*This is a guest post by researcher Audace Dossou-Olory of Stellenbosch University, South Africa.*

Consider the following question: **How many ways are there to connect $2n$ points on a circle so that each point is connected to exactly one other point?**

*This is a guest post by researcher Audace Dossou-Olory of Stellenbosch University, South Africa.*

Consider the following question: **How many ways are there to connect $2n$ points on a circle so that each point is connected to exactly one other point?**

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Friends of the Aperiodical, nerd-comedy troupe *Festival of the Spoken Nerd*, are currently on tour around the UK. As part of their show, questionably titled *You Can’t Polish a Nerd*, Matt Parker attempts to calculate the value of $\pi$ using only a length of string and some meat encased in pastry. He’s previously done this on YouTube, and the idea was inspired by the Aperiodical’s 2015 Pi Approximation Challenge, and in particular my own attempt to approximate $\pi$ with a (more conventional) pendulum.

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*In September, Katie and Paul spent a week blogging from the Heidelberg Laureate Forum – a week-long maths conference where current young researchers in maths and computer science can meet and hear talks by top-level prize-winning researchers. For more information about the HLF, visit the Heidelberg Laureate Forum website.*

At the start of his HLF lecture on Asymptotic Group Theory on Thursday morning, Fields medalist Efim Zelmanov described the ‘group’ as: “the great unifying concept in mathematics,” remarking “if you go for a trip, and you are allowed to take only two or three mathematical concepts with you, give serious consideration to this one.” Very loosely defined, a group is a set of things (its ‘elements’) that you can ‘multiply’ together, with this multiplication behaving in certain helpful ways. Think of numbers being added, functions composed together or rotations and reflections of a shape being carried out one after the other. I doubt any mathematician would accuse Zelmanov of overstating their importance in mathematics.

In his talk he discussed residually finite groups. These are groups which are infinite in size but still just a little bit finite-y. In technical terms, the group has a set of homomorphisms with finite kernels having trivial intersection. Although the group is too large to see all at once, as Zelmanov put it, we have “photos from all sides of the group”. He contrasted this to “hopelessly infinite groups”, for which no such photo album is possible.

A common way to look at a group is to find a set of ‘generators’: these are elements of the group which you can multiply together to create any element of a group (the elements ‘generate’ the entire group). Some infinite groups can’t be generated from a finite set — consider trying to find a set of rational numbers that you can multiply together to create any rational number. Those that can be generated from a finite set are unexcitingly called ‘finitely generated’. Of course, finite groups are also finitely generated.

Zelmanov considered under what circumstances finitely generated groups can be proved to be finite. One immediate way this won’t happen is if one of the generators is not periodic: if you keep multiplying it by itself you keep getting new elements forever, never ‘looping back’ to the original generator. (Imagine starting with 1 and continually adding 1…) The Burnside problem asks whether there are any other ways to make a finitely-generated, yet infinite, group. In 1991, Zelmanov proved that for residually finite groups, there aren’t. However, this isn’t the case for the ‘hopelessly infinite’ groups.

In his lecture Zelmanov, accompanied by his excellent hand-drawn slides, discussed this before moving on to related topics such as the growth of groups (if you start with a generating set, and create new elements by multiplying them together, how quickly does the set grow?) and ‘approximate groups’ (which, as the name suggests, are things that are like, but not quite, groups).

You may by now have seen the image below knocking around on Twitter and other social medias, in which a maths question appears to be almost a parody of itself:

The text reads:

An orchestra of 120 players takes 40 minutes to play Beethoven’s 9th Symphony. How long would it take for 60 players to play the Symphony? Let P be the number of players and T the time playing.

Well, once you’re done laughing, we’ve done some investigative journalism and found the origin of this question. And it turns out it’s quite nice!

I wrote this!! How did you get this??? I am a maths teacher in Nottingham UK. Wrote this 10 years ago. Here is the original whole worksheet pic.twitter.com/jYX55GSBKz

— Claire Longmoor (@LongmoorClaire) October 11, 2017

The question is from a worksheet developed by maths teacher Claire Longmoor (who is, based on current evidence, brilliant) ten years ago. Claire put together a selection of example questions with relationships in direct and inverse proportion, and deliberately included the orchestra question as an example of something where it doesn’t work that way. It’s a nice activity to help reinforce the difference, and in context the question works nicely.

Other examples on the sheet include a bricklaying example with creditably diverse gender representation, a car with terrifyingly low fuel efficiency, good cow names and a delightful insight into the bygone world of fruit picking.

Paul and I have spent this week blogging from the Heidelberg Laureate Forum, an international event for PhD/postdoc students and top-level maths and computer science researchers.

It was a long week of extravagant dinners, incredible talks and press conferences, (maths) celeb spotting, branded conference freebies, hilarious quotes and exceptional hospitality. Oh, and blogging. Here’s a round-up of what we wrote, in case you’ve missed it this week, as well as some of the other posts the rest of the HLF blog team wrote.

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The Open University and UK Mathematics Trust have teamed up to launch Perplex, a mobile app containing mathematical puzzles and games. It’s available for iPhone and Android, and can also be played directly on their website.

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*This week, Katie and Paul are blogging from the Heidelberg Laureate Forum – a week-long maths conference where current young researchers in maths and computer science can meet and hear talks by top-level prize-winning researchers. For more information about the HLF, visit the Heidelberg Laureate Forum website.*

A wonderful potted history of the theory of communication was capably presented by 2002 Nevanlinna Prize winner Madhu Sudan, who talked us through from the earliest mathematical thinking on the subject through to the present day, and his team’s work. It was also almost a love letter to one of his mathematical heroes, the father of information theory, Claude Shannon.

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