## Year 3 Module Reviews

##### MA359 Measure Theory

2018/2019

Nice module, good lecturer, easy assignments, hard exam. This module is a must do for anyone willing to do some more Analysis or interested in probability/stats. It certainly falls under the category of "if you want to do more maths in the future then you gotta know something about this". It is important to note it is a prerequisite for many other analysis modules. It essentially establishes basic material that will be fundamental for many other modules (e.g. ergodic theory, probability theory, etc...).

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2018/2019

I disliked measure theory a lot. That could be a surprising reaction given I quite enjoyed Functional Analysis 1, which has some crossover. Most of the proofs felt like they used random tricks pulled out of nowhere, which made it difficult to learn ways of proving things in the module. Beyond this, we got quite unlucky with the way the module was examined. If you revised off the notes made in lectures, you were at a significant disadvantage to those who revised using the recommended textbook. The exam paper was entirely example based and in the lectures, we didn't cover many examples at all so I feel the exam was an unfair reflection on the course that was taught. I would suggest that there is no need to take the module if you're on the BSc but maybe you'll like it, who knows? You don't have to listen to me, I'm not your mum.

##### MA3A6 Algebraic Number Theory

2017/2018

Out of all the modules I studied at Warwick, Algebraic Number Theory (ANT) was my favourite module by far. The module explored some really interesting results - for example, the Noetherian property and the Unique Factorisation of Prime Ideals. We also learnt some fascinating applications, such as how to compute the integral basis of a ring of algebraic integers. ANT also built upon previous ideas introduced in Algebra II and Intro to Number Theory - the perfect fusion of algebra and number theory! My favourite part of this module was learning how to solve Diophantine Equations! The lecture notes were very thorough and well put together - another credit to the brilliant lecturer. I would recommend the module to anyone who has an interest in algebra and/or number theory.

##### MA3D4 Fractal Geometry

2017/2018

This module would appeal to students with an interest in both geometry and analysis, in my opinion. Personally, I found it difficult to connect different components of the module together.

##### MA3D5 Galois Theory

2018/2019

Galois theory was born when Galois realised why the general quintic polynomial is not soluble in radicals. Basically, he realised that there exists quintic polynomials for which the permutation group of their roots was ''not correct" for the polynomial to be soluble in radicals. Galois theory builds up the algebraic machinery necessary to make sense of what I wrote above. Additionally, using this machinery, you prove other interesting results such as the impossibility of certain ruler and compass constructions and the classification of finite fields. I personally found the constructions laborious at first, but eventually the machine "starts to work" and you obtain very beautiful theorems and proofs. Understanding this machinery is not too hard, but will require some work. This is made easier by Samir's awesome lecture notes (and Ian Stewart's book on Galois theory for anyone interested to learn more). Clearly, this module has a lot of algebra. If you really hated Algebra 2, this module is not for you. However, if you are considering doing any number theory or algebraic geometry (or any serious algebra for that matter) in the future, you have to learn Galois theory at some point. This module therefore will be important for you to learn.

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2018/2019

A lot of people are put off this module because of the Algebra 2 prerequisite but I would recommend this module to everyone. The necessary algebra content is revisited at the start of the module and doesn't stretch much beyond definitions of rings, homomorphisms and ideals. This module was easily my favourite of the year. The content was a really nice balance of proofs and calculations and it was delivered by an enthusiastic and engaging lecturer. The module is accompanied by printed lecture notes written by Samir Siksek and a book by Ian Stewart so you will never be short of revision material. People who enjoy this module should definitely consider doing Algebraic Number Theory in the following term.

##### MA3F1 Introduction to Topology

2018/2019

Probably one of my favourite modules so far. The lecturer was really clear with his explanations and open to questions. His website was very helpful. Both TAs were great as well. So on the "logistic" point of view, this module was really well run. As for the content, as you may have guessed this module is about topology, so if you enjoyed metric spaces it’s a good one to take. You will also need some bits from Algebra II (quotient groups, equivalence relations), but nothing too complicated, so you can still take this module even if you didn’t like Algebra II. The content gets harder and more abstract in the last weeks, but again the lecturer is always happy to answer questions and the recommended book (Introduction to Topology by Hatcher) provides a lot of useful examples. Overall, I really enjoyed this module and I hope I have convinced some of you to try it.

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2018/2019

I found this module to be one of the hardest that I studied at Warwick. The concepts took a long time for me to grasp and the lecturer relied a lot on 'intuition' which I certainly did not have during the lectures. I found there was a real clash between the lecture and support classes to whether topology was all about rigour or intuition. However, once you start to get your head around the topic, this module contains some interesting results. Overall, I would take this module if you are genuinely interested in the subject and want to take it further but you should be prepared to battle with some new ideas and methods of working.

##### MA3F2 Knot Theory

2018/2019

As an advocate for quirkier mathematical concepts, I was really intrigued by the many properties and calculations which we can attribute to knots and links. It would help to have a basic understanding of Algebra 1 and maybe (but not required) Introduction to Topology, but otherwise not much else is needed. Though this module usually gets a bad reputation for its choice of lecturer, this year the lecturer was replaced, in which the new lecturer was very enthusiastic and encouraging, which was a massive help. Even if our lecturer hadn't been replaced, I reckon the exams are usually pretty straightforward, especially in our year of knot theory. Unless you disagree with the content of the module, I'd definitely recommend!

##### MA3G6 Commutative Algebra

2018/2019

An oversimplified description of Commutative Algebra would be that it is the study of commutative rings and algebraic structures associated with these rings. However, Commutative Algebra is VERY big and there are many open question that remain within the subject. The lecturer for my year was okay and his lecture notes were okay as well. The subject itself is quite deep with many hard problems. However, I think it is not too difficult to get started and understand most of the content in the module. In my year, we had 5 assignments for the module. Each assignment had a maximum of 3 marks; every correct answer received 0.5 marks and the final mark was rounded to the nearest integer. The assignments were not ridiculously hard, but they were not easy either. It was also important to attend the support classes (or get notes from someone who did!) as some assignment questions depended on the material covered in the support class. Moreover the lecturer we had is not a big fan of hard exams --- the exam we had was almost entirely bookwork. This module is an essential prerequisite for anyone wanting to study algebraic geometry in the future (indeed, the course is designed by algebraic geometers) and is also important for any future number theorists and general algebraists. To those who are interested in studying the subject deeper, I recommend the book ''Undergraduate Commutative Algebra" by Miles Reid. It contains most of the course; many exercises and is very clearly written. (A lot of people recommend Atiyah's book on commutative algebra, but I think that book is terrible.)

##### MA3G7 Functional Analysis 1

2018/2019

This module revisits a lot of concepts developed in second year such as normed and complete spaces, compactness and inner products. It then starts to tackle ideas like separability, adjoint operators and spectral theory. A perfect follow on module to Metric Spaces (and probably Norms, Metrics and Topologies?) and a good module to take alongside Measure Theory. A pretty typical Warwick analysis module.

##### MA3G8 Functional Analysis 2

2018/2019

This module does what it says on the tin. It's a follow on module from Functional Analysis 1, pushing most of the concepts you have already learned that little bit further. I found the module to be quite dry and very bookwork heavy but the lecturer was always clear, provided excellent lecture notes and relevant example sheets. If you like analysis, this module is for you.

##### MA3H2 Markov Processes and Percolation Theory

2017/2018

Pretty fun, decent lecturer, really hard assignments, quite difficult exam. Much easier if you've done Stochastic Processes.

##### MA3H3 Set Theory

2018/2019

Axiomatic set theory is integral to everything you have studied so far. This module is a great stand alone module for analysis and algebra lovers alike. Throughout the module you will explore the axioms upon which our understanding is built, understand cardinalities and finally define the natural numbers, integers, rationals and reals. Working with concepts such as induction, transitivity and recursion, you will reach the axiom of choice and Zorn's lemma. This module is very accessible and provides an insight in to the foundations of mathematics.

##### MA3H5 Manifolds

2018/2019

Manifolds was probably the hardest module I studied in my third year. In short, the course is about smooth manifolds. The theory of smooth manifolds is very rich: you will study vector bundles; differential forms; the generalised Stoke's theorem and many more interesting topics during the module. However, you will soon realise that these concepts are not easy to digest. Hard work is necessary! This is why I only recommend Manifolds to people genuinely interested in the subject and not looking for easy marks. The lecturer for my year was not good and his lecture notes were terrible. This is why I recommend Lee's "Introduction to Smooth Manifolds" to anyone intending to take the module in the future. It's an awesome book, covering all of the module and containing important prerequisites for the differential geometry module in the 4th year.

##### MA3H6 Algebraic Topology

2018/2019

Great fun, but also one of the most intense modules I've ever done. Would recommend for masochists only.

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2018/2019

Probably one of the hardest third year modules, but content is very interesting, and also very new (you've never seen anything like this before). The module is essentially an introduction to homology theory, and is a nice continuation of Introduction to Topology. This module will require you to get familiar with a lot of new concepts, some of which can seem unnatural and counter-intuitive at first, but eventually everything makes sense. The assignments are also quite hard. There is one per week with about 10-12 questions (of which 3-4 are assessed) and it is important to attempt all of them as some of the results they provide do come up later in the course. For all of these reasons you should expect to spend about twice as much time on this module as for any module (consider it a 30 CATS module).

##### MA3J2 Combinatorics II

2018/2019

This module has 7 chapters, each of which heavily differ from each other, so there is a wide range of content covered, most of which has some pretty satisfying results. If you didn't enjoy the enumerative side of MA241 Combinatorics, it doesn't appear much at all, so don't worry about having to delve into those results much more. The lecturer was quite slow to go through content, but was absolutely lovely and explained things very well. The exam set was also pretty manageable with no particularly difficult questions. If you learn the content well and have a solid understanding, the exam should be quite straightforward. The module is lectured using powerpoint slides, so that might put some people off, but not worth not taking if you enjoy the content.

##### MA3J3 Bifurcations, Catastrophes and Symmetry

2016/2017

I really enjoyed this module! It starts fairly simple and is an extension of the 1st year differential equations module (2nd theory of ODEs course is useful). Lecturer recaps everything and did a demo with a Zeeman Catastrophe machine to explain concepts. Second half of the course uses aspects from Algebra 2, so some memory of Cyclic and Dihedral is useful. Main disadvantage of this module is there is no online notes for the first half (so you have to be fairly good at note taking).