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.

I believe the best unconditional bound is currently 246, due to Polymath/Maynard, not 600.