Module Reviews

2017/2018
Hello! Variational principles is a term 3 module. This unfortunate fact does deter many people from taking the module. However, there are only 15 lectures and lectures happen 4 times a week. I personally found that doing this module gave me a break from revision and forced me outside (briefly!). Mathematically, this module is essentially an introduction to the calculus of variations. This calculus seeks to find functions that are extrema to a given integral, possibly subject to constraints. This sounds simple enough, but surprisingly the maths is very rich. This calculus is used in Hamiltonian mechanics, quantum mechanics, engineering and many other disciplines. In addition, I personally find the maths very beautiful! There are weekly exercise sheets but the final mark is your exam mark. There are no printed lecture notes but the lectures are very good. Make every effort to attend them (or make friends with someone who does!). To summarise, the module contains good mathematics and provides a welcome break from intense revision!

2017/2018
This module seems daunting at first – the idea of finding a topic, writing an essay and presenting it. But actually, once you’ve found a topic, it can be really interesting and become a nice break from other modules. Start thinking about your essay topic well before the deadline for the title as then you’ll have a topic you’re actually interested in. Also, try to keep on top of the work for the essay though as that’ll give you more time at Easter to focus on revision.

2020/2021
Do it early. It’s so much easier to get a solid amount of work done on your essay over summer, or over Christmas, rather than when you’ve got several other modules to keep up with too. In terms of the topic, try starting off with something quite broad (so you have enough sources to work from!) and see where it takes you. Also, you will need to write your essay in Latex. You don’t have to be amazing at Latex, but make sure you attend the course (especially if Andrew Brendonn-Penn is still leading it).

2017/2018
Hello! If you are reading this, you are probably interested in Geometry. And so you should be — geometry is a very beautiful area of maths. If I were only allowed to use one word to describe this module, that word would be “interesting”. Indeed, as well as learning euclidean and spherical geometry, you will be introduced to hyperbolic geometry, affine geometry and projective geometry. Hyperbolic geometry is analogous to spherical geometry done on a sphere of radius i (Ahhh!!!!!) and the latter two are important in algebraic geometry (4th year!). I must emphasise that this module is NOT EASY. The content is covered faster than most second year modules; there are weekly assignments and the maths may appear abstract when first encountered. Do not take this module if you’re looking for easy marks. In addition, the recommended textbook by Miles Ried is not the best textbook in the world. The lecture notes last year was the textbook. Even though I would still recommend getting this textbook, I would also recommend getting the textbooks by Coexter (title given in webpage) and also Ratcliff’s foundation of hyperbolic manifolds (available from Warwick library online) for the hyperbolic geometry section. However, despite the previous two paragraphs, I was satisfied by studying this module. Hence, my final advice would be to take this module if you like geometry and are seeking intellectual satisfaction.
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A very interesting module with a good teaching structure. People who are more algebra minded rather than analysis minded tend to like this module.
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2018/2019
The lectures were fascinating and honestly worth the three 9AMs. The weekly assignments were a reasonable amount of work in term 1 and the marking was a bit weird with assignments being out of 2. The module is quite tricky and easily one of the hardest parts of the year. Moreover, the exam was absolutely horrendous and made no sense as a two hour paper.

2020/2021
Definition, theorem, proof, repeat… a pretty typical analysis module focusing on integration, although the second half of the module is analysis with complex numbers, which is nicer. Jose is an excellent lecturer and the notes are good, but he set a ridiculously hard open book exam with far too much unseen material. Make sure to familiarise yourself with the key definitions and theorems and practice coming up with examples and counterexamples.

The module was majorly restructured starting in 18/19. The first part of the module rigorously introduces integration via the Darboux Integral (called the Riemann integral here, as it often is! It is equivalent though), and you look at integrals of functions that are not necessarily continuous. (where the Fundamental Theorem of Calculus does not apply). You prove the various properties of the integral from scratch, including FTC. The second part of the module introduces sequences of functions, their convergence and the notion of uniform continuity. This is used to develop integration a bit further, you prove Leibniz’s rule on differentiation under the integral sign, Fubini’s theorem on changing the order of integrals, and some sufficient conditions on integrating a sum termwise and on moving limits into integrals. These are all stuff you would have basically skimmed over and taken for granted in your earlier calculus work, so I quite enjoyed looking at this and being more confident in my manipulations. There was a bit at the end on absolute continuity and bounded variation which seemed a bit out of place and underdeveloped. The third part of the module introduces Complex Analysis, which is much nicer than Real Analysis. A major point is that complex-analyticity (has a Taylor expansion) and complex differentiability are equivalent in the complex functions, whereas not all smooth real functions are analytic. Mainly you focus on line integrals in the complex plane, which have some very powerful theorems associated (including Cauchy’s integral formula, and the residue theorem, though the latter isn’t covered in this course), and can relate back to real integrals to give cool ways of integrating functions like e^(-x^2), sin(x)/x or 1/(1 + x^n) from 0 to infinity. (differentiation under the integral sign mentioned earlier gives nice ways to evaluate the first two, as well, though that isn’t as explored in this course) You prove quite a powerful theorem called Liouville’s theorem, and apply this to show that smooth complex functions satisfying certain properties must be constant or assume a certain form, say a polynomial. (again, this is enabled by complex differentiability being much stronger than real differentiability) This module was lovely, and the assignments were fairly reasonable. There were quite a few questions available to practice. However, the final exam was anything but, and most people seemed to find that it was ludicrously time pressured because the lecturer essentially filled the exam with assignment-level questions. There were very few marks that you could get quickly, as there should be. It was so bad, Dave Wood had to reassure us that it would be considerably scaled. I would imagine next year’s will be easier. (though for the first years that start in 2022, the specification is due to be changed again to add more complex analysis and some fourier analysis so may be harder yet, sorry!)

2017/2018
This module follows on from Linear Algebra. The assignments were hard but not the worst of second year. I’d recommend looking at the Maths Dr. Bob videos on YouTube when trying to understand the matrix section. The last section of the course is similar to Algebra 2 content. My advice would be not to heavily rely on a Jordan matrix question in the exam as sometimes long fiddly algebra can lead to computational errors which cost marks.
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2018/2019
Whilst the lectures for this module were not the most engaging or coherent, the content was comprehensible and easy to self study, despite the fact that the lecture notes were unclear in places. The exam and the assignments were very doable, making this module one of the easiest this year. As such, I would highly recommend it to everyone who enjoyed Linear Algebra in first year, as it follows from this module very well, and was overall an enjoyable and relaxed module.

2020/2021
Algebra 1 follows on from Linear Algebra in that it is all about matrices; specifically, about changing basis in such a way that the matrices we end up with look “nice”, which means slightly different things depending on the context. A lot of the theory and proofs I found difficult to follow, but most of the module is quite computational and so the hardest part is avoiding silly little errors in your working. I strongly advise practising the computational questions as much as possible; there are tons of examples/questions in the notes, assignments, past papers, etc.

2017/2018
This module is very different from Algebra 1. If you liked Abstract Algebra, then this module follows on from that. The lecturer wasn’t very good so I relied heavily on the lecture notes and online resources to understand content. Also, the assignments were hard to understand so working with others was useful. The exam was VERY hard, but everyone found it hard (so don’t panic too much!). The content overlaps with Number Theory at times, which was helpful as it meant I’d seen some of the content before.
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2018/2019
Algebra 2 is an amazing module although it’s quite different from Algebra 1. The very beginning follows from Introduction to Abstract Algebra ,although the kinds of things you deal with end up being quite different. The module has a lot of content but, especially in the ring theory section, it’s all contained in fairly brief sections. There is a decent amount of overlap in the ring theory section with Introduction to Number Theory but it’s taught slightly differently and one definition actually changed. The lecturer for our year was outstanding and the lectures were an absolute joy to attend. The assignments were all fairly straightforward – although they were marked rather harshly – and as long as you kept on top of all the different definitions and concepts, the module isn’t too hard to follow. Moreover, the exam wasn’t very difficult in our year and was very similar to assignment questions with a few of the more important pieces of bookwork thrown in. Overall, this was certainly one of the most interesting parts of second year and had the most entertaining lectures I’ve had so far.

I really enjoyed Algebra 2, which follows on from Abstract Algebra rather than Algebra 1 (so no matrices here!). The groups section equips you with more tools to investigate and classify the structure of groups, while the rings section has a bit more of a theorem-proof style to it. The assignments are good practice and the lecture notes are extremely good.

2017/2018
I really enjoyed this module. The content was hard to understand at first, but once you get your head around it, the module isn’t too bad. The lecturer was really good and set a nice paper. I’d recommend going to the revision lecture as that was useful. This module is good for anyone interested in dynamical systems and anyone taking Intro to Maths Bio as there is some overlap. The exam is pretty standard and I’d recommend learning the diagrams for the specific examples given in lectures, as they are important.

2020/2021
This module revisits some of the material covered in ODES in first year, but with more rigour. Don’t judge it solely on the first 2 weeks, because it gets much more interesting after that! You’re introduced to a wide variety of tools to help you investigate features of ODEs, with the ultimate aim tending to be sketching a phase portrait, which when you get it right, is very satisfying. The exam questions tend to be a mix of ODEs from the notes and slight variations of them, and there are only so many things you can be asked to do, so it’s not too hard. The lecturer is very good, and there are lots of exercises to revise from. Overall, a very well-taught and interesting module.

2017/2018
I loved this module. There is an overlap with the theory content from Theory of ODEs so if you’re interested in dynamical systems and applications, then this is a good module. This was a term 3 module but the content wasn’t too taxing. I found the section on biological systems hard to get my head around initially, but that quickly fell into place. I would recommend doing the example sheets and attending the support class.

2020/2021
3/5 Fairly cool module – interesting and not too hard, but not brilliantly taught. Basically, if you like ODEs then you’ll LOVE this module- and if you take Theory of ODEs then take this for a nice easy 6 CATs. It’s a term 3 module which isn’t ideal but it does save having to actually revise it! In terms of content it’s a highlights tour of results in ODEs applied to biology, which in all honesty was quite fun. BUT (and this may be highly specific to 20/21) it wasn’t particularly well taught. There were no notes, only 2 very short examples sheets and some fairly unhelpful support classes. Due to the short (3 week) delivery of this module the lecturer wasn’t able to elaborate on any results or examples*. Personally, this module drove me mad because of this, but once you get your head around how to apply the results it’s not too bad. *He is actually involved with COVID modelling so he’s a complete legend, and I think in a normal year this module would’ve been a lot better. – Recommended if you like Differential Equations

2018/2019
Number theory is a beautiful area of maths, and I would recommend this module to anyone interested in number theory/algebra. The content is not always straightforward and some of the proofs are very complicated, but the exam set this year was very fair and most people did really well. There’s also a significant overlap between MA257 Number Theory and MA249 Algebra 2; taking Number Theory will significantly help with the ring theory content in Algebra 2, which I found to be very beneficial.

2019/2020
This module is essentially a whistle-stop tour of number theory, with the main tool being commutative rings and their ideals (hence there is a significant overlap with rings section of Algebra II.) It starts a bit abstractly, looking at different types of rings (mostly integral domains) but then you will apply the developed algebra to number theory problems, such as Pythagorean triples, sums of two/four squares, modular arithmetic and quadratic reciprocity.
This was by far the hardest 2nd year module I did. Some of the proofs are extremely difficult, and the assignments almost more so. The exams that the lecturer (Adam Harper) sets are very repetitive however and are essentially a list of proofs from the lectures with some standard calculations thrown in (e.g. a Chinese Remainder Theorem calculation or calculating a Legendre symbol). It is therefore imperative that you learn the lecture notes and bookwork proofs; it is not sufficient to only do the past papers and redo the assignment sheets just before the exam like you may have done in first year. In fact, this is a key lesson for most second year modules and beyond; past paper/assignment sheet revision may get you a first in first year, but it won’t even get you a 2.2 in second year and above.
In short, this module was really painful (in fact it was harder than nearly all of my third year modules), but if you learn the lecture notes you can still get a very high mark. The lecturer kind of just reiterated the lecture notes. You will prove some famous number theory results (Lagrange’s four square theorem, when can an integer be written as two squares etc.) but I didn’t find them very interesting.
 

2020/2021
5/5 stars – awesome module, with some really beautiful results. Hands down my favourite module in second year. Number theory is a great area of maths, and studying some seemingly simple questions with a range of tools is very rewarding. I found this module to be well-paced, with a range of interesting results and often nice proofs. There is a fair crossover of content from the latter half of Algebra 2 which is always nice. Adam did a better job than most lecturers of adapting the content to online learning – with very clear and easy to follow notes and thorough online (asynchronous) videos. Assignments aren’t always easy but have some satisfying questions. Recommended if you like Abstract Algebra

2018/2019
This module was amazing. The lecturer was excellent and made the fairly abstract concepts seem intuitive and comprehensible. As such, the lectures were really engaging and absolutely worth going to. The content is lots of small, fairly straightforward proofs which fall together very cohesively. The exam was fair and followed the pattern set by assignment sheets and the metric spaces papers from previous years. Overall, this module was one of the most fascinating, and enjoyable parts of my degree so far and I would definitely recommend it for anyone for whom this module isn’t core.

2020/2021
A huge analysis-flavoured module packed to the brim with definitions, theorems and proofs. In a way, it is easier than previous analysis modules, since a lot of the content generalises what has been seen before, so there are sometimes patterns you can spot in the proofs. There is a LOT of content though. Practice coming up with examples and counterexamples as this sort of thing featured heavily in the exam. There are no assignments, but you should do the exercise sheets at some point to check your understanding, and also as practice for unseen proofs.

2017/2018
I actually enjoyed this module. I’d heard people say it was awful, but I found it interesting and not too difficult. However, the exam was horrendous. But don’t let that put you off as everyone found it hard! This module is useful if you’re interested in stats and links nicely with some applications (e.g. maths biology). I’d recommend taking Intro to Math Stats before taking this module so that you can have a solid understanding of distributions (which is necessary). Overall, despite the awful exam, I am glad I took this module.

2017/2018
I enjoyed this module. If you’re interested in stats or some applied areas (e.g. maths biology), then I’d recommend taking this. This module follows on from Probability A and B, explaining some of the content from first year in better detail, and then moves on to hypothesis testing. The hypothesis testing section is hard to get your head around (as it is more theoretical and it had been a while since I’d done hypothesis tests!), so I’d recommend working through the examples given in the lectures. The lecturer was good and explained things well. There are optional assignments which I’d recommend doing as they are useful when trying to understand some of the content. The support classes are also worth attending, even if only to get the assignment solutions!

2018/2019
This module is all about game theory and, despite ‘mathematical’ being in the title, the maths involved is very basic. It also contains very few economics examples, and differs completely from EC106! The content itself is not too difficult, and is delivered by a fantastic lecturer. However, the exam set this year was absolutely horrendous (although an incredible amount of scaling was applied). I’d recommend taking this module if you would like a break from maths and are interested in game theory and/or economics itself. Also, completing the exercises in the recommended textbook will help you in the exam.

2020/2021
The content, all about game theory, I found extremely interesting. The maths involved is quite basic (and if EC106 is still listed as a prerequisite, then it shouldn’t be). However, there are no proper notes and minimal example sheet questions, so when the lecturer tells you that the Steven Tadelis textbook is absolutely essential, you should believe him. (This is the only maths textbook I have ever bought so far for uni and it was definitely worth it.)

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.

2019/2020
Algebraic Number Theory is the most enjoyable module I have taken during my degree. It is a very wide field of Maths so the course content tends to vary year on year, but I found that it was taught coherently with each chapter building on/linking to the previous one. This does mean that it is important to keep up with the lectures as it is not a module you can easily dip in and out of.
The lecturer was extremely clear with good pace, and the lecture notes had a fair balance of proofs and examples. The exercise sheets were of a sensible difficulty which enabled us to consolidate the lectured material, and they were a similar style to the exam questions and therefore acted as good preparation.
Unfortunately, the exam this year was much harder than the past papers as questions were worth fewer marks than expected, and many bookwork proofs and examples were given a twist which increased the difficulty significantly. However, I would still recommend this module to anyone interested in this field of Maths as it was incredibly interesting and enjoyable. It is worth noting that there is lots of overlap between Algebraic Number Theory and Galois Theory, so it is worth taking the two alongside each other.
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.

2019/2020
Fluid Dynamics is a good choice for anyone with a grasp of differential equations. A strong knowledge of spherical and polar coordinates is also incredibly useful. The lecture notes were exemplary, and the lectures were good for anyone requiring a detailed discussion of aspects they didn’t understand; the lecturer gave incredibly clear explanations, although the pace was often slow.
There were no assessed example sheets for this module although I would recommend completing the exercise sheets before the exam as these provide good practice for exam style questions. The exam tends to be very similar year on year (with some questions copied and pasted) so past papers are an essential revision resource.
It is worth noting that the second year Physics module ‘Physics of Fluids’, whilst not at all essential as a prerequisite for Fluid Dynamics, covers very similar content. Therefore, this module may seem fairly dry for anyone who has taken Physics of Fluids.
Overall I would recommend this module to anyone with an interest in applied Maths as it was fairly interesting and didn’t add stress to the term.

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.

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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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.

2020/2021
This module is basically field theory. You will use the theory of field extensions to answer number theory problems; the motivating examples include the insolubility of a quintic and the impossibility of squaring the circle, but you will mainly be applying it to irreducibility of polynomials. This module never fully engaged me. The theory is very abstract but the assignment sheets are quite concrete and rely only on a few key results, hence most of the theory is useless for the assignment sheets. Furthermore, the lecturer (Gavin Brown) went through a small amount of the notes in the lectures, mostly just the key results and worked examples. This meant that much of the theory just went by me and I put little time into the module, only enough to do the assignment sheets. The end result was that the module was a bit unsatisfying. The upside to this was that the assignment sheets were quite fun, and you may find that they’re a bit more “A-Level” (i.e. using results to solve problems) than other modules. I think this module will massively benefit from returning to normal, where traditional lectures will make you fully engage with the content. The exam was, for me, the perfect open-book exam. The questions were tough, but with the notes you could fight your way through them. This module is suitable for any algebraist since there are no non-Core prerequisites (Algebra II only). The ideas overlap significantly with Algebraic Number Theory (although you solve the problems with different machinery), so it may be interesting to do both of them together.

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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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.

2019/2020
The lectures for this module were fairly straightforward and the lecturer gave very clear explanations and drew lots of useful pictures. The content itself isn’t particularly challenging and the weekly assessed assignments are without a doubt the easiest ones I’ve seen in my degree to the point where they don’t really help teach the content so doing the optional ones is certainly a good idea. However, the exam this year was unreasonably long and made absolutely no sense as a paper. Actually finishing the exam was quite challenging as the questions gave incredibly few marks for a great deal of work. Some of the definitions asked were never properly stated in lectures or in the notes. Overall, the module is fairly interesting, and could make a nice addition to any selection of modules, but the exam could be quite the deterrent and if you’re not interested in topology may be too off-putting. It’s worth noting that, if you’re taking this module as a prerequisite to algebraic topology, some of the material on lifting properties of covering spaces is never taught and is only in the textbook.

2020/2021
Colin Sparrow lectured this module. The module was not lectured amazingly. However, the content was very interesting and would’ve been easy if the module was lectured better. This module is essential for a geometer / topologist. The key topics are:
Fundamental groups: of S^1, homotopy invariance, induced homomorphisms (and a bit of category theory)
Brouwer fixed point theorem
Borsuk-Ulam Theorem
Seifert-Van Kampen Theorem: wedge product, möbius strip and projective space, CW complexes
Generators and relations of groups, topological spaces whose fundamental group is isomorphic to a given group

The exam was quite easy.

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!

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).

2019/2020
Commutative Algebra is essential for anyone intending to study further algebra, in particular Algebraic Geometry. The content of this module was fairly interesting and easy to grasp despite an often unclear delivery from the lecturer. The notes were generally good although they did not always correspond to lectures since the order of teaching was different.
The exercise sheets were a reasonable difficulty and indicative of exam style questions, and the lecturer uploaded clear solutions which was very useful. The exam, whilst harder than previous years, was still very fair with a good balance of seen and unseen content.
Overall, for anyone interested in algebra, this module is a fine choice in term 2.

This module is “ideal” if you love algebra but don’t want to faff around with commutativity. However, there are a lot of methods and practice you need to do in order to get the gist of the questions. This isn’t just a copy-the-bookwork module, you need to understand and gain an intuition to the concepts taught, so that you can solve the homework and exam problems. If it’s the same lecturer, he’s not very keen on definitions, so the intuition is crucial.

2020/2021
This module is a whistle-stop tour of many areas of commutative algebra (the study of commutative rings). Topics include prime/primary/maximal ideals, primary decompositions, modules, ring extensions, ideals and varieties, and rings of fractions. This is very useful for any algebraist, especially those interested in algebraic curves, algebraic geometry or algebraic number theory. The ideas you use in Commutative Algebra will come up again in these modules. This is quite a difficult area of maths, and the content is hard despite being so broad. The proofs are very involved and complicated, but this isn’t a big problem since the lecturer (Chunyi Li) avoids bookwork in his assignment sheets and exam. However, he sets hard assignment sheets and exam papers, and is a big fan of “true or false” and “give a (counter)example” questions in the exam. He goes through a lot of this in seminars, so you’ll need to attend all of them and have in store all sorts of examples and counterexamples. He also leaves a lot to intuition and brief justifications, so you really need to understand the concepts, not just learn them. The lecturer is very good at explaining things and often went through some of the assignment questions in seminars, so don’t be afraid to speak to him. In previous years, people complained about the lecturer being hard to understand due to his heavy accent. It wasn’t so bad this year since he didn’t need to shout across a lecture hall, only speak into his mic, but it may be a problem again in future years.

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.

2020/2021
This module is a mix of analysis and infinite-dimensional linear algebra, and the content follows on nicely from Analysis III, Multivariable Calculus, Algebra I and Norms, Metrics and Topologies. You will study many different types of vector spaces (nearly all of which are normed spaces) and the maps you can put on/between them (e.g. norms, inner products, and linear operators). This module is completely accessible to any 3rd Year mathematician, including those who are less interested in analysis, and the content is fairly interesting and quite easy to grasp. The worst thing about the module this year was the exam, which was one of the most insanely hard exams I have ever taken. Even with it being open-book I couldn’t figure out how to apply the results from the lecture notes to the questions and I couldn’t begin to attempt multiple questions. Hopefully Vassili will tone things down from next year onwards. One final hint is that there hasn’t been a exam question yet on the measure theory section or the Sturm-Liouville problem. Whether you take that to mean they have to appear soon or are ignored by the lecturer is up to you.

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.

2017/2018
Pretty fun, decent lecturer, really hard assignments, quite difficult exam. Much easier if you’ve done Stochastic Processes.

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.

2019/2020
Set Theory is confusing to start with, but once you get a hang of what’s going on, it turns out to be a very straightforward module. You need to make it click, and when it does, it’s quite a nice module to do! The lecturer was awful though, I did well enough ignoring everything he said and just worked from the old notes. Definitely a module for self-teaching!

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.

2019/2020
The lectures for manifolds weren’t the clearest. The lecturer tends to mumble and his handwriting isn’t always legible. The module does have lecture notes and many good textbooks. The lecture notes can be unclear in places but for the purpose of getting used to definitions they get the job done and they have many examples. Be prepared for some very challenging content if you take this module; you’ll likely need to regularly go over all the definitions and the working following them. Moreover, the exercise sheets, particularly the first couple, are very challenging. However, many questions from them regularly come up in the exam and those that do tend to be the easier ones. Despite the module’s apparent difficulty, the exam is surprisingly approachable since it consists mostly of definitions and easier proofs. If you’re interested in doing any further maths, a knowledge of manifolds will be essential so this is a good module to do. I would encourage anyone taking it to try their best to avoid being put off by the difficulty of the material during term.

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).

2020/2021
A very challenging but reasonably rewarding module. Tbf I relied mostly on the book and youtube lectures. Chris Lazda is a lovely guy and all but I don’t think he made the content as interesting to learn as he could’ve. Certainly a lot of time was spent understanding the topics but Im glad I did!

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.

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).

2017/2018
Wide range of content (all of it interesting!), but requires understanding and recalling results from every other analysis module ever (I’m talking Analysis 1 to Differentiation to Fourier Analysis to Measure Theory) which I did not have, so I struggled. I’d also say that because there’s so much content, some of it is stated without proof or going into depth. Take that whichever way you want!

2019/2020
Ergodic Theory is Dynamical Systems with added Measure Theory. That’s it. That’s the entire module. So if you liked both of those, it’s a definite recommendation, since revising this is a lot of repeating DynSys definitions and examples. Plus there’s some interesting stuff on information theory near the end of it. However, if you didn’t like one or both of those (I’m guessing it’s the Measure Theory part which is more off-putting), perhaps avoid it, you don’t want to repeat that additional stress.

2017/2018
One of the most notoriously difficult modules on offer at Warwick, this took me as much time as my project and two other modules combined in term 1. Well worth the effort however, in my opinion the most interesting area of mathematics there is.

2019/2020
Algebraic Geometry is the study of Algebraic Varieties, which are basically zero sets of families of polynomials. However, despite this innocent statement, there is A LOT to learn in Algebraic Geometry. Thats what makes this course difficult : there is a lot of unfamiliar maths to cover in a short space of time. What also makes this course difficult are the biweekly assignments. These assignments are hard and will take most of your term 1.
Unfortunately, the literature for a beginner in Algebraic Geometry is not that great. The best source I know is Andreas Gathmann lecture notes on Algebraic geometry. They are basically a more detailed version of the lecture notes given at Warwick. Also, Miles Reid’s “Undergraduate Algebraic Geometry” is good to have as well.
The course is basically a subset of Joe Harris’s “Algebraic Geometry – A First Course”,with many of the problems taken from this book. However, in my opinion, that book is written very tersely. You may not want to depend solely on this book.
This module is important for anyone with an interest in Geometry, Topology and/or Number Theory.

2020/2021
Algebraic geometry is hard. Not just this module but the general consensus among mathematicians is that algebraic geometry is hard. We study zero sets of families of polynomials and develop tools to help solve problems related to these, you will spend a lot of time learning how to consider your zero sets as hyperplanes in projective space, which makes things easier. You need a good understanding of linear algebra and commutative algebra to do well here.In addition to the literature mentioned above, shafarevich 1 is good for supplementary material, Hartshorne is also brilliant if you want to study beyond the course to understand where the ideas come from.

2019/2020
The highlight and focus of the course is on maximal tori in Lie groups which is a genuinely very interesting concept yielding a nice analogue of Sylow’s theorem for finite groups. Unfortunately, this year’s exam didn’t cover tori at all so most of everyone’s revision was probably for nothing. On the bright side the TA is great!

2020/2021
Lie groups are essentially manifolds, which are also groups, and whose tangent spaces are Lie algebras – a pretty cool concept. The lectures were awkward to follow and the notes are rather dense, however they do have everything you need to learn the course. Be warned though- don’t try it if you aren’t comfortable with manifolds knowledge.

2017/2018
This module brings together lots of different modelling techniques which can be used to apply maths to real-world problems (no biology background is needed). The module starts with deterministic models (which overlaps a lot with MA3J3) and then looks at adding in stochasticity (usually Gaussian), age structure (PDEs) and spatial heterogeneity (distance kernels). Familiarity with the 2nd year Stochastic Processes module will be useful. The assignments (which aren’t for credit) tend to be much harder than the exam and are full of mistakes. This module is definitely appropriate for 3rd year students who have taken modules covering aspects of continuous-time markov chains and fixed point analysis for a systems of ODEs.

2018/2019
As a graph theory enthusiast, I really enjoyed this module. Graph Theory doesn’t quite focus on the many properties and results that many mathematicians have proved throughout the years, but focuses more on developing your techniques into understanding and proving results. If you like a lot of bookwork and factual recall, this isn’t the module for you. The lecturer likes to include many several mark questions which are heavily unseen content, even in the compulsory, which tests your ability to develop your own proofs. This isn’t an easy module, and I would only recommend it if you take the time and effort to properly understand not just the content but the general ideas on how the proofs work.

2019/2020
Group Theory is basically making a module solely dedicated to finding all simple groups of order less than 500. If you like groups and manipulating them using a variety of techniques, this is a fun module, though it’s difficult to visualize some of the concepts introduced. The last chapter is pretty much using every technique in the previous 6 chapters in order to find some sort of counterexample, and no doubt you’ll be asked to use exactly the same techniques to counter a simple group of order >500. As you’d expect, if you like groups, you’ll find this fun.

2020/2021
The module basically takes the group theory section of Algebra II and extends it to a full module. After a review of Algebra II, the new content includes nilpotent and soluble groups, blocks, matrix groups and the transfer homomorphism. The end goal is to classify the simple groups of order up to 500 using this machinery. The lecturer this year was Inna Capdeboscq, but the lecture notes and assignment sheets were originally written by Derek Holt. Inna was a very good lecturer, but I found the proofs to be very unintuitive which made them difficult to learn. Furthermore, the assignment sheets were extremely hard and tedious with even Inna admitting they “make you want to pull your hair out”. The worst thing is that the content itself was just bland with nothing really standing out. The section on matrix groups is absolutely awful, it goes on and on for ages and it’s pointless, but the section on nilpotent and soluble groups was quite nice once you get past all the definitions. Inna decided to set a fairly easy exam this year, but past papers (probably set by Derek) look much harder. It’s hard to say what future exams will be like. In my opinion, this module is only worth it as a 4th year CATS filler or for avid group theorists. But be warned: when people say, “Group Theory is an easy 4th year module”, they mean it’s easy relative to some other 4th year modules. No 4th year module is easy, and Group Theory was certainly harder than all the 3rd year modules I took this year.

2019/2020
Elliptic Curves has a heap of content, but a lot of it seems like waffle, when really you should be sifting through to find what you need to understand to solve the problems (here’s a hint, the complex chapter is useless). However, once you know the techniques, applying them is quite fun, and I’d suggest that this module is certainly closer to A-Level calculation-based maths rather than your boring repetitive bookwork nonsense. Also, there’s not really any need for prerequisites, and it pairs well with a Maths in Action project on cryptocurrencies, which at least 9 people figured out.

2019/2020
Dynamical Systems is basically taking functions and analysing their orbits on points in order to solve problems. This is a relatively nice module for a 4th year one, and has some interesting examples, such as sequence maps which depend on matrices, and maps on the unit circle. Very straightforward module, would definitely suggest it for anyone who doesn’t know what 4th year modules to fill up their CATS with. Also very accessible for 3rd years.

2019/2020
Judging by my housemates’ long struggles with their research projects, the Maths in Action project seems like the easier option, but is still a lot of work. Like with the 2nd year essay, make sure you decide what you want to do early and that will encourage you to work on it, in which the Christmas holidays is the perfect time to do so! You also get free pizza at the presentation sessions!

2019/2020
Honestly, I didn’t pay much attention to this module because the safety net had me covered. The lectures were rather dull and the content was a tad dry, but it incorporated graph theory into the geometry, which was a bonus. Would definitely recommend Intro to Topology to get an understanding of fundamental groups beforehand though.

2019/2020
A Riemann Surface is a one dimensional complex manifold. The module is about exploring the properties of these surfaces. This often involves generalising what you have learned in complex analysis to manifold theory and mixing in some topology.
It might not be surprising then that the results of this module are as ‘nice’ as complex analysis and the content of the module is of interest to many types of mathematicians. Riemann surface theory is important to Algebraic Geometers, Differential Geometers, Dynamical Systems people and even Algebraic Number theorists.
The module is taught very well, with a good lecturer and insightful example sheets. The content of the module is taken form Beardon’s Primer on Riemann Surfaces.
Overall, a good module which should be interesting to many people!

2019/2020
Algebraic Curves is the study of one dimensional Algebraic Varieties. Despite what is written on the module webpage, Algebraic Geometry is a NECESSARY prerequisite for the course. It is impossible to take this module without taking Algebraic Geometry.
In my opinion, the course was lectured very badly but the lecture notes were ok. The example sheets were tersely written, which made doing assignments a painful experience.
I would say the module is only useful for future Algebraic Geometers or Algebraic Number Theorists.
If you do intend to take this module, I recommend that you take a few books out of the library. This is because the content of this module is not contained in any one book. Miles Reid’s UAG and UCA are helpful.
In addition, Chapter 4 of Hartshorne’s Algebraic Geometry may help you understand the course better, even though you may not understand his proofs (which use schemes and sheaf cohomology).
Fulton’s Algebraic Curves is also useful.

2020/2021
In this year, algebraic curves was lectured by Diane Maclagan, so the course was very different to how Miles lectured it. A curve is a 1 dimensional algebraic variety, you’ll study field extensions, local rings of regular functions, function fields and so on. These things lend themselves to concrete examples which were taught well in general with plenty of approachable (non-assessed) exercises. All the of this year’s lectures are good to revise from if you have access to them

2019/2020
I will go as far as to say that Differential Geometry is BY FAR the hardest to understand mathematics module offered at Warwick. To put that into perspective, I also took Algebraic Geometry, Cohomology and Algebraic Curves in my final year.
The course is about Riemannian Geometry. There is a lot to say about Riemannian Geometry! Therefore, there is a lot to learn if you take this module.
I recommend Lee’s “Introduction to Riemannian Manifolds” and Jurgen Jost’s “Riemannian Geometry and Geometric Analysis” as textbooks for the module. You may also want to have a look at Lee’s “Intro to Smooth Manifolds” to remind yourself off the necessary manifold theory needed to understand this module.
This module is important to any future geometer and/or theoretical physicist (General relativity uses “pseudo-Riemannian geometry”). Obviously, only take the course if you are interested in differential geometry. It will be a lot of hard work!

2019/2020
The Advanced PDEs module is about the theory of existence and regularity of solutions to 2nd order linear elliptic PDEs.
It turns out that, in general, “solutions” to such problems are best formulated in spaces called Sobolev spaces. About 7 weeks of the module is devoted to developing the theory of these spaces and their associated embedding theorems. The rest of the course is about applications of this theory to some PDEs.
In my opinion, the module contains some very interesting analysis (you will need measure theory and functional analysis 1). You will also see applications of many results in functional analysis that you learned in the previous year, which might have otherwise appeared to have been unmotivated at the time.
The lecturer for my year was good, but I still think the module is quite hard, with many difficult example sheet questions (get support class notes!) and many hard to remember proofs. The bible for PDEs is given by Evans (the module is equivalent to chapters 5 and 6 of his book), and it is a very good book. His book should make things easier to learn.
Partial differential equations are important to wide range of disciplines: from fluid mechanics to stochastic analysis to differential geometry. Therefore, the content of this module will be important to most people reading this.