Course Title | ## Topology and Manifolds | ||

Course Code | ## MH3600 | ||

Offered | Study Year 2, Semester 2 | ||

Course Coordinator | Philipp Harms (Dr) | philipp.harms@ntu.edu.sg | 6513 7187 |

Pre-requisites | MH2100 OR MH1803 | ||

AU | 4 | ||

Contact hours | Lectures: 39, Tutorials: 12 | ||

Approved for delivery from | AY 2022/23 semester 2 | ||

Last revised | 3 Jan 2023, 17:42 |

This is a first introduction to topology and calculus on manifolds. The tools introduced in this course are the natural framework for the generalization of the ideas that you learnt in Calculus I, II, and III to infinite-dimensional and non-Euclidean spaces. These methods open the door to other fields in mathematics like algebraic topology, functional analysis, differential/Riemannian/symplectic/Poisson geometry, or Lie theory, to name a few. They also have strong ties with important applications in the physical sciences and engineering like dynamical systems, mechanics, symmetry analysis, or control theory.

The aim of this course is to enable you to formulate and solve mathematical problems using the ideas and the formalism coming from topology and global analysis.

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

- Understand the basics of point-set topology and to interpret important notions like convergence and continuity in topological terms.
- Handle topological notions like connectedness, compactness.
- Understand applications of topology to approximation and fixed-point theory.
- Formulate non-Euclidean spaces as manifolds.
- Handle differentiability problems in globally defined spaces as well as expressing dynamical systems as vector fields

Topological spaces. Basis of a topology. Subspace, product, box, and quotient topologies. Continuous functions. Homeomorphisms, embeddings. Metric topologies. Topologies induced by norms and inner products. Function spaces and uniform convergence. Connectedness and compactness. The Intermediate Value Theorem, the Extreme Value Theorem, Heine-Borel-Lebesgue, the Uniform Continuity Theorem. Countability and separation axioms. The Urysohn lemma and the Metrization Theorem. Approximation. The Stone-Weierstrass Theorems. Fixed Point Theorems and applications. Manifolds. Charts and atlas. Atlas equivalence and manifold topologies. Examples. Submanifolds, products, and mappings. The tangent and the cotangent bundle. Subimmersions, immersions, and transversality. Vector fields, flows, and dynamical systems. The Lie derivative. Vector fields as differential operators.

Component | Course ILOs tested | SPMS-MAS Graduate Attributes tested | Weighting | Team / Individual | Assessment Rubrics |
---|---|---|---|---|---|

Continuous Assessment | |||||

Tutorials | |||||

Assignment | 1, 2, 3 | 1. b, c3. a | 20 | individual | See Appendix for rubric |

Presentation | 1, 2, 3, 4, 5 | 1. b, c3. a | 10 | individual | See Appendix for rubric |

Project 1 | 1, 2, 3, 4, 5 | 1. b, c3. a | 10 | individual | See Appendix for rubric |

Examination (2 hours) | |||||

Short Answer Questions | 1, 2, 3, 4, 5 | 1. b, c3. a | 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

## 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 presenting homework as well as the results of a reading/research project at the end of the course. Students will get informal but individual feedback from the peers and from the course instructor for each of these presentations.

Common mistakes in the assignment will also be discussed in the provided solution set.

Lectures (39 hours) | Present the key ideas behind mathematical concepts. Illustrate the key ideas with examples. Present important steps used to solve different types of problems. |

Tutorials (12 hours) | Develop proficiency in problem solving skills. Reinforce concepts already covered in the lectures. Give an opportunity for weaker or more reserved students to clarify doubts. Students will lead the tutorial sessions - presenting their solutions to problems they have selected for discussion by the group. |

There are three reference books:

Munkres, James R. Topology, a First Course. Second edition. Pearson, 2014. ISBN: 978-1292023625

Willard. S. General Topology. Dover. 1994. ISBN: 978-0486434797

Abraham, R., Marsden, J. E., and Ratiu, T. S. Manifolds, Tensor Analysis, and Applications. Second Edition. 1988. ISBN: 978-1461269908

Attendance:

Attendance of lectures is not compulsory. However, you are highly encouraged to attend all lectures and keep handwritten notes.

Assessments:

1. A late submission for an assignment is not accepted

2. No make-up assignments or presentations will be arranged

3. A student who is late for an assignment or presentation without valid Leave of Absence will be given zero mark.

4. In case of a valid reason for absence, the total course marks would subsequently be rescaled to a base of 100%.

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.

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

Philipp Harms (Dr) | SPMS-MAS-05-41 | 6513 7187 | philipp.harms@ntu.edu.sg |

Week | Topic | Course ILO | Readings/ Activities |
---|---|---|---|

1 | Topological spaces. Basis of a topology. Subspace, product, box, and quotient topologies. | 1 | Lecture |

2 | Continuous functions. Homeomorphisms, embeddings. | 1 | Lecture and tutorial |

3 | Metric topologies. Topologies induced by norms and inner products. Function spaces and uniform convergence. | 1, 2 | Lecture and tutorial |

4 | Connectedness and compactness. The Intermediate Value Theorem, the Extreme Value Theorem, Heine-Borel-Lebesgue, the Uniform Continuity Theorem. | 1, 2 | Lecture and tutorial |

5 | Countability and separation axioms. The Urysohn lemma and the Metrization Theorem. | 1, 2 | Lecture and tutorial |

6 | The Tietze extension theorem. Paracompactness, partitions of unity, and Euclidean embeddings. | 1, 2 | Lecture and tutorial |

7 | Approximation. The Stone-Weierstrass Theorems. | 1, 2, 3 | Lecture and tutorial |

8 | Fixed Point Theorems and applications. | 1, 2, 3 | Lecture and tutorial |

9 | Manifolds. Charts and atlas. Atlas equivalence and manifold topologies. Examples. Submanifolds, products, and mappings. | 4 | Lecture and tutorial |

10 | The tangent and the cotangent bundle. Subimmersions, immersions, and transversality. | 4, 5 | Lecture and tutorial |

11 | Vector fields, flows, and dynamical systems. | 4, 5 | Lecture and tutorial |

12 | The Lie derivative. Vector fields as differential operators. | 4, 5 | Lecture and tutorial |

13 | Project presentations and revision | 1, 2, 3, 4, 5 | Lecture and tutorial |

Grading Criteria | Exceptional (9-10) | Effective (7-8) | Acceptable (5-6) | Developing (0-4) |

Accuracy | The interpretation is highly accurate, concise and precise. | The interpretation is mostly accurate. Some parts can be better explained or more succinct. | The interpretation is somewhat accurate. However, it contains some inaccuracies, missing points or ideas that are not related to the interpretation. | The interpretation are mostly inaccurate. |

Visual and Oral delivery | Slides are informative, good clear voice and constant eye contact | Slides are adequate, oral delivery is satisfactory and frequent eye contact | Slides are bare and somewhat disorganized, voice is sometimes inaudible and little eye contact | Slides are disorganized, voice it too soft, and no eye contact |

Presentation | Very clear and organized. It is easy to follow your train of thought | Mostly clear and organized. Some parts can have better transitions. | Somewhat clear. It requires some careful reading to understand what you are writing. | Mostly unclear and messy. It is difficult to understand what you are writing as there is no clear flow of ideas. |

Question and Answer (for each individual student) | Very clear and precise answers to all problems. Explain the problems from various different perspectives logically. | Correct answers to most of the problems. Explain the problems in an organized way. | Partially-correct answers to most of the problems. Explain the some of the problems . | Unclear and messy answers. Difficult to understand. |

Each student has to give a final presentation in the last week of class. Presentations may be related to research or reading projects. Presentations are graded individually in the style of an oral examination.

Grading Criteria | Exceptional (9-10) | Effective (7-8) | Acceptable (5-6) | Developing (0-4) |

Accuracy | The interpretation is highly accurate, concise and precise. | The interpretation is mostly accurate. Some parts can be better explained or more succinct. | The interpretation is somewhat accurate. However, it contains some inaccuracies, missing points or ideas that are not related to the interpretation. | The interpretation are mostly inaccurate. |

Visual and Oral delivery | Slides are informative, good clear voice and constant eye contact | Slides are adequate, oral delivery is satisfactory and frequent eye contact | Slides are bare and somewhat disorganized, voice is sometimes inaudible and little eye contact | Slides are disorganized, voice it too soft, and no eye contact |

Presentation | Very clear and organized. It is easy to follow your train of thought | Mostly clear and organized. Some parts can have better transitions. | Somewhat clear. It requires some careful reading to understand what you are writing. | Mostly unclear and messy. It is difficult to understand what you are writing as there is no clear flow of ideas. |

Question and Answer (for each individual student) | Very clear and precise answers to all problems. Explain the problems from various different perspectives logically. | Correct answers to most of the problems. Explain the problems in an organized way. | Partially-correct answers to most of the problems. Explain the some of the problems . | Unclear and messy answers. Difficult to understand. |