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.

The first will reward a well-made, delicious item; the second will reward the item which has been decorated the most beautifully and looks most like what it’s supposed to be; and the third will reward the most ingenious mathematical theming.

You can view the entries from this year on the MathsJam website.

The other regular competition is the Competition Competition. This invites attendees to submit a competition, which other attendees can enter. There are some rules, including minimum font size, paper size and maximum value of prize. To be clear, the rules say “any type of competition is permitted as long as it can be judged by the setter (or a winner randomly chosen from the correct entries)”.

Prizes are awarded for best competition, popular vote winner (the competition with the most entries) and “best attempt at circumvention of the rules while still strictly sticking to the rules”. Seeing people attempt the latter is quite delightful.

Chatting to people at MathsJam this year, I was reminded of my entry into the Competition Competition when it first ran in 2014. I invited entrants to write down an integer between 0 and 100, then I said that I would run a Shapiro-Wilk test of normality on the numbers people had written down. This tests the null hypothesis that the data come from a normally distributed population. The competitive element of the competition asked people to guess the p-value obtained from that test.

While we were chatting about this, The Aperiodical’s own Paul Taylor asked me what p-value came out as. I couldn’t remember, but I’ve looked it up. The numbers entered were as follows:

Number entered |
Number of people entering it |
---|---|

2 | 1 |

6 | 1 |

12 | 1 |

16 | 1 |

50 | 1 |

71 | 1 |

72 | 2 |

73 | 2 |

86 | 2 |

97 | 2 |

99 | 7 |

In a sample of 21 integers from 1-99 where only four numbers are below 50 and one third are precisely 99, it may not surprise you to learn that the statistical test gave strong evidence to reject the hypothesis that these data were from a normally distributed population. The p-value (from R) was 0.0002211865 and the winner was Francis Hunt, who guessed 0.0001.

You can find out about the MathsJam competitions and other side activities that take place on the MathsJam Extras page.

]]>New York maths museum MoMath is looking for support to bring the UK’s Maths Inspiration show, featuring Aperiodipals Matt Parker and Rob Eastaway, to the US for a Broadway theatre run. They’re looking for a commitment from a minimum number of schools to come and see the show next summer in order to secure funding. They say:

“The interactive daytime show, aimed at 14- to 17-year-olds, features a variety of top math popularizers from the UK demonstrating that there’s more to math than just taking exams. There will be plenty of humor mixed in with serious math content — your students won’t want to miss this!

We are looking at the first week in June as a possibility for this exciting event, which will take place at around 1 pm on a school day. Tickets are anticipated to cost about $30 per student. Please let us know if you would be interested in attending an event like this by registering at broadway.momath.org. No payment is required to register your interest now.”

So, if you’re a school in the US, and would be interested… do that I guess!

This is from back in August, but we spotted this post on the Royal Society’s website analysing this year’s A-level results. While they generally seem pleased with the uptake in maths and science subjects, they comment:

“…it still remains of concern that there is a regional disparity in the uptake of mathematics. Only around a fifth of pupils in England who choose to study mathematics or further mathematics are located in the north of the country, which is below average for the proportion of the total number of entries in the north.”

They’re blaming this on the most recent funding model for A-levels, and quote The Smith Review in which the importance of maths qualifications is emphasised.

The Royal Society comments on A Level results 2017, at the Royal Society Website

via Paul Glendinning on Twitter

I found this online colouring book called Illustrating Group Theory – it looks to be a nice introduction to symmetry and groups, and builds slowly from simple concepts. It’s available as a printable version and you can fill it in online as well. They’re currently seeking feedback to improve this current version (my two cents: the diagrams could be much bigger, and possibly slightly less ugly, but it’s a lovely concept).

Another slightly old story, from back in October – Dame Frances Kirwan has been elected to the Savilian Professorship at the University of Oxford. Dame Kirwan will be the 20th holder of the Savilian Chair, which was founded in 1619, and is the first woman to be elected to any of the historic chairs in mathematics at Oxford.

As well as being a Fellow of the Royal Society and a former chairperson of the London Mathematical Society, Kirwan researches algebraic and symplectic geometry at Oxford. She sounds pretty cool.

Frances Kirwan elected 20th Savilian Professor, on the Oxford Mathematical Institute Website

]]>The next HLF will take place in September 2018, and applications open today for Young Researchers who want to participate. If you’re a maths or computer science researcher and want to be invited on a trip to Germany with lots of interesting talks, delicious food and good company, you can apply on the HLF website from today.

]]>The game involves using the numbers 1 to 9, and twelve symbols (three each of +,×,-,**÷**). The challenge is to combine the symbols and numbers in the right way to get a higher score than your opponent. It requires fast calculation, strategic thinking and a bit of luck.

Their IndieGoGo campaign hopes to raise enough money to go into production, and they have 7 days left to take pre-orders and donations in return for goodies. It’s also possible to make a donation which results in not you, but a worthy school in rural India, getting a copy of the game.

Watch the video below for an idea of how it works!

]]>It’s one of very few things whose Wikipedia article falls into both categories: ‘doctoral degrees‘ and ‘dance competitions‘.

Submissions are collected in the form of videos, and judged by a panel of artists and scientists. The competition is now in its tenth year – and this year, one of the winners entered a mathematical dance!

The competition accepts entries in categories including biology, chemistry, physics and social science, and this year topologist Nancy Sherich has won the competition overall with her piece, Representations of the Braid Groups. It’s… interesting?

Announcing the winner of this year’s Dance Your PhD contest, at Science

]]>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.

]]>*To mark the end of the month, Dr Nira Chamberlain gave a lecture yesterday at UCL, and if you missed it, the event live-stream will be posted on the Chalkdust social media: Facebook / Twitter*

This October, we have been celebrating Black Mathematician Month. The objective was, first, to promote the work of black mathematicians from around the world, highlighting the crucial role they play and their successful careers, and secondly, to show how society can let down black mathematicians, often (but not always) without realising it: from the kid who at school is convinced by their teachers not to become a mathematician “because is not for them” to the professional mathematician who assumes that a black person in a conference “is part of the cleaning staff”, to more serious racism issues.

Over the past few weeks, we have spoken to mathematicians who have worked in Europe, North America and Africa. From an Oxford PhD graduate who works in the ‘sexiest field in science’, to a Nigerian professor who has set up a prize for women in mathematics in her name and from a globe-trotting number theorist to a woman who describes herself as a mathematician, professor and activist, everybody has had a different story to tell and a different perspective. But many ideas cropped up time and time again. A central theme was the publicising of science, the importance of being seen and engaging with communities that, for many reasons, might not otherwise see a career in mathematics as a viable option. Also, almost everybody that we spoke to, talked of the need for support at every level, from high school all the way up to early-career researchers. The importance of good and compassionate mentors is clear. From family and friends to teachers and professors, there is something that we can all do to help improve diversity in mathematics.

Fortunately, the mathematics community around the world is becoming increasingly aware of the small proportion of women doing maths and strong efforts are in place to try to fix this severe problem. Other minorities in our field, however, are often overlooked. Through the whole month, many people, associations and institutions have joint their efforts to promote Black Mathematician Month. There is strong evidence that suggests there are severe issues with representation of ethnic minorities in academia. Anyone who works in academia has noticed that the number of black people drops significantly at higher levels, with the numbers in the highest positions at universities being staggeringly low.

Although science and mathematics are about obtaining new results and pushing the frontiers of our knowledge, science is carried out by the scientists, theorems are proved by mathematicians and every single model or equation has a human story behind it. Therefore having a segregated community, one in which a black professor is forced to leave their job and city in the 2000s by the Ku Klux Klan, one in which a black woman is assumed to be part of the cleaning staff, one in which you have to be the lucky person not to be discouraged to do science, is not acceptable. We cannot be blind and care only about the outcomes of the scientific community, we need to actively care for the people involved.

The first Black Mathematician Month has finished, but hopefully, it leaves two lessons behind. First is that the awareness of potential racial issues is crucial. Being aware of the times in which minorities are treated differently, hearing their stories and what they have to deal with, perhaps on a daily basis, does bring light to an often-ignored problem. What have people of colour working in your school, laboratory or university experienced in their careers? What do they do to cope with discrimination? Secondly, Black Mathematician Month shows that we all actively need to construct a better mathematics community. We all need to destroy the intellectual stereotype. Regardless of our role, we can all contribute to a more equal society. Whether we are the empathetic classmate, the inspirational teacher, lecturer, the tolerant colleague in a conference and the organiser who strives to have a diverse range of speakers, we all play an active role in how our community currently is and we can all make it better.

We have come a long way since, less than one hundred years ago, the first black person was awarded a PhD in mathematics and his transcript had the word “colored” printed across it to mark the racial difference, but we are still far from a perfect community. The first Black Mathematician Month is now over, but the lessons that we’ve learnt during it can hopefully be part of building a more open community.

Although we will be back again in October 2018, we will be trying to put the lessons that we have learnt into practice throughout the year. Whether it is trying to provide people with role models, gathering better data on university applications or just going out and talking about how beautiful mathematics can be, we believe that there is something that we can all do to help build a more representative and fairer mathematical community, to the benefit of all.

]]>The closing talk of the HLF’s main lecture programme (before the young researchers and laureates head off to participate in scientific interaction with SAP representatives to discuss maths and computer science in industry) was given by Fields Medalist **Steve Smale**.

Speaking without slides, Smale shared with us some of his recent work in the crossover between mathematics and biology, but the central theme of his talk was one which goes to the heart of what mathematics is. Mathematics is embedded in science, and is used to describe and understand many aspects of scientific discovery.

The question was of whether mathematics is realistic or idealistic – do the mathematical models we use to solve problems and understand the universe give a realistic picture of how the world works, or is it all fantasy and we’re ignoring the fine details in order to get a model that works nicely? It’s a constant struggle, and Smale illustrated this with several historical examples.

**Alan Turing**’s theory of computation was an inspiring vision of how we can understand and use computers, and has influenced the whole field since. But was it realistic? For some applications, Turing’s approach is the correct model, but for others it fails. Modern study of NP-completeness in computability assumes an infinite amount of input, but obviously this doesn’t model the real world – it’s an idealisation.

Moving on to another giant of maths history, Smale turned to **Isaac Newton**. Newton’s work on physics, differential equations and mechanics was all an idealisation – to the extent that his calculations didn’t even include friction, which was added to the theory 100 years later.

**John Von Neumann**, who created the early models of quantum mechanics, introduced the concept of a Hilbert space – again, an idealisation. And even in other fields – **Watson and Crick** discovered the structure of DNA, but didn’t include the protein core of chromatin, later discovered to be a fundamental part of the more detailed structure.

This idealism is possible because this major work was often done without Newton, Turing or Von Neumann doing lab experiments – they used experimental data from other people’s work, but data which had already been ‘digested’ by the rest of the scientific community. Newton built on the work of Galileo, Copernicus and Kepler before him.

Smale’s current research is on the human body, and in particular the heart. How does it manage to beat in such a coordinated way, with all the myocytes, or muscle cells, acting together to create a regular heartbeat?

The synchronisation was compared to what happens in a crowd applauding for a long time, when the clapping falls into synchronisation almost accidentally. It’s a dynamical system, and can be understood through maths, much like the beating heart.

Smale is working towards a deeper understanding of the heart through mathematics, and in this case standing on the shoulders of Alan Turing. Turing’s final paper on morphogenesis – the biological processes that lead to stripes on a zebra, or the arrangements of seeds in plants – included some differential equations. These were again an idealisation of the reality, as they were based on what happens as the numbers of cells increases to infinity (obviously not the real situation – some biologists disliked Turing’s work for this reason, as they mainly worked with small quantities of cells).

The cells making up the heart are of the same **cell type** – they all behave in the same way. Smale is looking at how this happens, and outlined how if you consider the set of genes as a graph, with individual genes as the nodes, and directed edges indicating how each transcription factor controls the adjacent gene, elevating protein production, you can build a system of ordinary differential equations to describe how it works.

This system of equations can be seen to reach an equilibrium – a global basin of attraction where the levels of each gene work in exactly the right way to determine how the cell behaves. These basins define the cell types – so a liver cell, or a heart cell, knows exactly what quantities of each protein to produce based on these equations, which are hard-wired into the DNA. You can even consider stable periodic behaviour in the system, to understand the heartbeat’s regular cycle.

Of course, this is again all idealisation – but the workings of the heart are something that have been understood from the biological angle for some time, and now mathematics is providing new ways to model and understand it – which will hopefully lead to powerful medical advances we can implement in reality.

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