2020 Advanced topics in Geometry C

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Kalman Tamas 
Course component(s)
Lecture
Mode of instruction
ZOOM
Day/Period(Room No.)
Mon3-4(Zoom)  
Group
-
Course number
MTH.B403
Credits
1
Academic year
2020
Offered quarter
3Q
Syllabus updated
2020/9/18
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

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].

Student learning outcomes

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 that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Standard lecture course

Course schedule/Required learning

  Course schedule Required learning
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

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)

None required

Reference books, course materials, etc.

J. Milnor: Topology from the differentiable viewpoint
W. Fulton: Algebraic topology

Assessment criteria and methods

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 (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed [Geometry I], [Geometry II] and [Geometry III].

Office hours

to be determined

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