2023 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry G1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Tamas Kalman
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Fri (M-112(H117))
- Class
- -
- Course Code
- MTH.B507
- Number of credits
- 100
- Course offered
- 2023
- Offered quarter
- 3Q
- Syllabus updated
- Jul 8, 2025
- Language
- English
Syllabus
Course overview and goals
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.
Course description and aims
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
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
regular lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| 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 |
Study advice (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).
Evaluation methods and criteria
Homework assignments (100％)
Related courses
- MTH.B202 ： Introduction to Topology II
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
Prerequisites
Basic algebraic topology (homology, cohomology, and the fundamental group) and complex analysis (Riemann mapping theorem).
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.