▶Japanese
Post to SNS
- Home
- > School of Science
- > Graduate major in Mathematics
- > Special lectures on current topics in Mathematics D
- School of Science
-
School of Engineering
- Undergraduate major in Mechanical Engineering
- Undergraduate major in Systems and Control Engineering
- Undergraduate major in Electrical and Electronic Engineering
- Undergraduate major in Information and Communications Engineering
- Undergraduate major in Industrial Engineering and Economics
- Common courses
- School of Materials and Chemical Technology
- School of Computing
- School of Life Science and Technology
- School of Environment and Society
- 工学院，物質理工学院，環境・社会理工学院共通科目
- Liberal arts and basic science courses
- First-Year Courses
- School of Science
- School of Engineering
- School of Materials and Chemical Technology
- School of Computing
- School of Life Science and Technology
-
School of Environment and Society
- Department of Architecture and Building Engineering
- Department of Civil and Environmental Engineering
- Department of Transdisciplinary Science and Engineering
- Department of Social and Human Sciences
- Department of Innovation Science
- Graduate major in Technology and Innovation Management
- Common courses
- 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.)
- Intensive (H213)
- Group
- -
- Course number
- MTH.E634
- Credits
- 2
- Academic year
- 2016
- Offered quarter
- 3-4Q
- Syllabus updated
- 2016/4/27
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
-
Course description and aims
In this lecture series we introduce the concepts of knot, link, 3-dimensional manifold, Alexander polynomial and so on, and prove some of their basic properties. We also discuss Morse and Floer theory. To help better understand the material, practice problems will be provided and their solutions collected.
The aim of the course is to build a strong foundation in low-dimensional topology. thus enabling students to start doing independent research in the field.
Student learning outcomes
The main themes of the course are knot theory and Floer homology groups associated to 3-dimensional manifolds. Floer homology is currently one of the most advanced tools used in topology and related fields, with many possible applications. After covering the basics of low-dimensional topology, our main focus will be Heegaard Floer homology.
Keywords
Knot, link, 3-dimensional manifold, 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 format with homework exercises
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Knots, links, genus and fibredness. Formal properties of knot Floer homology. | to be specified in each lecture |
| Class 2 | Three definitions of the Alexander polynomial (infinite cyclic cover, Rolfsen's surgical view, Seifert matrix). Seifert's theorem (on the genus and the degree of the Alexander polynomial). | to be specified in each lecture |
| Class 3 | Neuwirth's theorem (on fiberedness and the leading coefficient of the Alexander polynomial), definition of the Alexander polynomial via Fox calculus. | to be specified in each lecture |
| Class 4 | Kauffman’s state model、Conway's skein relation for the Alexander polynomial. Grid diagrams. | to be specified in each lecture |
| Class 5 | Combinatorial definition of knot Floer homology, its grading and Euler characteristic. | to be specified in each lecture |
| Class 6 | The (combinatorial) proof of d^2=0 and invariance. Outline of general Floer theory. | to be specified in each lecture |
| Class 7 | Morse function, Morse lemma, changes in sublevel sets. Heegaard decomposition for 3-dimensional manifolds. | to be specified in each lecture |
| Class 8 | Gradient flow, transversality, moduli spaces and their orientation. | to be specified in each lecture |
| Class 9 | Compactness via broken flow lines, Morse complex, gluing. | to be specified in each lecture |
| Class 10 | The isomorphism of Morse homology and singular homology. | to be specified in each lecture |
| Class 11 | Symplectic geometry, Lagrangian submanifolds, action functional. | to be specified in each lecture |
| Class 12 | Holomorphic curves, Lagrangian intersection homology, Maslov index. | to be specified in each lecture |
| Class 13 | Heegaard diagrams, spin^c structures, Heegaard Floer homology for closed manifolds. | to be specified in each lecture |
| Class 14 | d^2=0 and invariance. The original definition of knot Floer homology. | to be specified in each lecture |
| Class 15 | Sutured Floer homology and the proof of genus and fibredness detection. | to be specified in each lecture |
Textbook(s)
none in particular
Reference books, course materials, etc.
Some of the lecture will follow recent survey papers by Juhasz (arXiv:1310.3418) and Manolescu (http://arxiv.org/abs/1401.7107). For the Morse theory part, Hutchings's lecture notes (http://math.berkeley.edu/~hutching/teach/276-2010/mfp.ps) will be useful.
Assessment criteria and methods
Evaluation will be based on homework. Details will be specified in the class.
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
- MTH.B341 ： Topology
- MTH.C301 ： Complex Analysis I
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Complex analysis (up to the Riemann mapping theorem), algebraic topology (homology and homotopy), and smooth manifolds (for example, integral trajectories of a vector field) will be useful. The most important thing, however, is to have an open and inquisitive mind!
