In September 2017, John Duncan, Michael Mertens, and Ken Ono published a paper announcing a connection between the Pariah group known as the O’Nan group (after Michael O’Nan, who discovered it in 1976) and another modular form. Like Monstrous Moonshine, the new connection is through an infinite-dimensional shape which breaks up into finite-dimensional pieces. Also like Monstrous Moonshine, the modular form in question has a deep connection with elliptic curves. In this case, however, the connection is more subtle and leads through yet another set of important mathematical objects: the quadratic fields.

It’s also possible to have fields of things that aren’t numbers, which are useful in lots of other situations — see Section 4.5 of *The Mathematics of Secrets* for a cryptographic example.

What mathematicians call a field is a set of objects which are closed under addition, subtraction, multiplication, and division (except division by zero). The rational numbers form a field, and so do the real numbers and the complex numbers. The integers don’t form a field because they aren’t closed under division, and the positive real numbers don’t form a field because they aren’t closed under subtraction.

A common way to make a new field is to take a known field and enlarge it a bit. For example, if you start with the real numbers and enlarge them by including the number *i* (the square root of -1), then you also have to include all of the imaginary numbers, which are multiples of *i*, and then all of the numbers which are real numbers plus imaginary numbers, which gets you the complex numbers. Or you could start with the rational numbers, include the square root of 2, and then you have to include the numbers that are rational multiples of the square root of 2, and then the numbers which are rational numbers plus the multiples of the square root of 2. Then you get to stop, because if you multiply two of those numbers you get

$$(a+b\sqrt{2})(c+d\sqrt{2}) = (ac + 2bd) + (ad+bc)\sqrt{2}$$

which is another number of the same form. Likewise, if you divide two numbers of this form, you can rationalize the denominator and get another number of the same form. We call the resulting field the rational numbers “adjoined with” the square root of 2. Fields which are obtained by starting with the rational numbers and adjoining the square root of a rational number (positive or negative) are called quadratic fields.

Identifying a quadratic field is almost, but not quite, as easy as identifying the square root you are adjoining. For instance, consider adjoining the square root of 8. The square root of 8 is twice the square root of 2, so if you adjoin the square root of 2 you get the square root of 8 for free. And since you can also divide by 2, if you adjoin the square root of 8 you get the square root of 2 for free. So these two square roots give you the same field.

This is the same *b*^{2} – 4*ac* as in the quadratic formula.

For technical reasons, a quadratic field is identified by taking all of the integers whose square roots would give you that field, and picking out the integer *D* with the smallest absolute value that can be written in the form *b*^{2} – 4*ac* for integers *a*, *b*, and *c*. This number *D* is called the fundamental discriminant of the field. So, for example, 8 is the fundamental discriminant of the quadratic field we’ve been talking about, not 2, because 8 = 4^{2} – 4(2)(1), but 2 can’t be written in that form.

After addition, subtraction, multiplication, and division, one of the really important things you can do with rational numbers is factor their numerators and denominators into primes. In fact, you can do it uniquely, aside from the order of the factors. If you have number in a quadratic field, you can still factor it into primes, but the primes might not be unique. For example, in the rational numbers adjoined with the square root of negative 5 we have

$$6 = 2 \times 3 = (1+\sqrt{-5})(1-\sqrt{-5})$$

where 2, 5, $1+\sqrt{-5}$, and $1-\sqrt{-5}$ are all primes. (You’ll have to trust me on that last part, since it’s not always obvious which numbers in a quadratic field are prime.)

We express this by saying the rational numbers have unique factorization, but not all quadratic fields do. The question of which quadratic fields have unique factorization is an important open problem in general. For negative fundamental discriminants, we know that *D *= -1, -2, -3, -7, -11, -19, -43, -67, -163 are the only such quadratic fields; an equivalent form of this was conjectured by Gauss but fully acceptable proofs were not given until 1966 by Alan Baker and 1967 by Harold Stark. For positive fundamental discriminants, Gauss conjectured that there were infinitely many quadratic fields with unique factorization but this is still unproved.

Furthermore, Gauss identified a number, called the class number, which in some sense measures how far from unique factorization a field is. If the class number is 1, the field has unique factorization, otherwise not. The rational numbers adjoined with the square root of negative 5 (*D *= -20) have class number 2, and therefore do not have unique factorization. Gauss also conjectured that the class number of a quadratic field went to infinity as its discriminant went to negative infinity; this was proved by Hans Heilbronn in 1934.

What about Moonshine? Duncan, Mertens, and Ono proved that the O’Nan group was associated with the modular form

*F*(*z*) = *e *^{-8 π i z} + 2 + 26752 *e*^{ 6 π i z} + 143376 *e*^{ 8 π i z } + 8288256 *e*^{ 14 π i z } + …

which has the property that the coefficient of *e*^{ 2 |D| π i z }_{ }is related to the class number of the field with fundamental discriminant *D *< 0. Furthermore, looking at elements of the O’Nan group sometimes gives us very specific relationships between the coefficients and the class number.

Mathematicians say a symmetry that gets you back where you started after you do it *n *times is a “symmetry of order *n*”.

For example, the O’Nan group includes a symmetry which is like a 180 degree rotation, in that if you do it twice you get back to where you started. Using that symmetry, Duncan, Mertens, and Ono showed that for even *D *< -8, 16 always divides *a*(*D*)+24*h*(*D*), where *a*(*D*) is the coefficient of *e*^{ 2 |D| π i z }_{ }and *h*(*D*) is the class number of the field with fundamental discriminant *D*. For the example *D *= -20 from above, *a*(*D*) = 798588584512 and *h*(*D*) = 2, and 16 does in fact divide 798588584512 + 48. Similarly, other elements of the O’Nan group show that 9 always divides *a*(*D*)+24*h*(*D*) if *D* = 3*k*+2 for some integer *k* and that 5 and 7 always divide *a*(*D*)+24*h*(*D*) under other similar conditions on *. *And 11 and 19 divide *a*(*D*)+24*h*(*D*) under (much) more complicated conditions related to points on an elliptic curve associated with each *D*, which brings us back nicely to the connection between Moonshine and elliptic curves.

Monstrous Moonshine showed that the Monster, and therefore the Happy Family, was related to modular forms and elliptic curves, as well as string theory. O’Nan Moonshine brings in two more sporadic groups, the O’Nan group and its subgroup the “first Janko group”. It also connects the sporadic groups not just to modular forms and elliptic curves, but also to quadratic fields, primes, and class numbers. Furthermore, the modular form used in Monstrous Moonshine is “weight 0”, meaning that *k* = 0 in the definition of a modular form given in Part II. That ties this modular form very closely to elliptic curves.

Umbral Moonshine also uses weight 3/2 modular forms.

The modular form in O’Nan Moonshine is “weight 3/2”. Weight 3/2 modular forms are less closely tied to elliptic curves, but are tied to yet more ideas in mathematical physics, like higher-dimensional generalizations of strings called “branes” and functions that might count the number of states that a black hole can be in.

That still leaves four more pariah groups, and the smart money predicts that Moonshine connections will be found for them, too. But will they come from weight 0 modular forms, weight 3/2 modular forms, or yet another type of modular form with yet more connections? Stay tuned! Maybe someday soon there will be a Part IV.

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This book is written to answer the question ‘when would you ever use maths in everyday life?’ It therefore focuses on applied maths, across a surprisingly wide breadth of applications. The book is organised into sections such as ‘the human world’, ‘the natural world’, ‘getting around’ and ‘the everyday’. Within each section there are approximately ten topics, for which the maths behind some facet of ‘everyday life’ is explained, with cheerful colour graphics and not shying away from using an equation where necessary.

As I obtained the book whilst on a visit to Bletchley Park, my attention is first drawn to the ‘technology’ section and the topic of Bitcoin, an application of cryptography. Bitcoin is a so-called *cryptocurrency* and world payment system, albeit one without a central bank, or similar, to regulate and guarantee it. To my surprise Bitcoin was only coined (sorry!) in 2008; the book describes it and the ‘blockchain’, along with the math behind Bitcoin. This rather pleasingly uses a graph that an A-Level student could visualise, \( y^2 = x^3 + 7\), alongside a delightfully named ‘hash function’, and requires ideas such as a line crossing a curve – again familiar to an A-Level, and probably a GCSE, mathematician – and introduces modular arithmetic, which probably is not. Bitcoin is a great example of a topic talked about in mainstream media which is totally dependent on mathematics.

As a Brit who also loves Barcelona, my attention was also drawn to the section on ‘the everyday’ and the topic of architecture. In this section, London’s St Mary Axe and Barcelona’s Sagrada Familia are used as examples of mathematics influencing architecture. The book uses St Mary Axe to illustrate some of the ways mathematics can make a structure cost-effective and reduce turbulence. The Sagrada Familia is used to introduce catenary and parabolic curves, using the hyperbolic cosine function and a quadratic equation respectively.

In the ‘human world’ section there is a fascinating topic called ‘cheating’ which looks at the mathematics behind detecting plagiarism, introducing techniques such as principal component analysis and stylometry. Stylometry is illustrated by comparing frequency analysis of words in both Shakespeare and Marlowe’s works. Later in the topic, we’re introduced to something that astounded me, Benford’s Law, which states that in a list of measurements in which the largest is at least 100 times bigger than the smallest (the size of the lakes in Michigan are used in the book), the first digit of the components of the list will be distributed according to the formula

\[\Pr(n) \sim \log\left(\frac{n+1}{n}\right) \]

This leads one to expect approximately 30% of such a list to begin with a 1, and only approximately 5% with a 9, which is contrary to the even spread that one imagines. Benford’s law has been used in forensic analysis, and one can only imagine its utility in detecting fraud and in areas such as ‘big data’. This law, astonishingly, is independent of the unit of measurement used, and the insight in the book leads one to find out more.

Throughout this book the illustrations are clear and are used to explain a range of applications of mathematics. Despite not requiring the reader to be aware of, and still less understand, the mathematical concepts described, I am confident that even knowledgeable mathematicians will learn something, and those who teach, tutor or merely communicate mathematics will find it invaluable in answering the question ‘when would you ever use maths in everyday life?’

The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>The American Math Society’s Joint Policy Board for Mathematics has announced the winners of its 2018 Communication award. This year’s winners are internet maths wizard/YouTube star Vi Hart, and Aperiodipal and Stand-up Mathematician Matt Parker.

Both produce brilliant, enjoyable and illuminating mathematical videos (Vi Hart, Matt Parker), as well as numerous other projects – Vi Hart has worked with Khan Academy, produced online interactive mathematical stories, and done some super work on hyperbolic/4D virtual reality, while Parker performs with science comedy team Festival of the Spoken Nerd, has started the MathsJam pub maths movement, has written a popular maths book, and appears regularly on TV and radio.

The award includes a prize of $2,000, and aims to encourage high-quality communicators of mathematics. We think they’ve made a good choice!

News post on the AMS website

About the AMS JPBM Communication prize

The IMA/LMS Zeeman Medal has been awarded every two years since 2008, to an individual, to “recognise and acknowledge the contributions of mathematicians involved in promoting mathematics to the public and engaging with the public in mathematics in the UK, and demonstrate that such activities are valued by the societies and the mathematical community at large and are a part of a mathematician’s roles and responsibilities”. The nomination process is now open for 2018, and details of eligibility and how to make a nomination are at the link below.

]]>**What’s The Calculus Story all about?**

It’s an introduction to calculus for an unusually wide readership, mainly through the story of how the subject developed. And this turns out to be something of an adventure, largely because of the way infinity comes in at almost every twist and turn.

**How is the book different to previous ones you’ve written?**

My previous book, 1089 and All That, was a fairly light-hearted look at maths in general, and became something of a bestseller, with 11 foreign translations. The new book takes the subject a bit further, and is, if anything, even more ambitious, because it tries to explain not only what calculus is, but how to actually start doing it.

Calculus is all about the rate at which things change, and this is how we often get to understand, through the laws of physics, how the world really works. But the subject contains many results, too, that can be enjoyed purely for their own sake, usually because they are surprising in some way.

**What do you hope the book will achieve?**

It is always easier to embark on calculus if you have some idea from the very beginning of the subject as a whole – some ‘big picture’, if you like. Without that, you can easily get bogged down in comparatively minor detail, lose direction, and – even worse – lose interest.

My main hope, then, is that The Calculus Story provides that big picture, and in an unusually readable way.

And if anyone were to read the book as preparation for a university or college course, they would – in my view – really hit the ground running.

**Why did you choose to write the book now?**

I didn’t. The book has been gently brewing, in a way, since 1962, when I first met calculus, at the age of 16. For once you’ve met it, your mathematical life is never quite the same again – calculus just opens so many new doors.

**What’s the most interesting fact you learned while writing the book?**

If you drill a hole of given depth straight through the centre of a sphere, the amount of material left over is independent of the radius of the sphere!

**Have you ever used calculus in ‘everyday life’?**

No. Calculus underpins much of modern life, but in a rather hidden way, through the laws of physics, chemistry, biology and economics. It tends to be ‘under the bonnet’, so to speak.

The most likely way that calculus might be used explicitly in a truly ‘everyday’ context is to solve some optimisation problem.

**Who do you think the book would best suit as a Christmas present?**

You’d better ask the New Scientist. They have just selected it as one of their choices for Christmas, claiming that (a) it will fit in a stocking and (b) it has ‘something for everyone’!

The Calculus Story, on Oxford University Press

The Calculus Story, on Amazon

Every one to two weeks a new chapter of the six-chapter story is released, and each chapter has a new cryptographic puzzle to solve. Teams consisting of up to four people can win prizes for being the first to solve each puzzle, and also for being randomly picked from all correct entries for each puzzle.

The Alan Turing Cryptography Competition begins on **Monday 15th January** 2018, with MathsBombe starting on **Wednesday 10th January 2018**. For more information and to enter, visit the Cryptography Competition website or MathsBombe website.

The day after last week’s budget, I logged onto the BBC News website and clicked on their budget calculator to find out if I was a winner or a loser. The questions are pretty simple: first off, it asks how much you drink, smoke and drive, and then it asks how much you earn, plus a few bits and bobs to cover technicalities. Then, it spits out an answer: did Phil leave you feeling flush, was it more of a hammering at the hands of Hammond? I came away £8 a month better off…and significantly angrier than I expected.

The first problem is that I don’t drink much, smoke or own a car. I know that makes me a bit weird, but still, budgets clearly aren’t aimed at me. Also, way to go successive governments, enshrining these items which are bad for our health and the environment as regular budget giveaways. But there’s a much bigger issue with this calculator: what it misses out.

Let’s ignore inflation. Ignore productivity. Ignore economic growth. Ignore any kind of context. Ignore other kinds of tax as though income taxes and booze duty are the only ways the government raises revenue. Perpetuate a ludicrous idea that personal wealth exists in an economic and societal vacuum—a narrow, individualistic notion of money where all that matters is the tax on your pay and your petrol.

Let’s talk about income. Productivity is a measure of how much money we make per hour worked, taken nationwide. The UK is currently experiencing a productivity crisis—for the last decade, productivity hasn’t improved, for the first time in *literally centuries*. Whether the budget includes policies which might attempt to revert productivity growth to historical norms will have a far, far larger effect on your long-term wages than a minor alteration to income tax brackets. Yet this ‘budget calculator’ makes no mention of it and, by omission, implicitly minimises hugely important issues like this.

Let’s talk about other taxes. Not only does tax affect you directly, but it shapes incentives around the whole economy, discouraging things which are taxed heavily and encouraging things which aren’t, promoting or sidelining economically beneficial and detrimental activities which will have huge effects on everyone’s future wealth. These not-so-subtle effects will have a far larger influence on the price of a pint than any alcohol duty alteration a chancellor would dare implement.

The calculator ignores benefits in kind—the NHS, roads, schools, scientific research—which the government provides. It ignores the fact that tax savings for you mean less money for them. The foregone cure for cancer and the pothole which gives your neighbour a hefty bill at the mechanic aren’t included in the calculator.

This little web form fails on its own narrow terms by ignoring wider economic issues which affect your wages and buying power, and fails to provide any meaningful analysis of the effects on you and society by ignoring what taxes pay for.

But its biggest sin is to give the illusion of understanding. Making sense of the economy requires numbers—lots of them. But faux-neutral calculators like this are part of a media circus around budgets which incentivises politicians to offer easily-understood trinkets rather than fix actual problems.

]]>*The new live DVD from science comedy trio Festival of the Spoken Nerd, Just for Graphs, is out now, and we’ve been sent a copy to review. We got together a pile of appropriately nerdy science fans to watch (left), and here’s what we thought.*

The latest Festival of the Spoken Nerd DVD/download is from the Nerds’ 2015 show Just for Graphs – it toured the UK in late 2015, and had a hugely successful Edinburgh Fringe run in August 2015. The show is themed around graphs, plots, charts and diagrams – as mathematicians, we were sad to see they’re not entirely using graphs made from nodes and arcs (although a few of those do make it in the show!)

As well as plenty of classic diagrams (Venn, Euler and otherwise) there’s also plots – Steve Mould plots the birth of his child, while Matt Parker plots a function that plots itself – and Helen Arney brings musical interludes and live demos, including an electrifying demonstration of how the graph of electrical voltage in a speaker cable can be transmitted through not just wires but people, and at one point a large section of the audience.

The show also contains some nostalgia for retro technology (which raised some cheers from us), and an interesting new way of plotting a graph of the pressure inside a tube of gas – by setting the gas on fire, of course.

While watching it on a screen doesn’t quite have the same ambience as seeing it live in a theatre, you still get a sense of how the live audience experiences it and the show is full of visual spectacles, which do come across well on screen. I was part of the production team for some of the tour shows and Fringe run, and it was just like being there for real (only smaller and more pixelated).

The show is full of science references and deliciously geeky jokes, and without spoiling too many of the punchlines/conclusions, if you’re into maths or science and want to be entertained, it’s a graph a minute.

*Just for Graphs is available on DVD, as a digital download and for some reason on VHS.*

I ended Part I with the observation that the Monster group was connected with the symmetries of a group sitting in 196883-dimensional space, whereas the number 196884 appeared as part of a function used in number theory, the study of the properties of whole numbers. In particular, a mathematician named John McKay noticed the number as one of the coefficients of a modular form. Modular forms also exhibit a type of symmetry, namely if *F* is a modular form then there is some number *k* for which

There are also some conditions as the real part of *z *goes to infinity.

$$F\left(\frac{az+b}{cz+d}\right)= \frac{1}{(cz+d)^k} F(z)$$

for every set of whole numbers *a*, *b*, *c*, and *d* such that *ad*–*bc*=1.

In 1954, Martin Eichler was studying modular forms and observing patterns in their coefficients. For example, take the modular form

I don’t know whether Eichler actually looked at this particular form, but he definitely looked at similar ones.

$$F(e^{2 π i z}) = e^{2 π i z} \prod_{n=1}^\infty\left[(1- e^{2 π i z n})^2 (1- e^{2 π i z 11 n})^2\right]$$

The coefficients of this modular form seem to be related to the number of whole number solutions of the equation

*y*^{2 }= *x*^{3} – 4 *x*^{2 }+ 16

This equation is an example of what is known as an elliptic curve, which is a curve given by an equation of the form

Elliptic curves are not ellipses!

*y*^{2 }= *x*^{3} + *ax*^{2 }+ *bx *+* c*

Elliptic curves have one line of symmetry, two open ends, and either one or two pieces, as shown in **Figures 1 and 2**. They are called elliptic curves because the equations came up in the seventeenth century when mathematicians started studying the arc length of an ellipse. These curves are considered the next most complicated type of curve after lines and conic sections, both of which have been understood pretty well since at least the ancient Greeks. They are useful for a lot of things, including cryptography, as I describe in Section 8.3 of *The Mathematics of Secrets.*

In the late 1950’s it was conjectured that every elliptic curve was related to a modular form in the way that the example above is. Proving this “Modularity Conjecture” took on more urgency in the 1980’s, when it was shown that showing the conjecture was true would also prove Fermat’s famous Last Theorem. In 1995 Andrew Wiles, with help from Richard Taylor, proved enough of the Modularity Conjecture to show that Fermat’s Last Theorem was true, and the rest of the Modularity Conjecture was filled in over the next six years by Taylor and several of his associates.

Modular forms are also related to other shapes besides elliptic curves, and in the 1970’s John McKay and John Thompson became convinced that the modular form

*J*(*z*) = *e *^{-2 π i z} + 196884 *e*^{ 2 π i z} + 21493760 *e*^{ 4 π i z } + 864299970 *e*^{ 6 π i z } + …

was related to the Monster. Not only was 196884 equal to 196883 + 1, but 21493760 was equal to 21296876 + 196883 + 1, and 21296876 was also a number that came up in the study of the Monster. Thompson suggested that there should be a natural way of associating the Monster with an infinite-dimensional shape, where the infinite-dimensional shape broke up into finite-dimensional pieces with each piece having a dimension corresponding to one of the coefficients of *J*(*z*). This shape was (later) given the name *V*♮, using the natural sign from musical notation in a typically mathematical pun.

Terry Gannon points out that there is also a hint that the conjectures “distill information illegally” from the Monster.

John Conway and Simon Norton formulated some guesses about the exact connection between the Monster and *V*♮, and gave them the name “Moonshine Conjectures” to reflect their speculative and rather unlikely-seeming nature. A plausible candidate for *V*♮ was constructed in the 1980’s and Richard Borcherds proved in 1992 that the candidate satisfied the Moonshine Conjectures. This work was specifically cited when Borcherds was awarded the Fields medal in 1998.

The construction of *V*♮ turned out also to have a close connection with mathematical physics. The reconciliation of gravity with quantum mechanics is one of the central problems of modern physics, and most physicists think that string theory is likely to be key to this resolution. In string theory, the objects we traditionally think of as particles, like electrons and quarks, are really tiny strings curled up in many dimensions, most of which are two small for us to see. An important question about this theory is exactly what shape these dimensions curl into. One possibility is a 24-dimensional shape where the possible configurations of strings in the shape are described by *V*♮. However, there are many other possible shapes and it is not known how to determine which one really corresponds to our world.

The “modular” in “Modular Moonshine” is related to the one in “modular form” because they are both related to modular arithmetic, although the chain of connections is kind of long.

Since Borcherds’ proof, many variations of the original “Monstrous Moonshine” have been explored. The other members of the Happy Family can be shown to have Moonshine relationships similar to those of the Monster. “Modular Moonshine” says that certain elements of the Monster group should have their own infinite dimensional shapes, related to but not the same as *V*♮. “Mathieu Moonshine” shows that one particular group in the Happy Family has its own shape, entirely different from *V*♮, and “Umbral Moonshine” extends this to 23 other related groups which are not simple groups. But the Pariah groups remained outsiders, rejected by both the Happy Family and by Moonshine — until September 2017.