MATH2701: Abstract Algebra and Fundamental Analysis

- 4 take home assignments (2 for Algebra and 2 for Analysis, 15% for each half),
- 2 take home exams (1 for Algebra and 1 for Analysis, 10% for each half),
- 1 final exam (50%).

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The course lived up to its difficulty and it makes sense considering this is the taster for everything that you do in pure maths. It's split into two halves: the first half being analysis (taught by Lee Zhao) in the first five weeks and the second half being algebra (taught by Catherine Greenhill) in the last five weeks, so considerable effort needs to be shared among both parts.

The content was really interesting and it definitely serves as a great bridging course between first year courses and third year pure math courses. In algebra, you are introduced to group theory, transformations (reflections, rotations, translations, etc), and end with projective geometry (which treats lines and points as the same element). On the other hand, in analysis, you learn the underlying concepts of limits and sequences (Cauchy sequences), construction of the reals (Dedekind cuts, Stevin's construction), which leads nicely into p-adic valuation (and subsequently p-adic numbers), as well as inequalities (Holder's inequality and Jensen's inequality) and norms/convex bodies.

The assignments were quite fun and interesting, the analysis assignments ended up being a grind while the algebra assignments were fairly breezy. I enjoyed thinking about the assignments and they definitely helped with preparation for the finals.
Overall, there really isn't anything I can fault about the course. As difficult as the course was, it was the most enjoyable I have had in a math course thus far. Definitely recommend doing the course if you're willing to put in the work for it.

2 x 2 hour lectures, 1 x 1 hour tutorial.

4.5/5.

Yes.

Dr. Catherine Greenhill (Algebra), Dr. Lee Zhao (Analysis)

Lecture notes are available before lectures.

5/5.

None prescribed.

20T3.

79 (DN).

Prerequisites:

A very very fun and very difficult course. The proofs in this course range a lot, but many of those covered in lectures are very difficult to reproduce. The course really helps you think abstractly, but a lot of people found it brutal, and went poorly. The assignments require a lot of thinking, so should not be left until the last minute. Dr. Zhao was a really good lecturer, and really helped develop my abstract thinking with how he explained the process of thought in developing the solutions to questions. Unfortunately, I feel as though Dr. Du was tasked with some of the more boring parts of this course, and the notes were difficult to follow at times. I am, however, an analysist before an algebraist, so I feel that might be some of my own bias coming into play. A good geometric intuition will help a lot with the algebra component of the course. I wouldn't recommend this course unless you really liked abstract thinking and a challenge.

1x 2hr, 1x 1hr Lecture, 1x 1hr Tutorial

5/5

No.

Dr. Lee Zhao and Dr. Jie Du

Lecture notes available, but the algebra notes were written by a past student and aren't properly edited. Past final exams and tutorial problems supplied.

4.5/5

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

2018 S2

89 HD

- Analysis half: 5 x small assignments (each weighted 2%), can collaborate with your peers and the internet on how to do the problems. 1 take-home test
(weighted 15%), must be done alone.
- Algebra half: A mixture of 10 minute quizzes and assignments (combined weighting of 25%)
- Final exam weighted 50%

The formal prerequisite is a CR in MATH1231/MATH1241/MATH1251 or enrolment in Science (Adv Maths) or Adv Science, but you really should have a bare minimum of DN in MATH1241/MATH1251 if you are considering this course.

This course is generally regarded as the pure mathematics "trademark" course. It is what distinguishes this major for the rest. It forms the bridge between the mostly computational nature of first year courses, and the extent of proof in the later pure courses. As implied multiple times above, it is divided into an analysis half, and an algebra half.

Analysis is the formalisation and extension of every idea used in modern calculus, whereas 'algebra' is the exploration of various structures that build and are used in mathematics. They generally involve quite different ways of mathematical thinking, but form the two main blocks (and debatably, pathways) of a pure mathematician.

Analysis is just intense by nature, but was something that I found quite neat and challenging. It is common to just spend hours at a problem and not get anywhere, and at the same time it's always a huge excitement when you figure it out. This half encourages you to draw upon ANYTHING you've been previously exposed to, and produce neat results out of it. Some topics include the big 'O' notation, inequalities and p-adic analysis.

The structures of abstract algebra are mostly groups and fields. Group theory is used in this section but to a small extent; the course's name feels like a misnomer as it's mostly focused on geometries (including projective geometry and transformations). Unfortunately, it really didn't work well with me for several months; I only managed to figure everything out at the end after receiving a lot of help. (There may have been other factors influencing this problem.)

Given the nature of pure mathematics, a bridge between first and third year is certainly necessary and this course serves that purpose quite well. However, whilst it may be easier than what's to follow, the content you learn can be a huge shock, hence the significantly lower candidature for the course. Most people do well in this course, but it's usually because they're just that capable.

3 x 1 hour lectures, 1 hour tutorial

5/5 - This course's difficulty is well beyond any other math course in the first two years.

No

Dr Lee Zhao, Dr Jie Du

Analysis half - Decently comprehensive lecture notes provided. Algebra half - The lecturer provides his notes, but they are hand-written and often hard to read. Notes written by a student also published but they are very brief. A few past papers provided; some more obtained through the lecturer.

3.5/5

Nil

2017/2

88 HD