MacTutor

1.1. From the Publisher.

In 2013, a little known mathematician in his late 50s stunned the mathematical community with a breakthrough on an age-old problem about prime numbers. Since then, there has been further dramatic progress on the problem, thanks to the efforts of a large-scale online collaborative effort of a type that would have been unthinkable in mathematics a couple of decades ago, and the insight and creativity of a young mathematician at the start of his career.

Prime numbers have intrigued, inspired and infuriated mathematicians for millennia. Every school student studies prime numbers and can appreciate their beauty, and yet mathematicians' difficulty with answering some seemingly simple questions about them reveals the depth and subtlety of prime numbers.

Vicky Neale charts the recent progress towards proving the famous Twin Primes Conjecture, and the very different ways in which the breakthroughs have been made: a solo mathematician working in isolation and obscurity, and a large collaboration that is more public than any previous collaborative effort in mathematics and that reveals much about how mathematicians go about their work. Interleaved with this story are highlights from a significantly older tale, going back two thousand years and more, of mathematicians' efforts to comprehend the beauty and unlock the mysteries of the prime numbers.

1.2. From the Introduction.

Until the nineteenth century, the Inaccessible Pinnacle was, well, inaccessible. A slender slice of basalt rising amongst the Cuillin mountains on the Isle of Skye, off the north-west coast of Scotland, it seemed unclimbable to the early pioneers of the nineteenth century who set out to explore the Cuillins. These days, it remains a challenging climb, but it is definitely not impossibly difficult: given fine weather, tour guides can take novices right to the top.

Mathematical exploration has much in common with this kind of adventuring. You stand looking at the sheer surface of your mathematical problem, searching for toeholds and crevices that might give a way up. After a long time looking, you start to make out an indistinct crack to the left, and a slight pattern in the rock up and to the right that reminds you of a climb you heard about once. Pulling together all the features you've noticed, you can sketch out a possible route up the rock face, although it's not quite clear whether that small ledge will make a good toehold and there's a pretty ambitious reach near the top that might well be a stretch too far.

Still, now that you have a possible route in mind, you can start climbing, and hope that the details will become clearer along the way. Perhaps that reach will be too big, but when you get a bit closer maybe there'll be a crack in just the right place for your fingers.

Unfortunately, when you're three-quarters of the way up a silver of rock breaks away, your toehold disappears from beneath your feet, and you drop back some way. Eventually, however, if you persevere you might reach the top.

Once someone has found their way to the top, suddenly the inaccessible becomes much more accessible. Once you know that someone has been up, you know that it can be done. If you have access to their notes, or know someone who heard them describe the route, then perhaps you can even follow in their footsteps. In some cases, what was at first a risky and demanding endeavour for pioneers becomes mainstream, and suitable for a weekend stroll even by those who have no climbing experience.

Now, there are many things about this that do not describe what it is like to do mathematics - and probably it's not such a great description of rock climbing either. But the analogy has its uses.

Rock climbing usually involves many people: often a challenge is taken on by a team rather than an individual, and also many teams will tackle the same rock face. In mathematics, there are romantic stories of individuals making heroic breakthroughs by themselves. Less well known are the collaborations, some of which in the twenty-first century involve very large numbers of people. This book has a romantic story of the best kind - an extraordinary breakthrough by an individual. It also has insights into these new large collaborations, and what they can reveal about what mathematicians do when they do mathematics. The Inaccessible Pinnacle for the mathematicians on this quest is the Twin Primes Conjecture, one of the most famous unsolved problems in the whole of mathematics. I'll tell you much more about it in the coming chapters .

I am no climber. If you are, then you will have detected this from the flaws in my description above! I do, however, love going on holiday to Skye, and when I am there I enjoy walking. I am inexperienced, and nervous about setting out on potentially hazardous trips by myself. However, I have had some wonderful days walking in the foothills of the Cuillins, seeking a personal challenge, enjoying the stunning scenery, catching glimpses of the peaks when they emerge from the clouds and marvelling at the skill, fitness and courage of those who reach the summits.

This book is for those who enjoy roaming the mathematical foothills. I hope to be the kind of guide that I would like for my trips to Skye: I want to show you the sights. In particular, I hope to give you glimpses of the summits, and to tell you tales of the people who climb them, while also leading you along my favourite routes in the foothills and pointing out some directions for adventures you might want to undertake in the future. Inevitably, this will present you with challenges. I have tried to select our routes carefully, and much of the journey is easy walking with spectacular views, but sometimes you will find that there is a tricky stretch, with some parts seeming out of reach. You have a big advantage over the hillwalker, though, because if a part is too challenging for you at the moment then you do not have to stop there: you can turn the page and skip over it in one bound! I'll try to highlight these parts, but I hope that you will skim through them to get some sense of the ideas involved, even if you do not want to go through them in detail on a first reading. This is, after all, how mathematicians usually read each other's papers and books.

1.3. Review by: Michael N Fried.
Mathematical Thinking and Learning 20 (3) (2018), 248-250.

... in May 2013, the problem of prime-twins suddenly shifted to the list of questions where mathematics is making progress - very rapid progress. In that month, a practically unknown mathematician, Yitang Zhang, showed that there is a finite number $M$ with infinitely many pairs of primes $p_{k}$ and $p_{k+1}$ satisfying $p_{k+1} − p_{k} ≤ M$. Naturally, he would have liked to have shown that that number is 2, but he had to settled for a somewhat larger figure ... 70,000,000! Still, it was a tremendous step forward. Indeed, if you zoom out from the number line by a factor of, say, 70,000,000, you would see an infinite number of pairs of primes very close together, a distribution similar to what you might see were the twin prime conjecture true. In any case, the importance of Zhang's discovery was recognised immediately and generated a flurry of mathematical activity, the result of which was to reduce the bound within a single year to 246 ! The book, Closing the Gap, by Vicky Neale gives an account both of that exciting year and many other fascinating questions from the theory of numbers, all discussed in a consistently entertaining and enlightening way.

The book has a rondo structure: A-B-A-C-A-D-A-E ... In the refrains, Neale discusses the progress on reducing Zhang's bound during the months from May 2013 to April 2014. In the episodes between the refrains, she describes and explains mathematical ideas connected with the problem as well as other related matters. For example, she discusses Waring's problem (posed in 1770) which asks to show that for any $k$, there is a number s such that every positive integer can be written as the sum of $s$ numbers (including 0) raised to the $k$th power. And she speaks about the remarkable theorem proven by Ben Green and Terence Tao in 2004 showing that the prime numbers contain arbitrarily long arithmetic sequences. Tao actually plays an important part in the story told by Neale.

But does this narrative, as fascinating and enjoyable as it is, have something to offer to professional mathematics educators, readers of this journal? I think there are at least three ways in which it does. First, although this can be debated (see Fried & Dreyfus, 2014), it seems to me nevertheless quite important that mathematics educators, both practitioners and researchers, deepen their knowledge of mathematics itself. For this, much is to be gained from this book. For myself, I learned a lot, even about subjects I thought I knew before. And then there were many things completely new to me, for example, Szemerédi's theorem which tells when a set of integers is big enough to contain an arithmetic sequence of any length. It was not Neale's intention to give thorough and detailed explanations of these things, but she explains enough well enough to make everything plausible and interesting and, therefore, enriches us mathematically.

This leads me to the second point. It is clear from every page in the book that Neale is a superb teacher. The book is full of visual or otherwise concrete devices, for example, which bring out theoretical ideas and questions. ...

Vicky Neale is a mathematician working in the area of number theory (Ben Green ... was her PhD advisor), and the clarity of her examples and explanations surely comes from the clarity of her own understanding. Deep knowledge of the subject may not be sufficient for good teaching but, as I remarked regarding the first point above, it is almost certainly necessary. Necessary too is enthusiasm for the subject - and this, I venture to say, also comes from understanding the subject. Neale is not lacking in enthusiasm. Having mentioned the fact that all primes greater than 3 are either one more or one less than a multiple of 6, she cannot contain herself and says, "Isn't that neat?" - and it is not the only time says something like that! Probably because her knowledge of the subject is completely believable, those interjections are believable and, I might add, completely infectious. These are things to think about when we think about good teaching and the relationship between mathematical knowledge and teaching of mathematics.

1.4. Review by: Nikoleta Kalaydzhieva and Sam Porritt.
Chalkdust Magazine (28 June 2018).

As a number theorist myself, people always ask me "Why are primes so interesting?" Next time it happens, rather than spending hours proving my case, I will just refer them to Vicky Neale's book Closing the Gap. Written in an engaging and inclusive way, it makes a perfect read for beginners but it also picks up the pace fairly quickly, so even enthusiasts like myself are bound to enjoy it. In particular, it starts by defining prime numbers, and yet somehow in the space of 160 pages, Neale manages to take the readers on a journey to cutting edge research mathematics. But enough rambling, let me tell you a bit more about what you might find inside the book and what we liked about it.

What is inside?
I will try and provide a quick summary, without giving too much away. In the introduction Neale provides an interesting way of looking at tackling a research problem, by comparing the experience to rock climbing on uncharted terrain. Henceforth the chapters alternate between the mathematics behind one of the oldest unsolved problems in number theory, the Twin Primes Conjecture, and the story of what followed from the latest breakthrough.

In the first few chapters, we are introduced to the notion of a prime, and twin primes, which are a pair of primes that differ by 2. As with many problems in number theory, the twin prime conjecture is easy to state but as you might guess from the name is still unsolved. Similarly, Goldbach's conjecture and the existence of infinitely many primes of the form $2p + 1$ where $p$ is a prime, called Germain primes, is not known.

And as Neale says "if primes are hard let's try something else." Thus in the chapters that follow she discusses problems similar, in some sense, to the ones before, that we know the answers to or that we can prove.

And let's not forget Vicky's mathematical pencil, which she uses to illustrate the distribution of primes.

Following, this brief introduction into the world of analytic number theory, she moves on to some more complicated results from Ramsey Theory and probabilistic number theory that were used in the most recent breakthrough. Moreover, she gives a very nice heuristic of why we expect the twin prime conjecture to be true.

And as this is a pop maths book, there are plenty of problems for you to puzzle over.

However, if the maths, in particular the log's become too much for you, or if you are already familiar with the topics covered, you can always turn to the other half of the book which deals with the recent history of progress in solving the Twin Primes Conjecture. And this story has it all! Started by a single mathematician working on his own, who managed to combine works that seemed unrelated to make a dent in the problem, improved by a global collaborative project (the Polymath project), and we are yet to see how this story will end. We do however get an intimate insight into the research world in recent years, and it is a lot more exciting and dynamic than one might expect!

What we liked
In an increasingly connected world, the internet is changing the way research mathematics is done. Closing the gap provides a valuable insight into this new world of blogs and massive online collaborations by telling the story of some recent ground-breaking results in prime number theory. Neale uses real conversations with, and comments from, experts in the field to bring the reader into this world and introduce them to the people behind the proofs. We like how Neale has done a good job of keeping the book lively by skilfully weaving the main narrative with entertaining and relevant mathematics for the reader to ponder and puzzle over.

1.5. Review by: Mark Hunacek.
Mathematical Association of America Reviews (2 December 2018).
https://www.maa.org/press/maa-reviews/closing-the-gap

Let's begin with some background. One of the more famous currently unsolved problems in number theory is the twin primes conjecture, which asserts that there are infinitely many pairs of consecutive odd numbers (such as 3 and 5, or 5 and 7) that are both prime. According to an article by Dana Mackenzie in What's Happening in the Mathematical Sciences, Volume 10, it is unclear when, and by whom, this conjecture was first proposed, but its roots go back at least as far as 1849, when Alphonse de Polignac published the more general conjecture that for every positive even integer $k$, there are infinitely many pairs of primes differing by exactly $k$. Not only has the special case $k = 2$ (which is, of course, the twin primes conjecture) proved to be exceptionally difficult; until recently there was no proof that there exists even one $k$ for which this result is true (the "bounded gap" problem).

In May 2003, however, Yitang Zhang electrified the number theory community by proving that there is in fact at least one such $k$, and that indeed $k ≤ 70,000,000$. Then, in the year or so following Zhang's announcement, there was a frenzy of activity in the area, and by April 2004 the bound for $k$ had been reduced from 70,000,000 to 246.

What makes this story doubly interesting from a human interest (as well as mathematical) standpoint is that Zhang was not, at the time of his result, a professor at a prestigious research university; in fact, he was a lecturer at the University of New Hampshire, having experienced some difficulty getting a job, likely as a consequence of a bad breakup Zhang had with his doctoral advisor. He was also in his mid-50s. (Take that, G H Hardy.) His story makes interesting reading: indeed, the New Yorker magazine published an article about him in its February 2, 2015 issue.

In this excellent book, author Vicky Neale attempts, with considerable success, to make the story of the bounded gap problem accessible to laypeople with little formal training in mathematics. In the process, she also touches upon a lot of other topics. These include other topics in substantive mathematics (examples: the prime number theorem, Goldbach's conjecture, sums of squares, Waring's problem, the Hardy-Littlewood circle method, quaternions, Fermat's Last Theorem and unique factorisation of integers) as well as issues about mathematical culture. She does a superb job, for example, of explaining the nature of mathematical research, and ways of communicating mathematics. The book discusses the concept of peer-review in journals, the ArXiv, and (quite extensively, since it is very relevant to the bounded gap problem) the Polymath project, along with practical implications for this, such as attribution of credit. (I am often surprised by how little some of my math-major students know about things like math journals and the publication process.)

The structure of the book is interesting. The title of every odd-numbered chapter is a date (e.g., May 2003) and that chapter covers developments in the bounded gap problem during that month. The even-numbered chapters develop the necessary mathematics, starting with the definition of a prime number and the well-known proof (from Euclid) that there are infinitely many of them. The last chapter of the book is entitled "Where Next?" and poses some questions and speculations about the future.

Following this, there is a good bibliography, listing books, published articles, and internet sites (with URLs correct at least as of January 2017). The New Yorker article referred to previously, for example, appears in this bibliography. Even better, the bibliography is annotated; Neale not only lists the sources but describes what is found there.

Writing a book about mathematics for a lay audience can be a surprisingly difficult undertaking, because there are competing objectives to be balanced. If you write in too elementary a manner, the mathematics becomes so watered-down as to be useless; write too precisely, however, and you confuse your target audience. Neale manages to thread this moving needle nicely. Her prose is clear but not patronising, precise but accessible. The result is a very enjoyable book that can be read with profit not only by laypeople but also by mathematics students and the people who teach them.

1.6. Review by: Tim Harford.
https://timharford.com/2018/02/books-for-people-who-love-numbers/

Vicky Neale's Closing the Gap is an excellent account of recent progress in prime numbers, but also one of the best accounts you'll read by a mathematician about how mathematics research is done and how it feels to do it.

1.7. Review by: Colin Beveridge.
The Aperiodical Reviews (2 February 2018).
https://aperiodical.com/2018/02/review-closing-the-gap-by-vicky-neale/

Did you read Cédric Villani's Birth of a Theorem? Did you have the same reaction as me, that all of the mentions of the technical details were incredibly impressive, doubtless meaningful to those in the know, but ultimately unenlightening?

Writing about maths, especially deep technical maths, so that a reader can follow along with it is hard - the Venn diagram of the set of people who can write clearly and the set of people who understand the maths, two relatively small sets, has a yet smaller intersection.

Vicky Neale sits squarely inside it, and Closing The Gap has gone straight into my top ten "books to give to interested students."

Here's a clever way to structure a maths book (I have taken copious notes): follow the development of a difficult idea or discovery chronologically, but intersperse the action with background that puts the discovery in context. That's not a new structure - but it's tricky to pull off: you have to keep the difficult idea from getting too difficult, and keep the background at a level where an interested reader can follow along and either say "yes, that's plausible" or better "wait, let me get a pen!." This is where Closing The Gap excels.

Neale takes as the difficult idea the Twin Primes Conjecture, and specifically the work that followed from Yitang Zhang's lightning-bolt discovery in 2013 that infinitely many pairs of primes are separated by at most 70,000,000 (which sounds like a lot ... but is very small compared to "no upper limit") - especially the Polymath projects and the work of James Maynard in reducing the bound to either 600 (unconditionally) or 12 (if the Elliott-Halberstam conjecture is true - a bound later reduced to 6 by Polymath8b).

The Elliott-Halberstam conjecture? What's that? Neale takes the time to explain, by way of a mathematical pencil, the flavour of the conjecture, without getting bogged down in the technical details; she tells us enough that the story makes sense, and enough that we could go and find out more if we wanted.

Because of Neale's position in the Venn diagram, she can pull off this kind of thing, making maths accessible without losing accuracy - she's meticulous about saying "there's more to this" when there's more to something.

This attention to detail is possibly overdone in places - I found myself rolling my eyes from time to time about in-text reminders that I met Terry Tao in a previous chapter, or that we'd hear more about such-and-such in a future one, which I suppose is an upshot of deciding to do without footnotes. This is literally my only mild criticism of the book; I'm even in thrall to the quality of the paper it's printed on.

Closing The Gap communicates the excitement, frustration and interconnectedness of top-tier mathematical research, including the relatively new approaches pioneered by Tim Gowers (among others) with the Polymath project. The book's introduction starts with an extended analogy comparing mathematics to climbing (we know a MathsJam talk about that!) - how something impossible gradually becomes possible, then difficult, then accessible to novices with the help of a guide. Neale sets herself up as this guide, and succeeds brilliantly.

1.8. Review by: Marianne Freiberger.
Plus Magazine (20 December 2017).
https://plus.maths.org/content/closing-gap

Closing the gap is among the clearest popular accounts of maths I've read in a while. It's about prime numbers, as the title suggests, but it's also a master piece in the art of weaving. Apart from exploring the mathematics, the book gives an intimate description of the process of doing maths as experienced by those who do it every day, and an account of a particularly exciting, and recent, period when prime number theory made some great leaps forward. And it's a look at a completely new way of doing mathematics: in large online collaborations that anyone can join.

So many strands are a recipe for a tangled mess (that's personal experience speaking) but Neale has turned the multi-layered nature of her story into a strong point of the book. Some of this is anchored in its structure. Chapters alternatingly look at the actual maths, with no interference from the real world, and at the very real events that unfolded between April 2013 and April 2014, during which some major advances were made. This way of structuring the book isn't a gimmick, as I feared at first, but probably the best way of getting both parts of the story across. Adding to the clarity in structure is Neale's calm, unhurried, and very personal voice, which holds your hand throughout.

Mathematically the book focuses in the famous twin prime conjecture which asserts that there are infinitely many pairs of primes whose difference is 2 (you can read more about this conjecture on Plus). The fact that the conjecture can be stated in a sentence illustrates one of the advantages of number theory when it comes to popular mathematics. Many of its central problems can be easily explained even to a maths phobe, and pretty much anyone can start playing with numbers to see how far they get. On some questions, you can get satisfyingly far with a relatively small mathematical tool kit. But on others, including the twin prime conjecture, you soon hit a brick wall.

The tricky nature of prime numbers, which form the building blocks of number theory, is explored in the chapters that focus on the maths alone. Neale introduces the subject from scratch and invites the reader to play. If you like puzzling over maths problems, or feel you need a break from the author's guiding hand, you can go away and scribble for a while, until you're ready to get back to the book. The problems and results are carefully chosen to illustrate the treacherous nature of the subject, and also to provide some surprisingly deep insights into the maths used by those at the cutting edge of the field.

If you'd rather give the puzzling a miss, then you can let yourself be guided through the maths and focus on those chapters that describe how recent advances on the twin prime conjecture came about (some of the harder maths can safely be skipped). As hard mathematical problems go, the twin prime conjecture is unusual in many ways, but in the context of the book it serves as a great example of how progress in pure maths comes about: not only through great theoretical leaps, but also through incremental improvements, the testing of boundaries, blind alleys and experimentation. Imagination, intuition, the ability to ask good questions and spot pervasive patterns, and the courage to get stuck are essential in this.

Closing the gap is firmly aimed at a general audience. A desire to share with non-mathematicians the pleasure, frustrations and excitement of doing maths, to shed some light on this all too secretive process, seems to have been one of Neale's main motivations for writing the book. She is unapologetic about the maths, so be prepared to think, and think hard in places. If you are already well-versed in mathematics, the book also has something to offer. It gives some insight into what's happened in number theory in recent years, at least as far as the twin prime conjecture is concerned. Above all, it will give you an interesting insight into the Polymath project, which has seen mathematicians bare all in public (metaphorically) to see if large and fast online interaction can bring the subject forward - with very interesting results.

I won't give too much away by saying that the gap between primes hasn't been closed sufficiently to prove the twin prime conjecture - not yet. For Neale this means that writing the book has been a bit of a gamble. Had the conjecture been proved just as she put the finishing touches to her manuscript, she would have had a whole lot of rewriting to do. As it stands, she has left us dangling from a cliff. As progress in mathematics goes, we may stay dangling there for a few decades. But if we're lucky, perhaps by next Christmas Neale will be able to provide us with a concluding sequel.

1.9. Review by: Deborah Chun.
London Mathematical Society Newsletter 482 (2019), 35-36.

Vicky Neale's book tells the stories of two problems in mathematics and along the way illustrates how mathematicians work and how mathematical progress is made. Her main plot-line, referenced in the title, is the quest to close the upper bound on the gap between successive primes. This book delves into the progress on the Twin Primes Conjecture, giving the startling result by Yitang Zhang, mentioning the history of the conjecture, and explaining the ensuing online collaboration. Besides this main plot line, Neale leads her audience to a technical result without requiring an understanding of calculus. This compact and well-researched book accomplishes quite a lot in 164 pages!

This book is appropriate for a general audience. Neale uses a warm, conversational tone throughout. She first introduces an analogy. The Inaccessible Pinnacle, a rocky peak on the Isle of Skye in Scotland, seemed at first unclimbable. The draw of the impossible proved enticing enough that eventually a climber reached the summit. This spot has since become a tourist destination for even inexperienced climbers. Similarly, mathematicians aim to make mathematics accessible, and Neale guides anyone interested through the world of mathematical research.

This framework for the two problem arcs shows nicely how new mathematics comes to be. Even numbered chapters cover aspects of the general study of mathematics, while odd numbered chapters lay out the progress of the Twin Primes Conjecture. Neale carefully leads the reader through a set of questions, minimising technical language and abbreviation. She shows how questions can lead to more questions as easily as they can lead to answers and explains how wrong ideas are just as vital to mathematics as correct ideas are. On page 57, she plainly states, "The process (of doing mathematics) is both messy and creative." She discusses maths as an individual pursuit and a collaborative endeavour. She illustrates the process of doing mathematics using the work surrounding Zhang's result.

Hardy and Littlewood show up early on in the Twin Primes Conjecture narrative. Later on, these prominent mathematicians steal the show for a tangential problem. Neale manages to conversationally approach and illuminate an asymptotic formula, which is quite a feat! She discusses Hardy and Littlewood's circle method used to address Waring's problem. In this story arc, we see how every whole number can be written as the sum of four squares (for example, $27 = 0^{2} + 1^{2} + 1^{2} + 5^{2}$) which generalises to a whole set of questions about writing integers as the sum of $s$ numbers to the $k$-th power. This compelling digression can additionally illustrate how problems can draw the mathematician's focus as she is working on another problem.

The main plot line is laid out chronologically by chapter title in Neale's odd chapters, from Chapter 3 "May 2013" to Chapter 15, "April 2014." In these chapters, she describes the progress on the Twin Primes Conjecture. She starts with Zhang's result that there are infinitely many pairs of primes that differ by at most 70 million, which was presented in a Harvard seminar on May 13, 2013. Neale discusses the origin of the conjecture, work done independently by key figures, and details the inception of Polymath, a huge, online collaboration that completed important work on this problem. She shows the successes of Polymath in this problem and clear limitations of the platform. She discusses generally the difficulty of appropriately crediting researchers for their work and explains the slow and painstaking labour of mathematics. On page 29, she writes, "Some improvements would be dramatic, reflecting additional new ideas from the authors, while others would be smaller, highlighting the difficulty of making any progress at all." She brings the reader up the very edge of the known progress on this conjecture.

Overall, this book has a lot to offer. The narration through the text is clear, friendly, and easy to follow. The time you would spend reading this book belies the complex journey you have been guided through. If you are looking for an introduction to the world of Polymath; if you are looking for the story of the Twin Primes Conjecture; if you are looking to show you friends and family what your life as a mathematician is; if you would like a bit of asymptotic mathematics explained to you plainly; if you would like a summary of Waring's problem; or if you just have a couple of hours and are looking for a nice diversion, then you have found it.

1.10. Review by: Bookshelf.
Notices of the American Mathematical Society 66 (5) (2019), 754.

This short book chronicles some of the recent spectacular developments in the study of prime numbers. It revolves around the explosive events of 2013-2014, which were initiated by Yitang Zhang's unexpected proof that there are infinitely many pairs of primes that differ by at most 70,000,000. Subsequent refinements and generalisations evolved at a rapid pace, with dozens of authors (individually and collectively) contributing in a short amount of time.

Although this is a "popular science" book about prime numbers, a basic level of familiarity with calculus and infinite series is assumed. Later on, the Hardy-Littlewood circle method approach to Waring's Problem is discussed, and there is even a short section devoted to unpacking the meaning of the corresponding singular series - some non-trivial mathematics! However, Neale always tries to explain things in a down-to-earth and friendly manner.

Neale interweaves recent events with historical background and related results. The book features a creative structure that lends itself well to the subject matter. Apart from the introduction, the odd-numbered chapters have titles such as "June 2013" and chronicle the relevant number-theoretic events that occurred in a given month. The even-numbered chapters discuss related number-theoretic topics at a leisurely pace. Results ranging from Euclid's Theorem and the Prime Number Theorem to Szemerédi's Theorem and Lagrange's Four-Square Theorem are explored in a conversational tone and with many illustrations.

The reporting begins in earnest in Chapter 3 ("May 2013"), which introduces one of the main characters in the drama, Yitang Zhang. Later chapters discuss the contributions of other mathematicians, in particular James Maynard, Terence Tao, and the Polymath8 group. A great deal of attention is paid to the role played by Polymath projects, in which groups of mathematicians collaborate on difficult problems online and in the open via editable wikis. For instance, a good four pages of this slender volume are devoted to the question "Is Polymath the future?"

A curious undergraduate mathematics major should enjoy this book and learn a great deal. For mathematicians who do not specialise in number theory but who are curious about the flurry of recent activity in the field, this book provides an excellent entry point.

1.11. Review by: Owen Toller.
The Mathematical Gazette 102 (555) (2018), 561.

Some readers of this review will have heard Vicky Neale's lecture on the current status of the Twin Primes Conjecture (it was given, for example, at the 2017 IMO Presentation evening). Such readers will be familiar with the content of the book. After some historical background, the recent story starts in earnest with Yitang Zhang's announcement in 2013 of a proof that there are infinitely many primes that differ by no more than 70 million. This number is the "gap" of the present book's title. Zhang's methods were quickly taken up and refined, but the core of Neale's story is the effect of the Polymath (online sharing) project with its ability to share ideas instantly worldwide. Within a year, the number 70 million had been reduced to 246, and one branch of the project has shown that if the Elliott-Halberstam Conjecture is true then there is a proof that there are infinitely many primes that differ by at most 6. However, Neale gives reasons for thinking that current methods are not going to be capable of reducing the number as far as the desired 2.

The book is clearly and enthusiastically written and beautifully presented. As with so many present-day mathematics books intended for as wide a readership as possible, the starting point is utterly basic, telling us what prime numbers are and what are their basic properties and patterns, and giving Euclid's proof that there are infinitely many. Subsequently there are digressions on the Prime Number Theorem, Goldbach's Conjecture, the Hardy-Littlewood circle method and other related topics. There is very little actual mathematical content in the book, but the Further Reading section at the end gives web links for most of the material mentioned: these rapidly become highly technical, so there is little content 'in between'. Passing insights into the life of a working mathematician are enjoyable: Julia Robinson is quoted as giving the following description of her working life:

Monday - tried to prove theorem. Tuesday - tried to prove theorem. Wednesday - tried to prove theorem. Thursday - tried to prove theorem. Friday - theorem false.

I'm not quite sure who is going to get the most out of this book. I personally was glad to hear about the progress of work and intrigued by the success of Polymath, but I don't see myself having an urge to reread it. Perhaps its main justification is to promote Polymath as part of the future of mathematical research.

1.12. Review by: Dominic Klyve.
Mathematical Reviews MR3751356.

Popular math books can take several forms. Some introduce readers to a variety of interesting questions and ideas in the field, and others use the length of the book to introduce non-mathematicians to a single theorem or mathematical question in detail. Books have introduced readers to the Riemann Hypothesis, the Birch and Swinnerton-Dyer Conjecture, Gödel's Incompleteness Theorem, and the Poincaré Conjecture, among many others. In Closing the gap: the quest to understand prime numbers, Vicky Neale attempts the same feat concerning recent work by Yitang Zhang and others on bounded gaps between primes.

She succeeds admirably.

Beginning with "What is a prime?," Neale walks the reader through a series of deeper questions and answers in mathematics. Writing in a supremely captivating and engaging style, she is unafraid to introduce technical terms and ideas. The reader is introduced to almost primes, the weak Goldbach Conjecture, admissible sets, Szemerédi's Theorem, and much more. Using pictures of a frog and a grasshopper, she proves that 3, 5, and 7 form the only prime triple in language so gentle that even the most math-phobic reader is unlikely to realise that something is being proved until the reader understands the argument in full.

The story is told loosely historically, setting the stage for Zhang's paper with the work of Goldston, Pintz, and Yıldırım. The importance of Zhang's first result (that there are infinitely many pairs of consecutive primes with difference less than 70,000,000) is carefully explained. There is an excellent and readable description of the Polymath project, including a fairly detailed description of Polymath8, which worked to reduce the size of this gap.
...
It's worth noting that the book sometimes leaves the story of prime gaps to introduce other ideas from elementary number theory, including integers that are the sum of two (or more) squares, Waring's Problem, and in a particularly bold (and largely successful) endeavour, the Hardy-Littlewood Circle method.

The reader will be left not only with a solid understanding of the original question on gaps between primes, but also with a broader idea of what it means to do research in mathematics. The book will be enjoyed by both practicing mathematicians and amateurs who want to understand more about our beautiful field.

1.13. Review by: Adhemar Bultheel.
European Mathematical Society (20 February 2018).
https://euro-math-soc.eu/review/closing-gap

Vicky Neale has a degree in number theory and is now lecturer at the Balliol College, University of Oxford. She has a reputation to be an excellent communicator. This also shows in this marvellous booklet in which she gives a general introduction to the advances made in the period 2013-2014 in the quest for a solution of the twin prime conjecture. But she also explains how mathematicians think and collaborate.

The twin prime conjecture is claiming that there are infinitely many prime numbers whose difference is 2 like 3 and 5 or 11 and 13. It is easy to explain what prime numbers are, and it is even possible for anyone to understand Euclid's proof that there are infinitely many primes. The twin prime conjecture is however still one of the long standing open unsolved problems: easy to formulate and understand but hard to solve. Several attempts and generalisations were formulated. For example it can be claimed there are infinitely many primes whose difference is an even positive integer $N$. The twin prime conjecture corresponds to $N = 2$.

And then, in April 2013, Yitang Zhang could prove that the latter generalization holds for $N$ equal to 70.000.000, a major breakthrough. Within a year $N$ was reduced to 246. Neale presents the different steps that were obtained in this reduction almost month by month as a thrilling adventurous quest.

Scott Morrison and Terence Tao, two mathematical bloggers quickly used Zhang's approach to reduce the $N$ to 42 342 946. Tim Gowers, another active blogger proposed a massive collaboration and a Polymath project was set up by Tao. This Polymath platform is a totally new way of collaboration between mathematicians that Gowers had proposed back in 2009. The blog is fully in the open and anyone who wants to take part can dump some guesses or partial ideas on the website. The results are published under the author name D H J Polymath and the website shows who has collaborated in the discussion. Neale spends some pages to discuss this kind of collaboration and comments on its advantages and disadvantages. The project on the twin primes was numbered Polymanth8 and it turned out to be particularly successful. The problem that had been out for so long now progressed quickly because already in June 2013, $N$ was down to 12 006. In July they reached 4 689.

But while in August 2013 Tao is announces to write up the paper with the Polymath8 result, another twist of plot occurs. James Maynard posted a paper on arXiv in November 2013 in which the bound $N$ is brought down to 700. Independently Tao announced on his blog on exactly the same day that he used the same method to obtain a similar reduction. Using the new method the old Polymath8 was renamed as Polymath8a and a new Polymath8b project was started. This resulted in April 2014 in bringing the bound down to 246. The bound can even be 6, but that requires to assume that the Elliott–Halberstam conjecture (1968) holds, which is a claim about the distribution of primes in arithmetic progression.

But Neale in this booklet brings more than just the account of this thrilling quest to close the gap. She also succeeds in explaining parts of the proofs and she also tells about similar related problems from number theory. For example the Goldbach conjecture: "every even number greater than 2 is the sum of the squares of two primes", or its weak version: "every odd number greater than 5 is the sum of three primes", are two famous examples. The generation of Pythagorean triples is another well known example. But there are other, maybe less known ones like Szemerédi's theorem proved in 1976, which proves as a special case a conjecture by Erdös and Turán: "The prime numbers contain arbitrary long arithmetic progressions." The Waring problem: "Every integer can be written as a sum of 9 cubes, or more generally, as a sum of $s$ $k$th powers, (where $s$ depends on $k$)," which triggered Hardy and Littlewood to count the number of ways in which this is possible. They proved the Waring conjecture by showing that there is at least one way of doing that. Neale also explains admissible sets which were used in a theorem proved by Goldston, Pintz and Yıldırım which was essential in proving and improving Zhang's bound on $N$. And there is some introduction to the prime number theorem and the Riemann hypothesis.

Neale cleverly interlaces these diversions with the progress on the twin prime problem, which has the effect that some tension is built up and new developments pop up as a surprise. Some of the notions and terminology that popped up in the other problems turn out to be related or at least to be useful in the twin prime problem.

Neale realises that she is writing for a general audience and carefully explains all her concepts. However, I can imagine that some of the mathematics, like for example the formulas for the asymptotics in the Hardy-Littlewood theorem involving a triple sum, fractional powers, complex numbers, and gamma functions will be hard to swallow for some of her readers. On the other hand, many of her "proofs" rely on visual inspection of coloured tables, and she has witty ways of explaining some concepts. For example admissible sets are presented as punched cards, a strip with a sequence of holes at integer distances, and the idea is that when this is shifted along the line of equispaced integers, then at least one (or more) primes should be visible in the punched holes. Modulo arithmetic she explains using a hexagonal pencil with the numbers 1-6 printed on its sides at the top, then 7-12 next to it etc. If you put the 6 sides of the pencil next to each other, you get a table of numbers modulo 6, and the primes in this table show certain patterns. Some of the graphics are less functional, yet very nice. On page 6 where prime and composite numbers are explained, a prime number $p$ is represented with $p$ dots lying on a circle, while composite numbers are represented by groups of dots arranged in doublets, triangles, squares, etc. which gives a visually pleasing effect. Other graphics are referring to a pond with frogs, grasshoppers, ducks, reed and waterlily leaves. These may be less instructive, but they are still a nice interruption.

Vicky Neale has accomplished a great job, not only in bringing the mathematics and the mathematicians to a broad audience. We meet some of the great mathematicians of our time like Gowers and Tao, both winners of the Fields Medal. We are informed how mathematical progress works, how new ideas are born. This can be through novel communication channels such as the Polymath, but it can still be a loner who works on a completely different approach who comes up with a breakthrough. Sometimes we can gain from results slumbering in mathematical history, but often it relies on coincidences when someone connects two seemingly unrelated results. And when the time for an idea is ripe, then it happens that two mathematicians independently from each other come up with the same result simultaneously.

2.1. Vicky describes why she wrote it.
Plus (10 February 2021).
https://plus.maths.org/content/why-study-mathematics-0

Maths is a versatile subject, with different flavours that appeal to different people with different tastes. It equips graduates with skills that employers value. It's full of fascinating ideas and powerful applications, and the process of understanding a new mathematical concept or solving a problem using maths is enormously satisfying. Whatever your priorities - whether you're looking to help other people, to earn a lot, to explore a creative subject or to make a difference in society - maths has something to offer you. The study of maths is rewarding in and of itself, and it gives you lots of options for the future.

At school, there are standard requirements about what students have to learn: there's a national curriculum. This isn't the case at university. Universities have significant flexibility when it comes to how they organise degree programmes. There's a huge variety of courses - offering differences in mathematical content and emphasis, in teaching style and in assessment methods - that can lead you to a maths degree. This means that it's really important to research the available options before you apply, to find courses that'll suit you.

There's a lot of flexibility about which topics are covered in a maths degree. In the UK, the only topics specifically mentioned in the QAA benchmark statement (which sets out expectations for maths degrees) are calculus and linear algebra. Beyond that, it's up to individual universities to design appropriate degree courses.

Different people have different mathematical tastes. Some like nothing more than to get their hands on a large data set, to interrogate it in order to see what conclusions they can draw, and to consider the robustness of those conclusions. Others are motivated by a particular application and spend time exploring which mathematical tools can help to answer the questions they find exciting in that area. Others still are fascinated by the beautiful, fundamental questions in this subject (it's surprising just how many fundamental questions there are for which we still don't have complete answers) and devote themselves to curiosity-driven maths.

Within a maths degree, you're likely to have opportunities to experience all of these facets. However, degree programmes do differ in the emphasis they place on each aspect of maths, as well as in the number of options they offer to students, so it's really worth thinking about what style of course will suit you. Are you motivated by the use of maths in industry and other applications, and therefore interested in building a mathematical toolkit for that purpose, or would you relish delving into the background theory of how and why the tools work? You're not restricted to one or the other: many courses combine elements of both. But when you're researching courses, it can be helpful to consider the extent to which each might be described as "theory-based" or "practice-based", because viewing courses through that lens might help you to focus on the ones that'll suit you.

You might have heard people talking about there being a jump in difficulty from school maths to university maths. While there are certainly differences between the two, this isn't the same as the latter representing a massive step up in difficulty. Some of the differences might be to do with teaching approaches, with the style of questions you're being asked to tackle, or with the amount of independent study you're being expected to undertake. Others might be due to the fact that you're meeting new topics.

It's important to remember that universities know the move from secondary to higher education involves a transition to new material and new ways of working: degree programmes are designed to support students making this transition. Universities are familiar with the profiles of their incoming students and what they can expect them to know or to be able to do, and they tailor their programmes accordingly. Having said that, you'll still need to adapt to new ways of working, to be willing to persevere and to ask for support when you need it. There's no need to panic, though. Over the course of a three- or four-year maths degree, you'll become increasingly independent and develop your study skills and mathematical sophistication. You don't need to have done all this before the first day of your degree!

Deciding which subjects you'd like to pursue is just one of the many factors you'll need to consider when choosing a course of study. For instance, there are a variety of ways these subjects can be combined to form a degree that suits your interests. And there are many other factors to consider, such as how teaching and assessment are structured, what opportunities you'll have to interact with staff and your fellow students, and what the career possibilities are.

I wrote Why Study Mathematics? in the hope that it will help you to find out more about maths at university. In Part I (from which the extract above is taken), we explore the practicalities of a maths degree. What's involved in studying a maths degree? What topics might you study? What teaching methods and types of assessment might you encounter? How do you choose between the wide variety of maths degree courses on offer? What makes a good maths student? What careers are open to maths graduates?

In Part II, we look more closely at some of the topics you might study at university, providing a taste of the theoretical underpinnings of maths and offering insight into its diverse applications: in medicine and health care, in digital communication, in engineering, in tackling climate change, and more. One chapter concentrates on topics that are common to pretty much all maths degrees, illustrating the relevance of differential equations to modelling the spread of disease, connections between linear algebra and JPEG image compression, and the surprising use of non-real complex numbers to very real problems in aerodynamics. Other chapters illustrate further aspects of the mathematical sciences, from the use of operational research to facilitate kidney donation to the role of data science in genetics and retail, from compressed sensing to improve medical imaging to non-Euclidean geometry and uncountably infinite sets. Don't worry if you haven't heard of all these mathematical ideas - these are topics you might meet in a maths degree, not ones you need to understand before you study one!

If you're a student looking to make a decision about university, or you are supporting someone who is making that decision, I hope that this book will give you a clearer picture of why a maths degree is a good option for many people. And if you are the one who's thinking about embarking on this adventure, then I would like to wish you all the very best with your mathematical studies.

2.2. From the Publisher.

Are you considering studying mathematics at university, having fallen in love with the subject at school? Are you ready to develop a variety of practical skills that employers need? Are you keen to have a wide range of career options after you graduate? Studying any subject at degree level is an investment in the future that involves significant cost. Now more than ever, students and their parents need to weigh up the potential benefits of university courses. That's where the Why Study series comes in. This book, aimed at students, parents and teachers, explains in practical terms the range and scope of mathematics at university level and where it can lead in terms of careers or further study. It will enthuse the reader about the subject and answer the crucial questions that a college prospectus does not.

2.3. From the Introduction.

Maths is a versatile subject, with different flavours that appeal to different people with different tastes. It equips graduates with skills that employers value. It's full of fascinating ideas and powerful applications, and the process of understanding a new mathematical concept or solving a problem using maths is enormously satisfying. Whatever your priorities - whether you're looking to help other people, to earn a lot, to explore a creative subject or to make a difference in society - maths has something to offer you. The study of maths is rewarding in and of itself, and it gives you lots of options for the future.

There are many factors to consider when choosing what to study at university and where to study it (and this follows a careful decision about whether your next step is to attend university or to pursue another path). You might already know from your experience at school or college that you have a particular interest or strength in a certain subject, or you might be choosing between a few options. Or, as some subjects are available at degree level but not at school or college, you might be researching these as well as looking into how subjects that you've previously studied develop and change at university. (As you'll see in this book, maths at university includes a broad range of topics, many of which you won't have encountered at school.) In addition, you might have a particular career path in mind, or you might be looking to ensure that you keep your job options open.

This book will help you to find out more about maths at university. In Part I, we'll explore the practicalities of a maths degree. What's involved in studying a maths degree? What topics might you study? What teaching methods and types of assessment might you encounter? How do you choose between the wide variety of maths degree courses on offer? What makes a good maths student? What careers are open to maths graduates?

In Part II, we'll look more closely at some of the topics you might study at university, providing a taste of the theoretical underpinnings of maths and offering insight into its diverse applications: in medicine and health care, in digital communication, in engineering, in tackling climate change, and more. My choice of subjects is inevitably centred on mathematical nuggets that I find fascinating, and I've tried to pick examples of topics and applications that don't often come up at school or college in order to give you a glimpse of further horizons, rather than to remind you of things you already know well. Don't worry if you don't follow all the mathematical ideas - these are topics you might meet in a maths degree, not material you're supposed to understand already!

I've included some suggestions for further reading at the end of this book. These include books that you might want to read before starting a maths degree as well as websites to inspire you and to help with your decision making. There's a lot out there for aspiring mathematicians, from engaging YouTube videos by Numberphile to stimulating maths problems by NRICH and the UK Mathematics Trust, from case studies by Maths Careers to biographies of mathematicians throughout history by MacTutor. Perhaps you're reading this book as a student looking to make a decision about university. Or maybe you're reading it because you're supporting a family member making this decision, or because you're a teacher working with students trying to choose a university course. Whatever your situation, I hope that this book will give you a clearer picture of why a maths degree is a good option for many people.

If you're the one who's thinking about embarking on this adventure, then I would like to wish you all the very best with your mathematical studies.

Last-minute note

I'm putting the finishing touches to this book in the spring and summer of 2020, under 'lockdown' in the UK because of the Covid-19 pandemic. As you'll see in a few pages' time, Chapter 5 starts with a discussion of the use of maths to study the spread of disease. Believe it or not, this was always the plan for this chapter, long before Covid-19 emerged, and the first draft now looks uncomfortably prescient.

The vital role that mathematical modelling has to play both in predicting how this disease might unfold and in simulating the effect of different strategies is being featured on the national news, along with discussions about 'flattening the curve' and the reliability of statistical data on cases so far.

In addition to modelling the spread of this disease, maths graduates have been developing software and planning logistics for the National Health Service (NHS) and supermarkets; teaching and safeguarding young people; analysing and managing risk in all areas of business; keeping the finance sector open; and supporting the economy, all of which are making a powerful difference during this pandemic.

The world around me as I write is strange and unfamiliar. I'm not allowed to leave my house, except to take a daily walk or to buy groceries. My students are scattered around the world, most having left Oxford at the end of term, just before lockdown was introduced. They're doing an extraordinary job of continuing to study maths and make excellent progress under difficult circumstances. I teach them from my kitchen table, using technology to bridge geographical gaps that now seem very wide.

No doubt you're wondering what this means for the future of university study and specifically of maths degrees. While university planning is well underway for the 2020/21 academic year, we don't really know what this will look like yet. I'm certain that it will have to involve a mixture of online and face-to-face teaching and learning. I hope that, in the long term, we'll take the best aspects of the former and adapt them to improve our maths degrees in the future, using technology to enhance our lessons, but not to replace those activities best done in person. Maths lecturers from different universities are already coming together to exchange ideas about how best to organise teaching and to support students' learning in light of our changing circumstances.

While all university courses might look a little different over the next few years, maths degrees will still offer a stimulating, inspiring, satisfying and rewarding programme of study as well as a great platform on which to build a fulfilling career.

2.4. Review by: Colin Beveridge.
https://aperiodical.com/2020/11/review-why-study-mathematics-by-vicky-neale/

I can pinpoint the exact moment it became clear I would study maths at university. Parents' evening, year 12, I mentioned to my French teacher that I was thinking about a French degree. He looked at me as if I was stupid and said something like "you're good at French, but you're GOOD at maths. Besides, a French degree isn't much use." Alright, fine. Maths it is. He was spot-on. I never looked back.

For others, the decision about whether to study maths is less clear-cut. For those people, Why Study Mathematics is an extremely useful tool in making an informed decision. In the first part, Neale looks at the ins and outs of a maths degree - what you'll study, how courses differ, how students differ, and where it can take you; the second part takes a deeper look at the kinds of things a mathematician thinks about.

I think the early sections on the different flavours of maths degrees are especially valuable: up to A-level, Mathematics looks like a bit of a monolith and (almost) everyone covers (almost) the same material. Setting out that (almost) every maths degree will cover some linear algebra and some calculus but beyond that it's a free-for-all prepares students for the wide range of courses available, and for the sometimes baffling decisions that need to be made.

The one thing I felt was missing from the book was a section on reasons not to study mathematics. It's a tricky thing: we want mathematicians! We want everyone to know and love maths! Evangelising about its beauty and rewards is absolutely right - but at the same time, maths isn't for everyone, and picking the wrong subject, or doing it for the wrong reasons, can be a ticket to misery.

Why Study Mathematics is tremendously engaging and clearly-written (I enjoyed Neale's first book, Closing the Gap, for the same reasons). The author articulates her enthusiasm for the subject beautifully, and it takes an inordinate amount of work to make it look so effortless.

I say it would make an excellent gift for the mathematically-inclined teenager in your life, and an invaluable addition to any school library.

2.5. Review by: Chalkdust.
Chalkdust (4 March 2021).
https://chalkdustmagazine.com/blog/why-study-mathematics/

Vicky Neale is a mathematics lecturer at the University of Oxford. Why Study Mathematics? is a book that looks into studying mathematics at university.

Style
Why Study Mathematics? is a book aimed at 16-18 year-olds considering studying maths or related subjects at university. The book is split into two parts. The first part discusses what is involved in the study of university mathematics, why you should consider studying maths, and what people go on to do after a maths degree. The second part looks at a few university-level mathematical topics.

Control
This book gives a very informative view into what studying maths is like, and a very readable introduction to the mathematical topics it covers in the second half.

Damage
This book is a must-have for every school library, as it will be an invaluable source of information for any students considering a mathematical degree.

Aggression
If you already have a maths degree (especially if you got it reasonably recently), this book isn't for you. But if you don't have one and are considering studying for one, I could not recommend this book more highly. I really wish this book has been written a while ago, so I could've had it when I was 17 and looking at universities.

2.6. Review by: Finley Ilett.
London Mathematical Society Newsletter 498 (January 2022), 54-55.

Why Study Mathematics? is a fantastic guide to this most wonderful of subjects, giving readers the ultimate guided tour of maths, with many aspects to the book that make it ideal for any A-Level student who wants to know more about the paths available for those who wish to
study it beyond secondary school.

Right from the opening, there is a clear focus on maths being a 'versatile subject' - already a firm response to the question posed by the title. What's more, the whole book contains an excellent level of detail relating to all aspects of maths-based higher education, as well as a sample of the many career opportunities there are. The result is a book built around clarity, and one that assumes little prior knowledge - this particularly comes into play in the chapter that explains how degrees (and the many titles, qualification names and accreditations) work, because most students are probably not very familiar with much of the terminology (in spite of it being rather important when making decisions about university, as Neale wisely points out). All of this collectively means that this book is perfect for learning what maths as a subject has to offer, and is definitely worth reading even for those students who are only vaguely interested in post-secondary mathematical study.

One part of the book that is much more relevant than many people may think is the often-overlooked content of maths degrees themselves. Some may assume that all degree courses are too similar for this to be of any importance, but that viewpoint is firmly opposed here, ensuring that readers are encouraged to actually look more closely at exactly what they are applying for. While the reality of this may be surprising for students who are used to the GCSE and A-Level system, with relatively little variation between a few exam boards (especially in maths and sciences), the way course content at university works is clearly explained - and a useful point of reference for students is the in-depth section on the four general areas of mathematics that are almost guaranteed to turn up in a degree (and the rough breakdown of each is the perfect way to make readers aware of the range of topics they are likely to encounter). This is important when many students spend too great a proportion of time just looking at post-university career prospects, and then risk regretting their choices at an earlier stage (perhaps by not having a clue what to do when given a choice between modules while at university, for example!).

Something that Neale never forgets is the importance of putting the maths that readers may study at university into a real-world context. There is nowhere near enough focus on this for secondary school maths, so it is a great asset to the book that this information is included throughout - it gives the subject relevance, in the eyes of the reader, thus providing a level of inspiration to those who had perhaps not considered the usefulness of the maths they study as deeply as they probably ought to have done - the section on applied maths within degrees being especially useful in this regard. However, this book is not solely to do with the study of the subject - the second part, aptly titled 'Maths In Action' looks instead at the uses of maths. These include some demonstrations and proofs that serve to remind readers of some of the beautiful qualities of the subject, along with some intriguing applications that few readers will have considered before. This offers a refreshing change from the exam-focused, theory-based type of maths that students are more likely to have already encountered. Leading on from this, emphasis is placed on the way that although some students know what they want to do, other may not - this is especially important for the many readers tired of all the university and careers advice excessively targeted at people who have already chosen their path, when they, being as-yet-undecided, cannot benefit as much from this advice themselves. It means that they can stop worrying about this, with Neale reminding the reader that in fact, many maths students end up going into a field they didn't even know existed before studying it at university. This might well also be something of an eye-opener to those who thought that they had already encountered all the mathematical fields there are to study.

One other element of this book in particular that must be recognised is the chapter on 'Further Applications Of Mathematics'. Ultimately, some will see this as the most inspiring part of the book, because this is where the importance of maths in our everyday lives is showcased, and the subtext of this chapter is the most valuable message the book has to offer - that of the sheer diversity of applications of maths as a general, all-encompassing whole. The examples of topics - ranging from medicine, computer science and cyber-security, to retail and climate science (as well as the all-important field within any form of maths of studying and reducing errors!) - really show the true scope of opportunity for someone who chooses to go on to study maths, or one of its related sub-groups, at university or beyond.

So overall, this book will encourage students to look further into the many possibilities maths makes available to those who choose to study it, and it can therefore be thoroughly recommended to anyone who is thinking of studying maths after their A-Levels - informative, engaging and inspiring, it serves to help readers make their post-secondary decisions from a much better vantage point - and one which they might not otherwise have imagined existed at all!

2.7. Praise for Why Study Mathematics? by: Nick Higham.
https://londonpublishingpartnership.co.uk/why-study-mathematics/

Why Study Mathematics? is an insightful guide for anyone consider­ing studying mathematics at university. It explains the sort of maths you can expect to find and how it will be taught, and highlights the wide variety of career options that a maths degree opens up. It also includes important examples of where maths is used in the real world. I recommend it to all prospective maths students and their parents.

2.8. Praise for Why Study Mathematics? by: Kerry Burnham.
https://londonpublishingpartnership.co.uk/why-study-mathematics/

The mystery of a mathematics degree - what it is, where it leads and why it's useful - is unlocked in this easy read. Detailed, accessible and broad ranging, Vicky Neale refines the  complex and varying nature of high-level mathematics into an understandable, useful and relatable form. An ideal guide for A level maths students when pondering their next steps.

2.9. Praise for Why Study Mathematics? by: James Grime.
https://londonpublishingpartnership.co.uk/why-study-mathematics/

An essential read for anyone considering studying mathematics at university. Vicky Neale takes you through what to expect in your studies, and explains the practical uses and beauty of a mathematics degree.

2.10. Praise for Why Study Mathematics? by: Rebecca Blazewicz.
https://londonpublishingpartnership.co.uk/why-study-mathematics/

Why Study Mathematics? explores in depth the various options that a maths degree has to offer as well as providing expert guidance on what to expect from a maths degree. Neale addresses all the questions that an enthusiastic mathematician considering a maths degree might have. She refutes the myth that studying maths beyond school/college is restrictive or overwhelmingly challenging. Neale's book is both informative and engaging: it made me want to study a mathematics degree all over again, or at least revisit certain topics that are discussed. I highly recommend this book to any student or maths enthusiast wanting to study mathematics at university. Being a mathematics teacher myself, this book is definitely a resource I shall be directing A level students to.

2.11. Praise for Why Study Mathematics? by: Kevin Houston.
https://londonpublishingpartnership.co.uk/why-study-mathematics/

Another great book from Vicky Neale. An extremely useful guide for students (and their advisors!) on studying mathematics at university.

2.12. Praise for Why Study Mathematics? by: Jason Hudson.
https://londonpublishingpartnership.co.uk/why-study-mathematics/

Why Study Mathematics? is superb! It will be the very first book that I will recommend to students who wish to study mathematics at university, as it provides a very easy-to-follow guide to the many fascinating areas and branches of mathematics that could be studied during a mathematics degree and to the ever-growing number of careers that require one. Neale provides compelling insight not just into how useful mathematics is in today's modern world, but also into what an essential and integral element of everyday life the subject is. The book provides wonderful examples of how mathematics is used in today's ever-evolving world, from its involvement in online shopping and JPEG compression to the way it's used to analyse climate change. The beauty of mathematics is also revealed through Escher's paintings, many elegant equations and the intriguing world of infinities, all of which should inspire students to go and find out more.

2.13. Praise for Why Study Mathematics? by: Jamie Frost.
https://londonpublishingpartnership.co.uk/why-study-mathematics/

This book is impressively thorough in its treatment of the factors that a student might consider, both in navigating the intimidating variation of choice in mathematical courses and in exploring the essence of what maths is and the subject's implications in a wide variety of industrial and research settings. Vicky Neale is clearly someone who not only has a deep knowledge of maths, from SIR models to JPEG compression, but also has extensive experience in helping students both in the lead up to university and beyond. I highly recommend it to any student, including both those who are just exploring their university options and those who are already set on a mathematics degree and want to explore the short-term and longer-term implications of their choice.

2.14. Praise for Why Study Mathematics? by: Sophie Carr.
https://londonpublishingpartnership.co.uk/why-study-mathematics/

Why Study Mathematics? is awesome! It's absolutely the book I wish I'd had as a sixth former, when, truth be told, you don't really know what studying a maths degree is. It's beautifully written and really engaging, and it represents a great starting point from which to explore all the different maths courses that are available. Crucially, it helps you formulate the questions you need to ask the tutors on those different courses to help you work out where will be the best place for you to study.

2.15. Praise for Why Study Mathematics? by: David Ireland.
https://londonpublishingpartnership.co.uk/why-study-mathematics/

This book is essential reading for A level students who are thinking about a maths degree. The reader is given an accurate picture of what to expect on a maths degree course and what employment opportunities may follow. The descriptions and examples are perfectly chosen and do a great job of showing why the subject is so interesting and enjoyable.