COURSE OUTLINE: MH3600

Course Title

Knots & Surfaces: Introduction to Topology

Course Code

MH3600

Offered Study Year 3, Semester 1
Course Coordinator Fedor Duzhin (Dr) fduzhin@ntu.edu.sg 6513 7469
Pre-requisites (MH2100, MH2200)OR(MH1803, MH2200)
AU 4
Contact hours Lectures: 39, Tutorials: 12
Approved for delivery from AY 2020/21 semester 1
Last revised 12 Aug 2020, 14:17

Course Aims

This course aims to introduce basic ideas and applications of topology. The emphasis is on beautiful, intuitive material such as surfaces, graphs and knots, rather than technicalities of point-set topology.

Intended Learning Outcomes

Upon successfully completing this course, you should be able to:

  1. Prove simple statements in point-set topology by interpreting definitions of a topological space, basic of topology, continuous function, homeomorphism, embedding, isotopy, homotopy, compactness, connectedness, fixed point property.
  2. Identify a surface with or without boundary from its plane model.
  3. Analyze embeddings of graphs into surfaces.
  4. Perform basic constructions and calculations of knot theory: tri-colorability, linking number, Seifert surface, link polynomials (Conway, Alexander, Jones, HOMFLY).
  5. Prove the Brower Fixed Point theorem and Nash Equilibrium Theorem and apply them to derive simple statements in economics.
  6. Calculate the fundamental group of a surface and derive and apply functorial properties of the fundamental group.
  7. Perform calculations in braid groups.
  8. Communicate mathematical ideas in the form of a typed LaTeX document,

Course Content

Topological spaces and continuous functions.

Surfaces. Graphs and maps on surfaces. Invariants of surfaces: Euler characteristic, orientability, the number of boundary components. Coloring number of a surface.

Knots and links. Link isotopy and Reidemeister moves. Link invariants: tri-colorability, linking number, link polynomials. Links and surfaces: two constructions of a surface whose boundary is the given link.

Brouwer's Fixed Point Theorem, its application to economics. Kakutani's Fixed Point Theorem and Nash equilibrium.

Degree of a map from a circle to itself. Homotopy and fundamental group.

Braid groups.

Assessment

Component Course ILOs tested SPMS-MAS Graduate Attributes tested Weighting Team / Individual Assessment Rubrics
Continuous Assessment
Tutorials
Homework 1, 2, 3, 4, 5, 6, 7, 8 1. b, c, d
3. a
20 both See Appendix for rubric
Mid-semester Quiz
Short Answer Questions 1, 2, 3, 4, 5, 6, 7 1. b
20 individual See Appendix for rubric
Examination (2 hours)
Short Answer Questions 1, 2, 3, 4, 5, 6, 7 1. b, c
60 individual See Appendix for rubric
Total 100%

These are the relevant SPMS-MAS Graduate Attributes.

1. Competence

b. Formulate mathematical statements precisely using rigorous mathematical language

c. Discover patterns by abstraction from examples

d. Use computer technology to solve problems, and to communicate mathematical ideas

3. Communication

a. Present mathematics ideas logically and coherently at the appropriate level for the intended audience

Formative Feedback

There is a formal formative feedback designed as follows: 20% of the course assessment is for doing homework. Specifically, for each homework question of your choice that you type in a shared LaTeX document, you get 1% of the total course mark. The number of attempts is not limited, i.e., you can type an incorrect solution, get feedback from the course instructor, then re-type your solution etc until you get it right. While you are working on the solution, you constantly get formative feedback that allows you to improve.

Besides, tutorials are organized as groupwork and hence, during tutorial sessions, you get informal feedback from your peers and from the course instructor.

Learning and Teaching Approach

Lectures
(39 hours)

These are traditional whiteboard lectures with occasional topology-themed videos and informal discussions with the class whenever appropriate.

Tutorials
(12 hours)

Here, you get to work in small groups. Each group works on one tutorial question and you are required to present your solution on the whiteboard and to type it into a shared LaTeX document. These solutions are then discussed with the whole class.

At home, you continue typing solutions to tutorial questions into the same shared LaTeX document.

Reading and References

The main reading is the complete set of lecture notes and tutorial questions (in fact, this is an unpublished textbook) by Dr Fedor Duzhin that can be downloaded from his website:
https://sites.google.com/site/fedorduzhin/

There are two reference books:
1) Goodman, Sue E. Beginning topology. Vol. 10. American Mathematical Soc., 2009.ISBN-13: 978-0821847961.
2) Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. No. Sirsi) i9780131848696. Upper Saddle River: Pearson Prentice Hall, 2008.

Course Policies and Student Responsibilities

Attendance:

Attendance of lectures is not compulsory and all material can be found in lecture notes. However, you are highly encouraged to attend all lectures and keep handwritten notes.

Absenteeism:

If you miss a quiz, you need to submit a valid leave of absence to the SPMS general office - then that quiz will not be included into the final score calculation, i.e., your score for quizzes will be the average of 4 rather than all 5 quizzes. Failing to submit a valid leave of absence will automatically mean that you get 0 for that quiz.

Online homework:

You are required to submit online homework typed in LaTeX in a document shared with all students in the course. You have unlimited attempts and each time you submit a solution, your instructor will give you immediate feedback on logic, phrasing, and typesetting of your solution. Although each problem set has a few very easy questions, you are encouraged to choose harder questions. It is a good idea to make mistakes and to learn from those mistakes.

Academic Integrity

Good academic work depends on honesty and ethical behaviour. The quality of your work as a student relies on adhering to the principles of academic integrity and to the NTU Honour Code, a set of values shared by the whole university community. Truth, Trust and Justice are at the core of NTU’s shared values.

As a student, it is important that you recognize your responsibilities in understanding and applying the principles of academic integrity in all the work you do at NTU. Not knowing what is involved in maintaining academic integrity does not excuse academic dishonesty. You need to actively equip yourself with strategies to avoid all forms of academic dishonesty, including plagiarism, academic fraud, collusion and cheating. If you are uncertain of the definitions of any of these terms, you should go to the Academic Integrity website for more information. Consult your instructor(s) if you need any clarification about the requirements of academic integrity in the course.

Course Instructors

Instructor Office Location Phone Email
Fedor Duzhin (Dr) SPMS-MAS-05-23 6513 7469 fduzhin@ntu.edu.sg

Planned Weekly Schedule

Week Topic Course ILO Readings/ Activities
1

Classification of surfaces.

Lecture

2

Topological spaces. Basis of a topology. Subspace, product, and quotient topologies.

1, 8

Lecture and tutorial

3

Continuous functions. Homeomorphism, embedding, ambient isotopy.

1, 8

Lecture and tutorial

4

Surfaces. Invariants of surfaces: Euler characteristic, orientability, the number of boundary components.

2, 8

Lecture and tutorial

5

Graphs and maps on surfaces. Coloring number of a surface.

3, 8

Lecture and tutorial

6

Knots and links. Link isotopy and Reidemeister moves. Link invariants: tri-colorability, linking number. Links and surfaces: two constructions of a surface whose boundary is the given link.

4, 8

Lecture and tutorial

7

Link polynomials.

4, 8

Lecture and tutorial

8

Brouwer's Fixed Point Theorem, its application to economics.

5, 8

Lecture and tutorial

9

Kakutani's Fixed Point Theorem and Nash equilibrium.

5, 8

Lecture and tutorial

10

Degree of a map from a circle to itself.

6, 8

Lecture and tutorial

11

Homotopy and fundamental group.

6, 8

Lecture and tutorial

12

Braid groups.

7, 8

Lecture and tutorial

13

Revision

1, 2, 3, 4, 5, 6, 7, 8

Lecture and tutorial

Appendix 1: Assessment Rubrics

Rubric for Tutorials: Homework (20%)

There are unlimited attempts. Each attempt:

0 - not fully correct

1 - fully correct

"Please note that in practice each group member would receive the same score. However, it may vary should there be evidence that you have not contributed meaningfully to your team."

Rubric for Mid-semester Quiz: Short Answer Questions (20%)

There are five quizzes, each quiz is 4% of the total course mark. Each question has several parts and, for each part we give either the full score if the part is fully correct or 0 of the part is not fully correct.

Rubric for Examination: Short Answer Questions (60%)

Each question has several parts and, for each part we give either the full score if the part is fully correct or 0 of the part is not fully correct.