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 |

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.

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

- 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.
- Identify a surface with or without boundary from its plane model.
- Analyze embeddings of graphs into surfaces.
- Perform basic constructions and calculations of knot theory: tri-colorability, linking number, Seifert surface, link polynomials (Conway, Alexander, Jones, HOMFLY).
- Prove the Brower Fixed Point theorem and Nash Equilibrium Theorem and apply them to derive simple statements in economics.
- Calculate the fundamental group of a surface and derive and apply functorial properties of the fundamental group.
- Perform calculations in braid groups.
- Communicate mathematical ideas in the form of a typed LaTeX document,

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.

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, d3. 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

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.

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. |

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.

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.

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 NTUâ€™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.

Instructor | Office Location | Phone | |
---|---|---|---|

Fedor Duzhin (Dr) | SPMS-MAS-05-23 | 6513 7469 | fduzhin@ntu.edu.sg |

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 |

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."