# HLF Blogs: Efim Zelmanov’s Desert Island Maths

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).

# Stupid-looking maths question does the rounds, isn’t stupid

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:

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!

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.

# Ritangle student maths competition open now

Ritangle, a maths competition aimed at A-level and equivalent maths students in the UK, is open for registration. The first set of preliminary questions has already been released, but the main competition starts on 9th November and there’s still time to register a team.

Comprising 21 questions over 21 days, the competition requires no maths beyond A-level and the winning teams gets a hamper and a trophy.

Ritangle website

# The Sound of Proof

Marcus du Sautoy has tweeted about a mathematics and music project he’s involved in, called The Sound of Proof. Five classical proofs from Euclid’s Elements have been interpreted by composer Jamie Perera into musical pieces, and they’ve put together an app/game to see if you can work out which one corresponds to which.

They’ll be announcing the results at an event as part of Manchester Science Festival in October. The project is a collaboration with PRiSM, the research arm of the Royal Northern College of Music in Manchester.

The Sound of Proof, at RNCM PRiSM

# 2017 London Mathematical Society Popular Lectures now online

The London Mathematical Society Popular Lectures present exciting topics in mathematics and its applications to a wide audience. The 2017 Popular Lectures were Adventures in the 7th Dimension (Dr Jason Lotay, University College London) and The Unreasonable Effectiveness of Physics in Maths (Professor David Tong, University of Cambridge).

The Lectures are now available on the LMS’s YouTube channel, along with many of the previous years’ videos.

# Petition to update UK traffic signs to use a geometrically plausible football

Aperiodipal and number ninja, Stand-up Mathematician Matt Parker, has set up a petition on the UK parliament petitions website to change the awful, awful tourist board official symbol for a football ground (US readers: imagine I’m saying ‘soccer stadium’). In Matt’s words:

The football shown on UK street signs (for football grounds) is made entirely of hexagons. But it is mathematically impossible to construct a ball using only hexagons. Changing this to the correct pattern of hexagons and pentagons would help raise public awareness and appreciation of geometry.

To end this madness, Matt needs 10,000 signatures for the petition to be responded to by the government (and 100,000 for it to considered for debate in parliament). It’s currently around the 3,000 mark – so it’s plausible that he might do it. It’s also had coverage in The Independent already, and Matt’s YouTube video on the campaign already has over 100,000 views.

To sign, you simply need to be a British citizen or UK resident, and fill in your details on the site (you’ll need a valid postcode). Ban this hexagonal filth!

Update the UK Traffic Signs Regulations to a geometrically correct football, on UK Parliament Petitions

# Progress on billiard table problem

Quanta Magazine reports progress on what its headline calls the “Infinite Pool-Table Problem”. The problem is explained in the article as follows:

Strike a billiard ball on a frictionless table with no pockets so that it never stops bouncing off the table walls. If you returned years later, what would you find? Would the ball have settled into some repeating orbit, like a planet circling the sun, or would it be continually tracing new paths in a ceaseless exploration of its felt-covered plane?

The article describes progress on the problem via study of ‘optimal’ billiard tables, “shapes whose particular angles make it possible to understand every billiard path that could occur within them”.