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School of Environment and Society
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- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kalman Tamas
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Fri5-6(H117)
- Group
- -
- Course number
- MTH.B507
- Credits
- 1
- Academic year
- 2019
- Offered quarter
- 3Q
- Syllabus updated
- 2019/3/18
- Lecture notes updated
- -
- Language used
- English
- Access Index
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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 | knots, links, their genus and fibredness, overview of knot Floer homology | Definitions and properties |
| Class 2 | Alexander polynomial (infinite cyclic cover, Rolfsen’s surgical view、Seifert matrix), Seifert's theorem | Definitions and properties |
| Class 3 | Neuwirth's theorem, Fox calculus | Definitions and properties |
| Class 4 | Kauffman’s state model、Conway skein relation, grid diagrams | Definitions and properties |
| Class 5 | combinatorial definition of knot Floer homology, its degree and Euler characteristic | Definitions and properties |
| Class 6 | d^2=0 and invariance, outline of Floer homology in general | Definitions and properties |
| Class 7 | Morse functions, Morse lemma, sublevel sets, Heegard splittings | Definitions and properties |
| Class 8 | gradient flow, transversality, moduli spaces and their orientations | Definitions and properties |
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
Homework assignments (100％)
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) and complex analysis (Riemann mapping theorem).
Other
I welcome any questions regarding this course.
