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### Careers Officer&ensp;<small class="text-muted">Daphne Lavi</small>

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title: Module Reviews
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date: 2024-06-27
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The mathematics courses at Warwick offer a wide variety of optional modules -- there are no compulsory modules in the third year -- but this can make it difficult to choose which modules to take. Below, you can find a collection of reviews of various maths modules to help you make your decision.

# Year 1

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<large>MA117 Programming for Scientists</large>
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<large class="text-muted">22/23</large>
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This is a 100% coursework coding introductory Java course, with a maths module code but taught by the DCS. The module assessment consists of three programming assignments, worth 25%, 35% and 40% of the module mark respectively -- these assignments are harder than anything in MA124 Maths by Computer, but are very doable if you put in the time and don't leave them until the night before they are due! The assignments are mainly focused on using Java to solve computational maths problems, rather than programming in Java for its own sake.
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For anyone who has already coded in Java (or has any significant programming experience) I would absolutely recommend this module more than any other first year optional, its 100% coursework nature means that it is very possible to get almost all, if not full, marks and it is one less exam to worry about in the summer.
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Second years can also take this module, so if you are confident in Java you might want to save it for then to increase the degree credit you get from it (as second year is worth more than first). I'd still recommend it to anyone who hasn't done much programming but enjoyed MA124 in term 1, provided you are prepared to put in the time to learn Java and do the assignments alongside your other modules. The lectures for this module are entirely online and have PowerPoints that you can read through too, there are also in-person drop in labs run by TAs too -- personally I found the latter a lot more helpful, but both are potentially useful resources. In the end though, like maths, coding is not a spectator sport -- you'll get the most out of this module by putting in the time and trying the assignments!
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<small class="text-muted">23/24</small><br/>
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# Year 2

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<h2 class="m-0">
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<large>MA241 Combinatorics I</large>
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<large class="text-muted">22/23</large>
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This module is split into two parts: the first half covers the basics of enumerative combinatorics, up to and including a short section on generating functions; and the second half covers some basic extremal graph theory, mostly on vertex and edge colourings and Kuratowski's theorem on minimal forbidden subgraphs.
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In the first half, we learn about bijective proof techniques, using them to count the classical problem of placing balls into boxes (or counting functions between two discrete sets, or other equivalent formulations) up to various symmetries, which leads into integer partitions, Stirling numbers of the second kind, Catalan numbers, and a brief introduction to generating functions.
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The second half of the module pairs well with <i>CS137 Discrete Maths 2</i>, <i>CS254 Algorithmic Graph Theory</i>, and to a lesser extent, with <i>MA252 Combinatorial Optimisation</i>, but serves well as a first course on graph theory alone. Graph traversal problems (Eulerian/Hamiltonian graphs) are introduced, followed by graph packing problems (independent sets, matchings, etc.), before covering some chromatic problems. The very last portion of the module covers some Ramsey theory and introduces some probabilistic proofs, similar to those from <i>CS137 Discrete Maths 2</i>.
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The notes are well written and readable, though somewhat slow-paced, and the lecturer is also engaging and responsive throughout the course. The assignments and exam questions are also all interesting and require some amount of thought and creativity rather than mechanically applying rules and theorems.
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Overall, the module is certainly on the easier side, and a high grade can be expected without too much hassle. Getting to grips with combinatorial thinking and theorems, as well as graph theory, is also greatly important, particularly in algebra, topology, geometry, and probability theory, as many results in these areas are combinatorial in nature. The main focus of this module is on enumerative combinatorics, but the second half of the module introduces some existential or extremal combinatorics as well, and you will find that these ideas pop up more and more as you progress through other modules.
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<div class="card-header" id="headingMA243">
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<large>MA243 Geometry</large>
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<large>MA250 Introduction to PDEs</large>
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<large>MA256 Introduction to Mathematical Biology</large>
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<div class="card-header" id="headingMA252">
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<large>MA252 Combinatorial Optimisation</large>
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<large class="text-muted">22/23</large>
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The first half of this module is occupied with an introduction to asymptotic notation and Landau symbols, with some basic algorithm analysis, up to and including the Master theorem for recursive algorithms. Next, graphs are introduced, and we explore algorithms for solving various classic graph theory problems, such as graph traversals, minimum spanning trees, and shortest path. Networks and flows are then introduced, leading up to the Ford-Fulkerson algorithm, the max-cut min-flow theorem, and flow decomposition. Then, we cover the graph-theoretic version of Hall's condition/Hall's theorem and algorithms for stable matchings.
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The second half of the module covers linear programming; converting linear programs to standard form, and transforming various graph optimisation problems into linear programs, before covering the geometric and tabular simplex methods.
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In the year I took this module, the notes were handwritten and unreadable, and the lectures were dull and unengaging; one some occasions, mistakes would be found near the end of an example, upon which the lecturer would erase the board and then repeat the last 20 minutes of the lecture with the mistake amended. The notes and lecturer have changed (improved) in the 23/24 academic year, but the content has remained largely unchanged.
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If you are interested in algorithms and graph theory, which the bulk of this module concerns itself with, you would be better off taking <i>CS260 Algorithms</i> and <i>CS254 Algorithmic Graph Theory</i>; and if you are interested in complexity theory (NP-hardness, complexity classes, etc. which <i>used</i> to be in this module before being removed), then I recommend taking <i>CS301 Complexity of Algorithms</i>, or alternatively for computability theory, <i>C259 Formal Languages</i>, <i>PH345 Computability Theory</i>, and <i>PH349 Logic III: Incompleteness & Undecidability</i>.
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<div class="card-header" id="headingMA254">
<h2 class="m-0">
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<large>MA254 Theory of ODEs</large>
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This optional module is a great choice for anyone who liked and did well in previous analysis modules and Modelling 1! It also goes well with the core MA260 Norms, Metrics & Topologies that takes place in the same term. There is a decent balance between proof and computation -- the module begins more proof orientated, proving results such as the contraction mapping theorem and Picard theorems (all of which are also seen at the end of MA260) and later is more computational, such as determining the stability of fixed points and drawing their phase portraits. There are also sections on Lyapunov functions, Hamiltonian systems, Lotka-Volterra equations and the Stable Manifold Theorem, which will interest those who enjoy physics and multivariable analysis.
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The notes are initially dense, but are well written with lots of worked examples in the later sections. They are supported by the accompanying worksheets which are good for understanding the material -- the notes explicitly signpost to specific questions on these sheets throughout. The lectures are on the fast side but are engaging and complement the notes, the lecturer is also enthusiastic and good at answering questions both in the lectures and on forums/over email. The 23/24 exam for this module favoured computation, but this was the first year it was taught by the current lecturer so future papers might be more balanced. The content hasn't changed much under the new curriculum, so most past papers are still useful.
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As someone who likes analysis and modelling, I enjoyed this module and would recommend it to anyone who found themselves asking under what circumstances we can guarantee a well-posed solution to the initial value problems from Modelling 1, as well as to those considering exploring 3rd/4th year modules in analysis, dynamical systems, physics and mathematical biology!
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<div class="card-header" id="headingMA257">
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<large>MA257 Introduction to Number Theory</large>
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<large class="text-muted">23/24</large>
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This optional module continues directly on from the number theory seen in Foundations, but without the focus on cryptography and more that of a general introduction. The first half mostly recaps number theory seen in Foundations, so joint degree students who did Sets & Numbers will not be at a disadvantage. There is some non-Foundations content on quadratic residues and Legendre notation towards the end of this half. The second half is about Diophantine equations, introducing results like Minkowski's theorem and the two, three and four square theorems. There is also some overlap with Algebra 3 with Gaussian primes and unique factorisation in the Gaussian integers. The end of the module also looks at Waring's problem and results in Diophantine approximation such as the Dirichlet approximation theorem, which you may have seen in Analysis 1.
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The notes are short and concise, relative to other second year modules; cover material well, and have lots of brief mentions of extension material/theorems for those who are keen to explore beyond this module. Many of the exercises seen on the worksheets and assignments are well-known 'classical' proofs and problems, so those who have read around number theory or tried olympiad number theory problems will likely have seen some of them before -- but this is far from necessary, and the questions are good practice at developing intuition, applying theorems, and spotting tricks.
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The lectures were well-paced and added more detail and useful explanation as the notes were quite short. The exams for this module in 22/23 and 23/24 were nice, with a good mix of computational questions and (mostly) bookwork proofs. The content has not changed much under the curriculum changes from 23/24 onwards, so most past papers are still useful. This module is a great choice for anyone who enjoyed the number theory in Foundations and/or for anyone who has read around number theory in their own time!
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<large>MA263 Multivariable Analysis</large>
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<div id="collapseMA263" class="collapse" aria-labelledby="headingMA263" data-parent="#accordionY2">
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<large class="text-muted">YY/YY</large>
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<large>MA269 Asymptotics and Integral Transforms</large>
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# Year 3

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<large>MAXYZ</large>
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<large class="text-muted">YY/YY</large>
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