MATH3701: Higher Topology and Differential Geometry

Prerequisites:

I ended up adoring this course afterwards. The start of the course is somewhat slow, but by the end you're doing some really cool things linking geometry and topology together, and I found it really interesting. This course is considered the hardest of the three main pure maths courses, and it certainly lives up to that (Dr. Chan ended up cutting some content and dumbing bits of the course down for us). Some parts of the course, before hitting manifolds, are quite computational, and the cross product makes its ugly return, but after introducing manifolds most of that disappears. The content relies heavily on your geometric intuitions, and I felt that a fair bit of the first third of the course was trying to build up our intuition before we hit the nitty gritty. Although the course name is "topology and differential geometry", Dr. Chan has taken the course and focused on the differential geometry aspect, with the topology more as a means to achieve things. Towards the end of the course, topology starts to become more important, but is still heavily related to the differential geometry aspect of the course, which I personally preferred. Neither MATH3611 nor MATH3711 are prerequisites, but the topology knowledge from MATH3611, and the properties of quotient groups from MATH3711 are very useful. I would highly recommend this course to anyone with a pure maths bent (not that you have a choice, it's mandatory). Dr. Chan is a great and engaging lecturer, and motivates all of his examples well. If it weren't for the computational aspect of some of the course, it's be an easy 5/5 rating.

2x 2hr Lecture, 1x 1hr lecture (second hour of the second lecture was a tutorial hour)

4.5/5

No.

Dr. Daniel Chan

Lecture notes (handwritten) and problem sets (typed) online.

4.5/5

Note: I don't use textbooks and can't comment on their usefulness. None prescribed, but useful references:

2019 T3

99 HD

30% Midsem Exam, 40% Assignment, 30% Finals Exam

12 UoC of Level 2 mathematics courses is required, but within that 12 UoC it is expected that 6 UoC comes from one of the following: MATH2111 or MATH2601 or MATH2011 CR or MATH2501 CR. However, I would recommend bare minimum DN in both MATH2111 and MATH2601, because whilst comparatively little linear algebra and calculus are used explicitly, a lot seems to get mentioned in passing.

MATH3611 or MATH3711 completed beforehand is highly recommended to understand the more abstract concepts, but not necessary.

Please be advised that this review is subject to becoming outdated immediately. The course structure in 19 t3 is expected to be different under the new lecturer.

This course is one of the core courses for a major in Pure Mathematics.

Topology is sort of a bridge between analysis and algebra - it uses concepts in both. Whilst its roots seem to stem from elementary set theory, it adds in various algebraic and analytic structures. It is of course, the field of maths that talks about the coffee cup and the donut. Whereas differential geometry is, as I like to summarise it simply, putting some kind of calculus (differential structure) on curves and surfaces that we already know of. (Although it does touch on manifolds, which generalises upon surfaces.)

I had a lot of trouble with this course throughout the semester. I found it difficult dealing with the fact that I had to forfeit any sense of mathematical rigour to understand the concepts, because many rigourously defined things were just too bizarre (although fortunately not examinable either). Eventually I relied heavily on two things - intuition and rote learning. Intuition can be helpful especially when dealing with topology because trying to formalise something built on algebraic topology can be hard, but being able to picture a loop or some shit in your head turns out to be sufficient most of the time. Same goes for certain aspects of differential geometry, most notably the basic definitions. On the other hand the rote learning was disappointing - I still don't fully understand Van Kampen's theorem or geodesics (or other unlisted stuff) yet. I've mostly been taking for granted how to use it.

I definitely do advise "intuition" as something to develop and utilise in this course. It pushed me through a fair lot of the course despite constantly being in agony over it.

What was certainly a bit of a shock was that this course had some very nasty computations. I wasn't expecting that for a level 3 pure mathematics course and it can get quite annoying, so that's something to keep in mind about. But other than that, the proofs in this course didn't require full detail (or so I felt). So long as you displayed some understanding of the concepts, something wishy-washy wasn't necessarily so bad it seems.

The teaching quality was something I was really uncomfortable about with this course. Occasionally concepts made sense, but at other times they were just skimmed with insufficient intuition. On the other hand things I found the consultations were a lot more life-saving (the 1-on-1 help helps a lot) and I am very thankful for both the 40% assignment and the generous marking across all tasks. I believe that the lecturer's personality should be praised.

3 x 1 hours of lecture, 1 hour of tutorial

4/5

No

Dr. Mircea Voineagu

Handwritten notes. Have cons but the handwriting is very legible. A reasonably abundant supply of past papers is provided.

2.5/5

- Topology (2nd Edition) by James Munkres - This one can be hard to read, but I've been reported that it's really useful.
- Elementary Differential Geometry by Andrew Pressley - This one is awesome

18 s2

89 HD Potentially subject to change, but if this comment is not edited out then assume no change was necessary