floer homology and related fields.
For Morse homology, see:
- Morse Theory and Floer Homology, Part I, Michèle Audin, Mihai Damian
- Monopoles and Three-Manifolds, Section 2, Peter B. Kronheimer, Tomasz S. Mrowka
- Morse Theory, John Milnor
Further supplements (optional):
- Analysis: functional analysis, ODEs, PDEs, elliptic PDE; for foundational analysis, see Harvard Math 55B
- Algebra: homological algebra, Lie groups and Lie algebras, representation theory; for foundational algebra, see Harvard Math 55A
- Geometry: Riemann surfaces, Riemannian geometry, symplectic geometry, contact geometry, geometric analysis
- Topology: homotopy theory, differential topology, fiber bundles and characteristic classes
For low-dimensional manifolds, see:
- An Introduction to Geometric Topology, Bruno Martelli
- 4-Manifolds and Kirby Calculus, Robert E. Gompf, András I. Stipsicz
For an introduction to knots, see:
- Grid Homology for Knots and Links, Chapter 2, Peter S. Ozsváth, András I. Stipsicz, Zoltán Szabó
- Notes of Jacob Rasmussen, Sections 1–2
Differential Geometry (PKU course: Differentiable Manifolds)
Course Page | Differential geometry is a branch of mathematics that studies geometric structures on manifolds and is fundamental to understanding Floer homology. This course introduces basic concepts such as manifolds, tangent bundles, Riemannian metrics, and curvature, laying the foundation for subsequent study of topology and geometry.
Introductory References:
- Introduction to Differentiable Manifolds (in Chinese), Weihuan Chen
- Lectures on Differential Geometry (in Chinese), Shing-Shen Chern, Weihuan Chen
- Differential Geometry: Bundles, Connections, Metrics and Curvature, Chapters 1-16, Clifford H. Taubes
- Differential Forms in Algebraic Topology, Raoul Bott, Loring W. Tu
Algebraic Topology (PKU course: Homology Theory)
Course Page | Algebraic topology studies the properties of topological spaces through algebraic methods and is an important theoretical foundation for Floer homology. This course introduces basic concepts such as homology groups and cohomology, as well as important computational tools such as CW complexes, Mayer-Vietoris sequences, and universal coefficient theorems.
Introductory References:
- Homology Theory (in Chinese), Boju Jiang
- Lectures on Algebraic Topology, Chapters 1-3, Haynes Miller
- Algebraic Topology, Chapters 2-3, Allen Hatcher
- An Introduction to Homological Algebra, Charles A. Weibel
The following are advanced courses that can be studied as needed.
Floer Homology Overview
Course Page | Floer homology is a generalization of Morse homology to infinite-dimensional spaces and is an important tool in low-dimensional topology and symplectic geometry. This course introduces the basic theory of sutured 3-manifolds and Floer homology, including Heegaard Floer homology, monopole Floer homology, instanton Floer homology, and more.
Knots, SU(2), and the Pillowcase
Course Page | This short course introduces the relationship between knots, SU(2) representations, and the pillowcase, focusing on SU(2) non-abelian representations of 3-manifolds and applications of instanton Floer homology. The course starts from a problem in Kirby’s problem list, understanding the properties of Dehn surgery manifolds by studying submanifolds in the SU(2) character variety of the boundary torus of knot complements (homeomorphic to the pillowcase).