2020 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry C
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Tamas Kalman
- Class Format
- Lecture (Zoom)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon (Zoom)
- Class
- -
- Course Code
- MTH.B403
- Number of credits
- 100
- Course offered
- 2020
- Offered quarter
- 3Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
The aim of this lecture course is to familiarize students with the basic language of, and some fundamental theorems in differential topology. This course will be succeeded by [MTH.B404 : Advanced topics in Geometry D].
Course description and aims
As an outcome, students are expected to
・understand the notions of homotopy group, cobordism ring, the degree of a map etc.
・be familiar with the classification of surfaces, the method of smooth approximation, general position arguments, the chain complex for the homology group of a CW complex etc.
Keywords
vector field, rotation, genus, homotopy group, degree, immersion, cobordism, transversality
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | vector fields and their rotation (Whitney index) | Details will be provided during each class session |
| Class 2 | classification of closed surfaces | Details will be provided during each class session |
| Class 3 | smooth approximation of continuous maps | Details will be provided during each class session |
| Class 4 | homotopy groups | Details will be provided during each class session |
| Class 5 | immersions, submersions, transversality | Details will be provided during each class session |
| Class 6 | degree of a map | Details will be provided during each class session |
| Class 7 | homology groups of CW complexes | Details will be provided during each class session |
| Class 8 | cobordism rings, Pontryagin--Thom construction | Details will be provided during each class session |
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)
None required
Reference books, course materials, etc.
J. Milnor: Topology from the differentiable viewpoint
W. Fulton: Algebraic topology
Evaluation methods and criteria
Evaluation will be based on exams and homework. Details will be provided during class sessions.
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
- MTH.B404 ： Advanced topics in Geometry D
Prerequisites
Students are expected to have passed [Geometry I], [Geometry II] and [Geometry III].
Office hours
to be determined