Home Page for MATH 4B03

Term 1, 2014/15


Table of Contents

Table of Contents

Announcements and Update

Course Description
Grading Scheme
Policy Notes

Announcements and Updates

I will return the last assignment on Wednesday, December 3rd during the last lecture period (I will not be lecturing, but you can ask me questions). You can also pick it up during my office hours on Thursday.

Extended office hours before the Exam: Tuesday, December 9th : 2:00 to 5:00 pm Thursday, December 11th : 2:00 to 5:00 pm

Read my notes (and my lips!)

Good Luck and Happy Holidays!


Assignment #5  was due on Wednesday, November 26th, 2014 (extended to Monday, December 1st!) Please hand it to me at the beginning of the lecture period.


Thursday. November 27th:      Shuai Zhang:    "Arbitrage as curvature"

Wednesday, November 26th:   Stephanie Ciccone:   "Flag Manifolds"

Monday, November 24th:        Miles Couchman:   "Classical Mechanics via differential forms"

Thursday. November 20th:      Savannah Spilotro: "Minimal Sufaces"

Wednesday, November 19th:   Sean Ridout:  "Chern classes"

Monday, November 17th:         Darij Starko:   "Dirac spinors and the Hopf fibration"

Thursday. November 13th:      Lee van Brussel: "Introduction to Morse Theory"

Wednesday, November 12th:   Dillon Doherty:  "Linking Numbers"

Presentations (30 to 40 minutes each) began on Wednesday, November 12th.

Reports are due a week after the presentation

There was a short 20-minute quiz during the lecture period on Monday, November 10th 

Assignment #4  was due on Wednesday, November 5th, 2014.

Click  here  for a video about the circles in the Hopf fibration.

Click  here  for a paper explaining how and where the Hopf fibration shows up in modern theoretical physics.

Click  here  for my introductory notes about calculus on differential manifolds  (which will be expanded, revised and updated regularly during the term)These notes cover the material of the first four weeks of the course. Please read them carefully before you come to the lectures

Click  here  for a much better set of notes by Professor Nigel Hitchin from Oxford University.

You should also read at least one of the recommended textbooks

There will be a short 20-minute quiz during the lecture period on Monday, October 27th (postponed to November 10th)

Assignment #3  was due on Monday, October 20th, 2014.

Assignment #2  was due on Monday, October 6th, 2014. 

There was a short 20-minute quiz during the lecture period on Monday, September 29th.

Assignment #1 was due on Monday, September 22nd, 2014. Please hand it to me at the beginning of the lecture period.

Click  here  for another beautiful lecture on 3-manifolds by a master topologist, the late William Thurston.

If you are interested, click  here  for a very "impressionistic" but deep lecture by Misha Gromov, one of my "favourite" mathematicians.

If you are interested, click  here  for my notes (I didn't sell them as courseware!) of an introductory course on real analysis for second year students (Math2AB3) that I taught a few years ago, where I even introduced topological spaces and gave succinct proofs of many basic results, such as the fundamental theorem of calculus, inverse function theorem, contraction mapping principle, results involving compactness, connectedness. I also gave a precise definition of the Riemann integral (but not Lebesgue). Have a look. This course died because some of my departmental colleagues thought that I was trying to teach a graduate level course in second year!

The lectures are scheduled as follows: Mo, We, Th  17:30 - 18:20  in HH/217 .

This is a course, where students are expected to study some of the material on their own by reading the relevant material from my notes, the reference books and/or any other resources (freely available on the internet) in preparation for the lectures. This sort of thing is known in the modern teaching/learning jargon as " self-directed, inquiry-based, blended, experiential learning".
Although this is a course in Pure Mathematics, I will avoid the obsolete but still ubiquitous "definition-theorem-proof" approach. My lectures will be more impressionistic, concentrating more on motivating and explaining basic concepts with examples and giving you the bigger picture rather than spending most of the time on boring gory details (the epsilon-delta, point-set-topology kinda stuff!) since these details can be found in almost any book on the subject or in the notes that I will post.

It is more important to know how to ride a bike than to  know the definition of a bike or watch someone else ride a bike. _______________________________________________________________________________________________________________________________________________________

Course Description

Description from Undergraduate Calendar

Review of multivariable calculus, basic properties of manifolds, differential forms, Stokes' theorem, de Rham cohomology and applications. 

Course Objective:

To understand, calculate and use differential forms.

This is a course, where students are expected to study some of the material on their own and also do a 20 minute presentation on a topic that is chosen with the instructor's approval.

Students are responsible for reading the relevant material from the reference books, my notes and/or any other resources (freely available on the internet) on their own (known as self-directed inquiry-based blended experiential learning)  in preparation for the lectures.

There are no required textbooks although I strongly recommend the students to get a copy and read at least two of the following books regularly:

Other books:

  • "The Geometry of Physics"  by  T. Frankel, Cambridge Univ. Press (related to physics)
  • "An introduction to manifolds" by Loring Tu, Springer-Verlag (explains a lot of things in detail, a bit slow-going for my taste!)
  • "Geometry of manifolds" by L. Nicolaescu (one of my former post-docs!), World Scientific (covers a lot of material, including elliptic operators)
  • "Differential Forms in Algebraic Topology" by R. Bott and L. Tu , Springer-Verlag (a lot of algebraic topology is done using differential forms)
  • "Mathematical Methods of Classical Mechanics" by V.I. Arnold, Springer-Verlag  ( a classic, one of my favourite books)

  • Some interesting pages on the internet:


    Lectures    Mo, We, Th   17:30 - 18:20  in  HH/217

    Grading Scheme

    There will be 5 assignments (due dates will be announced during the first week of classes). Doing the assignments regularly is paramount to understanding the material. Please hand the assignments to me at the beginning of the lecture period on the due dates. Late assignments will not be accepted.
    Here are the tentative dates for all the assignments:
    The  two 20-minute quizzes will be done in class (the exact date will be announced one week before the quiz).

    Every student is also required to do a 20-minute presentation chosen from a list of topics suggested by the instructor. Presentations will be done in November.
    The report (5 to 10 pages) on the presentation is due within a week after the presentation.

    Academic Dishonesty:  

    You are expected to exhibit honesty and use ethical behaviour in all aspects of the learning process. Academic credentials you earn are rooted in principles of honesty and academic integrity. Academic dishonesty is to knowingly act or fail to act in a way that results or could result in unearned academic credit or advantage. This behaviour can result in serious consequences, e.g. the grade of zero on an assignment, loss of credit with a notation on the transcript (notation reads: Grade of F assigned for academic dishonesty), and/or suspension or expulsion from the university. It is your responsibility to understand what constitutes academic dishonesty. For information on the various types of academic dishonesty please refer to the Academic Integrity Policy, located at http://www.mcmaster.ca/academicintegrity

    The following illustrates only three forms of academic dishonesty:

    1.  Plagiarism, e.g. the submission of work that is not one's own or for which other credit has been obtained.

    2.  Improper collaboration in group work.

    3.  Copying or using unauthorized aids in tests and examinations.

    Other Policy Notes:

    MSAF policy:

    Please check the following link for MSAF policy:

    When using the MSAF, also report your absence to me (the course instructor M. Min-Oo) within 2 working days by email (minoo@mcmaster.ca) and contact me in person to learn what relief may be granted for the work you have missed, and relevant details such as revised deadlines, or time and location of a make-up exam. Please note that the MSAF may not be used for term work worth 30% or more, nor can it be used for the final examination.


    Only the standard McMaster calculator Casio fx 991 can be used for the tests and the final examination.

    Important Notice: 

    The instructor and the university reserve the right to modify or revise information contained in this course during the term. The university may change the dates and deadlines for any or all courses in extreme circumstances. If either type of modification or revision becomes necessary, reasonable notice and communication with the students will be given with explanation and the opportunity to comment on changes. It is the responsibility of the student to check their McMaster email and course websites weekly during the term and to note any changes.

    Schedule of Topics (tentative)

    Week 0 (04/09 to 05/09):  First lecture: Introduction

    Week 1 (08/09 to 12/09):  What is a differential form in local coordinates?  How do we change coordinates? What is the wedge product? How do we differentiate differential forms? What is the exterior derivative?

    Week 2 (15/09 to 19/09):  What is a differential manifold? What is a vector bundle? What are vector fields? What is the Lie derivative? Cartan's magic formula.

    Week 3 (22/09 to 26/09):   How and what do we differentiate and integrate on a differential manifold?  Poincare Lemma, Stokes' Theorem.

    Week 4 (29/09 to 03/10):   deRham cohomology,  Mayer-Vietoris sequence, Euler characteristic, K√ľnneth formula, cohomology with compact support, degree of a map, winding numbers, linking numbers.

    Week 5 (06/10 to 10/10):  
    Riemannian metrics, Hodge duality, Laplacian, Hodge theory, Poincare duality.

    Week 6 (13/10 to 17/10): 
    Applications to Physics: Hamiltonian mechanics, Maxwell's equations, Potential Theory, Yang-Mills equations,

    Week 7 (20/10 to 24/10):   Geometry of surfaces (extrinsic and intrinsic) using differential forms a la Cartan, Gauss-Bonnet Theorem, Poincare-Hopf Index theorem. 

    Week 8 (27/03 to 29/03):  
    Theorem of Frobenius with applications, examples of Lie Groups and Lie Algebras

    Week 9 (03/11 to 07/11):   What is a connection? What is curvature?

    Week 10 (10/11 to 14/11):  Presentations

    Week 11 (17/11 to 21/11):  Presentations

    Week 12 (24/11 to 28/11): Presentations

    Week 13 (01/12 to 03/12): Introduction to Morse Theory