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School of Environment and Society
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- Common courses
- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kalman Tamas
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Fri5-6(M-112(H117))
- Group
- -
- Course number
- MTH.B508
- Credits
- 1
- Academic year
- 2023
- Offered quarter
- 4Q
- Syllabus updated
- 2023/9/14
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
We will cover important subjects in low-dimensional geometric topology, such as knots, links, three-manifolds, the Alexander polynomial, and Morse theory, all with a view toward developing Floer homology. Floer homology is central to modern topology and related fields, with many manifestations. In this course we concentrate on Heegaard Floer homology and its applications.
Student learning outcomes
We aim to prepare students for research in low-dimensional topology.
Keywords
knots, links, three-manifolds, Alexander polynomial, genus and fibredness, Morse theory, Floer homology
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
regular lecture course
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | compactness via broken flow lines, gluing, Morse complex | Definitions and properties |
| Class 2 | equivalence of Morse homology and singular homology | Definitions and properties |
| Class 3 | symplectic geometry, Lagrangian submanifolds, action functional | Definitions and properties |
| Class 4 | pseudoholomorphic curves, Lagrangian intersections, Maslov index | Definitions and properties |
| Class 5 | Heegaard diagrams, spin^c structures | Definitions and properties |
| Class 6 | lHeegaard Floer homology of a closed three-manifold | Definitions and properties |
| Class 7 | d^2=0, invariance, the original definition of knot Floer homology | Definitions and properties |
Out-of-Class Study Time (Preparation and Review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
No textbook
Reference books, course materials, etc.
Survey papers by Juhasz (arXiv:1310.3418) and Manolescu (http://arxiv.org/abs/1401.7107), plus online lecture notes by Hutchings (http://math.berkeley.edu/~hutching/teach/276-2010/mfp.ps).
Assessment criteria and methods
based on homework assignments
Related courses
- MTH.B202 ： Introduction to Topology II
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Basic algebraic topology (homology, cohomology, and the fundamental group), complex analysis (Riemann mapping theorem), and the previous quarter of this class.
Other
I preserve the right to change the listed topics. If the audience overlaps with the same course from last year, then I definitely will change topics. Possible alternatives are, for example, the Homfly polynomial and more advanced development of Heegaard Floer theory.
