Recently, an acquaintance from my days as a researcher at Liverpool University alerted me to the existence of the Museum of Mathematics in New York. My acquaintance, Janet West, was a PhD student when I was at Liverpool and is now involved in the museum. There’s plenty of stuff online to look at but I would like to draw your attention if you have not already seen it to a lecture by Colin Wright. Colin is well-known in the mathematics communication community as he probably does more mathematics talks in schools around the country than anyone else. His main talk is about the mathematics of juggling. You can see him talking on to the BBC about it by clicking this link. (Note that the headline says he is a teacher whereas in fact his job is in marine navigation!) Colin gave a talk at the Museum of Maths in New York which was recorded and is on YouTube. You can even buy a DVD version. Teachers: If you are interested in seeing his talk at your school, then go to his...

## What is the Best Proof of Cauchy’s Integral Theorem?...

posted by Kevin Houston

Today’s post may look as though I’m going all Terry Tao on you with a long post with lots of mathematical symbols. It’s really about the learning and teaching of Cauchy’s integral theorem from undergraduate complex analysis, so isn’t for everyone. If it’s not your cup of tea/coffee, then pop over here for some entertainment. Cauchy’s Integral Theorem Cauchy’s Integral Theorem is one of the greatest theorems in mathematics. There are many ways of stating it. Here’s just one: Cauchy’s Integral Theorem: Let be a domain, and be a differentiable complex function. Let be a closed contour such that and its interior points are in . Then, . Here, contour means a piecewise smooth map . In my years lecturing Complex Analysis I have been searching for a good version and proof of the theorem. My definition of good is that the statement and proof should be short, clear and as applicable as possible so that I can maintain rigour when proving Cauchy’s Integral Formula and the major applications of complex analysis such as evaluating definite integrals. Many of the proofs in the literature are rather complicated and so time is lost in lectures proving lemmas that that are never needed again. Here’s a version which I think has a good balance between simplicity and applicability. I’ve highlighted the difference with the version above. Cauchy’s Integral Theorem (Simple version): Let be a domain, and be a differentiable complex function. Let be a simple closed contour made of a finite number of lines and arcs such that and its interior points are in . Then, . Here an important point is that the curve is simple, i.e., is injective except at the start and end points. This means that we have a Jordan curve and so the curve has well-defined interior and exterior and both are connected sets. With this version I believe one can prove all the major theorems in an introductory course. I would be interested to hear from anyone who knows a simpler proof or has some thoughts on this one. Proof of Simple Version of Cauchy’s Integral Theorem Let denote the interior of , i.e., points with non-zero winding number and for any contour let denote its image. First we need a lemma. Lemma Let be a simple closed contour made of a finite number of lines and arcs in the domain with . Let be a square in bounding and be analytic. Then for any there exists a subdivision of into a grid of squares so that for each square in the grid with there exists a such that for all Proof of Lemma The set up looks like the following. For a contradiction we will assume the statement is false. Let and divide into 4 equal-sized squares. At least one of these squares will not satisfy the required condition in the lemma. Let be such a square. Repeat the process to produce an infinite sequence of squares with .By the Nested Squares Lemma (which is just a generalization of the Nested Interval Theorem) there exists . As is differentiable there exists such that for . But as the size of the squares becomes arbitrarily small there must exist such that is contained in the disk . This is a contradiction. Main part of proof Given there exists a grid of squares covering . Let be the set of squares such that and let be the set of distinguished points in the lemma. Define by Then as is differentiable, is continuous (and hence integrable). Without loss of generality we can assume that is positively oriented. Let be the union of positively oriented contours giving the boundary of . Since is made of a finite number of lines and arcs will itself be the union of a finite number of lines and arcs. For such that , is just the boundary of a square. On we...

## Mary Cartwright article...

posted by Kevin Houston

Mary Cartwright is fairly well-known amongst mathematicians in the UK but less widely known amongst the general public. A recent BBC online article about her and her work may be the beginning of a change in this situation. There is even a Radio 4...

## Estimating large numbers the Fermi way...

posted by Kevin Houston

Here’s a really good introduction to estimating numbers and includes Enrico Fermi‘s famous “How many piano tuners are there in Chicago?” You can see more like this at...

## Matt Parker Number Ninja...

posted by Kevin Houston

This is mainly one for those local to Leeds (although see the bottom of the post). As part of the Leeds Festival of Science, Matt Parker will be giving a talk at the University. From the advertising: Direct from BBC Radio 4’s Infinite Monkey Cage with Brian Cox and Robin Ince, with research featured on QI, Leeds will be welcoming stand-up Maths comedian Matt Parker. Expect everything from debunking number nonsense and flagrant sudoku abuse to the mysterious patterns in the locations of ancient monuments and defunct Woolworths stores. Suitable for ages 16 plus. Entry to the show is strictly by ticket only. Book early as places are limited. Tickets can be booked via the University of Leeds website I’ll be doing my own little bit for Leeds Science Week with a visit to at least one school. See the Schools Programme. Teachers and pupils: Although it’s too late to book me for science week, if you want me to visit your school, then get please get in touch. I’ll travel further afield than just...

## Maths problems tweets...

posted by Kevin Houston

I’m not a big user of twitter as I prefer to read material that has more than 140 characters. Nonetheless, often something interesting appears. Dan Meyer set off an intriguing set of responses to his call for maths problems that can be posed within Twitter constraints. The results can be seen here: Tweet-sized...

## Mathematics of the mosh pit...

posted by Kevin Houston

Rock music fans are aware of the phenomenon of the mosh pit at concerts — a riotous area, usually close to the stage, where concert-goers, in lieu of learning complicated dance steps, slam violently into each other (and hence an alternative name for moshing is slam dancing). Until now the fascinating mathematics of this emergent property of crowds has been unjustly ignored. However, this has changed with the advance publication on the arXiv Collective Motion of Moshers at Heavy Metal Concerts by Silverberg, Bierbaum, Sterna and Cohen. The abstract states Human collective behavior can vary from calm to panicked depending on social context. Using videos publicly available online, we study the highly energized collective motion of attendees at heavy metal concerts. We find these extreme social gatherings generate similarly extreme behaviors: a disordered gas-like state called a mosh pit and an ordered vortex-like state called a circle pit. Both phenomena are reproduced in flocking simulations demonstrating that human collective behavior is consistent with the predictions of simplified models. The authors model the behaviour of the mosh pit (and circle pit), provide source code and Java script interactive moshpit The conclusions will be surprising to many: If we increase the self-propulsion coefficient (decrease ), we find the motion, though random, is no longer fit by a Maxwell-Boltzmann distribution. Instead, collisions between active and passive MASHers [Mobile Active Simulated Humanoid] on the boundary of the simulated mosh pit removes energy faster than collisions among active MASHers can rethermalize the system. Consequently, measurements in silico show a radial temperature gradient is established with a higher effective temperature at the core of the simulated mosh pit and a lower effective temperature at the edge A valuable and timely addition to the literature. Ok, let’s...

## Archimedes and 3D printing...

posted by Kevin Houston

3D printing seems to have some sort of turning point in the last few months. Lots of people know what it is and many are experimenting with it. This week I began teaching on my History of Mathematics module; yesterday’s lecture was about Archimedes. I would never imagined that the two could be combined but today I saw a preprint on the arXive on the combination of Archimedes and 3D printing. The paper has a brief history of Archimedes and his works (and quite a few elementary typos) and focusses on how to make models of his inventions, for example his water screw, and more mathematical objects using a 3D printer. I don’t have access to 3D printing but would be intrigued to give some of the Archimedes models a go. Certainly, it would be great to have a supply of Platonic and Archimedian solids to show students. Has anyone out there tried something...

## Mathematics of Lego

posted by Kevin Houston

Obviously the first post of the year should be a serious one about the year ahead, New Year’s resolutions and such. Not this time. This time I urge you to have a look at the mathematics of Lego via Wired. Read the article here. To be honest I’m not sure if this really is a good example of a log-log plot (could I use it in class) or if it even tells us something deep about Lego but I think it’s worth a...

## End of the year…...

posted by Kevin Houston

It’s the end of the year. It seems only like yesterday we were all saying “What? I can’t believe it’s the end of January already”. So what was the year like? I feel I’ve had a good year – I’ve been having a lot of fun trying to understand discrete surfaces and if I hadn’t succumbed to a virus just shortly before Christmas I might have written the bulk of my first paper on the subject. I also had fun with the Beatles story which garnered a lot of attention. Interestingly the most read blog entry was on the passing of Vladimir Zakalyukin. In the world of science the big story was not in mathematics but in the discovery of the Higgs boson. Given that the evidence is hardly overwhelming, it wouldn’t surprise me if the big story in 2013 was its ‘undiscovery’ and the scientists admit a mistake or a misreading of the data. For 2013 one of my top resolutions is to finish my next book. Feel free to write and remind me of that at the end of 2013 if it remains uncompleted. It will be tricky as this coming semester I have a lot of teaching (if there are any of my History of Maths students reading, then watch The Two-Thousand-Year-Old Computer, there are two days left to watch.) Have a good...

## Nate Silver Talk at Google...

posted by Kevin Houston

Nate Silver has come to prominence recently through the seemingly simple method of predicting election results through statistics. I’ve just finished reading his book, The Signal and the Noise and it’s certainly one of my favourite books of the year. It covers all sorts of predictions, in baseball, chess, politics, earthquakes, terrorism, stocks, and more besides. If I were to recommend a Christmas book for a maths geek, then this would be the one. He recently gave a Google talk which you can see...

## Scott Young on Lectures...

posted by Kevin Houston

I follow a number of education writers and those who write about learning. One of these, Scott Young, is a high achieving recent graduate who seems to be making a living by giving learning advice. His latest claim to fame is his MIT Challenge where he studied a complete 4-year MIT course in one year as a way of learning new skills but also, presumably to prove the effectiveness of his methods. My opinion is that some of the methods are debatable as they don’t foster really deep understanding of a topic, a number are just ways of quickly mastering the necessary ideas to pass an exam. As an example, in one of the key documents he uses to advertise his Learning on Steroids programme he shows how someone used his methods to learn the mathematical concept of limits. There is a scan on page 7 of the notes. The student writes in a section headed “What is a limit, REALLY?” that “A limit is like a stalker, forever getting close to the target. Forever trying to close the distance between it and the target, but rarely ever succeeds.” This analogy is deeply flawed. Worse than that, leaving aside the word stalker, it is precisely what I have to stop students thinking a limit is. When I set an exam question on the definition of a limit, many students will give rather incoherent answers about “numbers getting close to other numbers but never quite reaching”. However, what I want students to realize is that the limit of 0,0,0,0,0,0,… is 0, i.e., every element of the sequence is equal to the limit — the sequence doesn’t get closer and closer to the limit. Another good example is interweaving this sequence with 1/n. I.e., 0, 1,...

## Lancelot Hogben

posted by Kevin Houston

No post last week – I seem to have spent the last two weeks on trains or motorways going from one part of the country to the next on various academic related missions so have not had the time to write a post. So back to business. Dara O’Briain’s Science Club is a new TV series that takes a Top Gearish approach to scientific topics (I often wondered why no one had attempted such an approach before). Last night one of the guests was Lucy Cooke who I’d never come across before but was very entertained by. As her unsung scientific hero she selected Lancelot Hogben as he pioneered the use of the African clawed frog in scientific investigation, including an early method of testing for pregnancy in humans. I had not known this side of him, I know him as the author of Mathematics for the Million. I bought a copy of this book from the thirties from a second-hand book shop in Ilford when I was an undergraduate. In its time it was a popular maths best-seller, though I doubt such an equation-packed book would do so well now. To give some idea of the content it starts with mathematics in ancient antiquity and gets up to calculus, modern algebra and probability. I have to confess that I have not read it cover to cover but have read chapters, dipped in here and there and used some material for my own courses. I would love to be able to teach the chapter on Mathematics for the Mariner as it covers a lot of interesting spherical geometry. Maybe some...

## XKCD on the maths of elections...

posted by Kevin Houston

Too much going on this week to post anything sensible so I’ll point in the direction of the latest xkcd:

## William Tutte and Tommy Flowers...

posted by Kevin Houston

On Monday night I unexpectedly came across a TV programme about the code-crackers at Bletchley Park during World War II. Usually such a programme focuses on Turing and the cracking of Enigma but this was about two of the lesser known players: William Tutte who was a mathematician like Turing and Tommy Flowers, a Post Office engineer who arguably built a programmable computer before anyone else. Seeing Tutte was a coincidence as just last week I was trying to understand his graph embedding theorem. [I’d met the theorem before but had forgotten all about it until I was trying to understand the “No Free Lunch Theorem” for discrete Laplace operators by Wardetzky et al, see here. Well, I think I understand the theorem but I don’t understand the proof. In my attempt to understand the basics of discrete Laplace-Beltrami operators I set aside a day last week to understand the proof and perhaps attempt to give a different one. Unfortunately after three days I still didn’t understand all the details of their proof and didn’t have a version of my own either! If anyone knows the details, then get in touch. But I digress…]I found a good description of the theorem on a site by Graham Farr at Monash. You can find it here. It’s a bit long but the important bits are in the first part. I liked the programme, for a start it didn’t assume you were stupid (even though it assumed I wouldn’t have heard of Tutte). The codebreakers programme is of course available for the next few days on the...

## The number of spots in a deck is 365?...

posted by Kevin Houston

Today’s TED video is an augmented reality card trick by Marco Tempest. Now, I’m a fan of card tricks but this was a bit too cutesy for me. However, I draw your attention to his computer’s claim (at 3:20) that if you add up all the spots on the cards in a deck then you get 365 which is the number of days in a year. I make the same claim in one of my school talks but I follow it by pointing out that obviously it can’t be right. (Obvious in the sense that you don’t need to get a deck and count but just use some simple maths.) Usually at least one of the pupils gets it. Anyhow, here’s the...

## Tails you win: The Science of Chance

posted by Kevin Houston

Last night David Spiegelhalter presented a TV programme on the science of chance. As usual it’s available on the BBC’s iPlayer for seven days. Spiegelhalter is an entertaining speaker, I’ve seen him give talks a number of times, but I felt this programme was a bit pedestrian. There’s lots in it but that just makes it too “bitty”. For example, there is a brief bit about Apple making iPods less random and an overlong bit about a cricketer who was “unlucky” to break an ankle and so on. Nothing was dealt with in depth. Upon reflection I think that the problem (leaving aside that I am not the target audience, but hey, I can have an opinion) is that there is a lack of drama and tension in the programme. It’s just one thing after another. So although the programme did progress through the history of chance there was no sense in these historical pieces of dramatic fights to uncover the truth. In fact, I felt that the only attempt at tension was rather misguided. To create drama Spiegelhalter jumps out of an aeroplane after explaining how at his age the risk of dying is in some sense less than that of a younger person as he is more likely to die in the next few years. As he heads toward the ground the screen freezes, a caption asks “what happened next?” and the programme moves on to another topic, returning much later. Are we supposed to think that perhaps the parachute didn’t open and he tumbled to his death? Well the tabloids would have told us: “Risk Prof dies in ‘unlikely accident'” or the announcer would have dedicated the programme to his memory. So of course there was no tension or drama in this piece. Particularly since the programme titles showed the parachute opening. Anyhow, there were many bits I did like such as the explanation of average age until death and the graphics used. Mercifully, he didn’t do the Monty Hall problem which has rather been done to death. And of course, as I said, Spiegelhalter is an entertaining speaker so at least he wasn’t putting me to...

## Robin Ince on science communication...

posted by Kevin Houston

This one is slightly outside my maths remit but I felt that Robin Ince made some serious points (and maybe some not so serious points) about science communication on this Saturday’s Saturday Live on Radio 4. You can hear it here for the next few days provided you can use the BBC’s iPlayer where you are. Ince’s website lists upcoming performances should you wish to see him live in the UK. He is good – I saw him doing non-science when he was promoting his Bad Books Club. (I have a signed copy of the book. I persuaded him to break the spine of the book so that it falls open at the frostbitten hands picture. Read the book to find out...

## In Liverpool

posted by Kevin Houston

I’m back in Liverpool. Rob Sturman and I were reprising the British Science Festival talk for the Liverpool Mathematics Society and today I’m giving a research talk at the University. I had hoped to write about all this last night when I was in my hotel, the famous Adelphi Hotel, but they don’t have internet access in the rooms, only in the lounge. I can’t remember when the last time I stayed in a hotel without free wifi access was. Anyhow, I’m currently in an office in the Mathematical Sciences Building. It may even have been the office I had here back in the 90s but I’m not sure maybe it was the one next door. A lot has been happening since I last wrote. Did you see that the University of Leeds held a talk by, Iain Lobbam, the director of GCHQ on Alan Turing? If not then see the Guardian article or the Telegraph...

## Criminal Network

posted by Kevin Houston

I’ve been really busy the last few weeks – the usual effect of the academic year starting! I’m running a new course called Maths at Work which is about how maths gets used in the “real world”. One of the projects for this involves the mathematics of networks, such as Twitter and Facebook. As part of my investigations I came across a use of Steiner trees in fraud detection. The University of Alberta press release called Math tree may help root out fraudsters gives the non-mathematical side of the story, the mathematical side can be found in the paper Social network meets Sherlock Holmes: investigating the missing links of fraud. It would be interesting to see if this actually works! Anybody have any...

## Wie man mathematisch denkt!...

posted by Kevin Houston

Today is my blog’s anniversary (happy birthday blog!) and, by coincidence, the official release date of the German translation of my book, How to Think Like a Mathematician although I believe it’s been available for at least a month now. It’s called Wie man mathematisch denkt which I think translates more or less as “How one thinks mathematically”. It’s published by Springer no less and you can buy it on Amazon.de. My thanks to the translator, Roland Girgensohn, who put a lot of effort into making sure that the translation was...

## Mystery Chord Update

posted by Kevin Houston

The story of the A Hard Day’s Night Chord appeared in numerous papers since my talk with Rob Sturman and Ben Sparks at the British Science Festival. I was also asked to appear on BBC Radio Merseyside which you can hear at http://www.bbc.co.uk/programmes/p00xn5jh for the next 6 days if you are in a territory where iPlayer programmes are available. Start at about...

## The Beatles’ Magical Mystery Chord

posted by Kevin Houston

I’m giving a talk at the British Science Festival on Saturday. Details are here. To accompany it I’ve made a video on how mathematics can be used to analyze the opening chord of A Hard Day’s Night by the Beatles.

## Maths projects

posted by Kevin Houston

Last week I updated my personal and work web sites for the first time in far too long and I changed the look of this blog. It doesn’t work perfectly yet as some of the colours and spacing are wrong. And I should get rid of that banner at the top – it’s looking a bit old. On a different topic, I’ve been criticized over on the TES forum for my mechanic analogy. I thought it was good analogy, but PaulDG disagrees: As with all analogies, and having only read those few paragraphs of his work, I’m afraid it looks to me as if he’s the sort of “expert” who’s got us into the problems we’re currently in. Ouch. He then gives a straw man a thorough thrashing and finishes with Houston appears from his analogy to be in favour of understanding without skills. This seems rather illogical: Houston wants students to do more of A and less of B therefore he doesn’t want students to do any B. To be honest I can’t see how anyone could have such a reading of what I said. I was tempted to jump into action with a shout of “Someone is wrong on the internet, I must intervene” but after some reflection I decided to just read the rest of the thread. So, the thread itself is about a very important topic — maths projects in schools. As someone who is trying to prepare a projects module for second year students I am aware how difficult it is to prepare good projects. The problem at all levels, school or university, is that it is very hard to set “medium strength” projects – they are either too easy or too hard. I think the key difficulty is...

## Painting by numbers – restoring frescoes...

posted by Kevin Houston

You may have seen in the news this week that an elderly amateur attempted the restoration of a religious fresco. The result is pictured below. I took the picture from the BBC article that you can read here. By coincidence this week I saw an article by Carola-Bibiane Schönlieb on the mathematical technique of inpainting. This allows restoration of damaged pictures by using some mathematical process to fill in the damaged part. A particular common method is to use diffusion equations – the idea is that we use something like the heat equation. Imagine the ink on the border of the damaged region as being like heat and the heat travels into the region through conduction. Compare the two images below take from a preprint by Schönlieb and others. The top one is a vandalized picture and the bottom is the version restored through a diffusion equation method. Most people probably wouldn’t realize that the picture had been repaired unless they’d been told. Schönlieb and others have worked on restoring the following Austrian fresco Here’s a picture of the proposed restoration (unfortunately only in black and white): The pictures come from a paper which contains the details. The article I saw this week is more accessible and can be read in the Matlab Mathworks...

## The algorithm that runs the world...

posted by Kevin Houston

A colleague in the School of Mathematics, Richard Elwes, has the front cover story of New Scientist. You can read the full article on the Simplex Algorithm...

## Hannah Fry: Is life really that complex?...

posted by Kevin Houston

Mathematics talks at TED are rather rare so I was keen to link a recent talk from TEDxUCl by Hannah Fry which was about the applications of mathematics to complex problems such as human behaviour. After watching it I hummed and hawed. On the one hand this is a very well presented and explained talk (and this is even more impressive as the speaker is a fairly newly-minted PhD). On the other hand, it doesn’t feature much maths and crucially, for me at least, I wasn’t sure that anything has been proved. It wasn’t clear to me that any models have been developed. The wording at the end is fairly vague. Saying “Once we have done this…” leaves open the possibility that it has not been done yet. She also says “we can almost begin to talk about…” which is again too vague. Anyhow, in the end I decided to link to it. It’s only 10 minutes long and does give you some idea where things are headed. The end of the talk is mostly about predicting crime, something I looked into a few years ago when a local policeman contacted the School of Maths for help after watching the TV programme Numbers about a crime fighting mathematician. Unfortunately I was unable to help him with his enquiries as at the time the mathematical models for crime prevention and detection were very poor. The current video sort of claims they have been improved. I...

## A Mathematician Comes of Age by Steven Krantz...

posted by Kevin Houston

I’m a big fan of Steven Krantz. His book on teaching mathematics is the one I recommend most to beginning maths lecturers. The second edition is a must have since it contains short replies to his book from other lecturers, some of them highly critical. Currently, I’m reading his recent book A Mathematician Comes of Age. This is concerned with how a mathematician becomes mathematically mature. From the back cover: “It describes and analyzes how a student develops from a neophyte who can manipulate simple arithmetic problems to a sophisticated thinker who can understand abstract concepts, can think rigorously, and can analyze and manipulate proofs.” I must admit I’ve been diving in and out of the book at random and although I don’t agree with everything he says (or agree with the inclusion of certain topics – why are North Americans so concerned about “Math anxiety”?) there are thought-provoking passages every page or so. The parody of the Evolution of Teaching Math on page 49 is very funny and includes 1980s: A framer sells a bag of potatoes for $10. His production costs are $8 and his profit is $2. Underline the word “potatoes” and discuss with your classmates. Leaving aside the humour, I particularly like the section on Reading and Thinking (p95): It has been observed [No reference given – KH] that the key things that a good teacher does are engage the students in the learning process pace the students teach the students to read Now, reading mathematics is something I’ve thought about and try to get my students to do, see Chapter 2 of How to Think Like a Mathematician (follow the link for free samples of chapters 3 and 4). Like in HTTLAM Krantz mentions the importance of reading with...

## Maths compulsory until age 18

posted by Kevin Houston

The House of Lords just published a report recommending that students in the UK study mathematics until the age of 18. This recommendation cropped up last year in the Vorderman review. The Lords report can be found online in html and in pdf format. It’s a fairly long report but my response is as simple as my response to the Vorderman review: Where are the teachers going to come from? We don’t have enough teachers at the moment to provide the best-quality maths education. Giving the current ones more to do would lead to catastrophe. The Lords’ report even states The Department of Education, recognising the role of teaching in increasing the progression of students to A level STEM subjects,… has introduced a number of initiatives to increase the number of specialist teachers (such as, golden handshakes and bursaries), but, by their own admission, “the targets set by the previous Government for numbers of specialists teaching physics and maths will not be met”. Is the best that the DoE can come up with is golden handshakes and bursaries? It is unsurprising that the targets won’t be met. There will be no single solution to the problem of mathematics education in the UK, it will require many areas to be tackled but the solution must surely include higher pay for maths and science teachers. The simple reality is that my students can finish their degrees and get a job that pays them an average starting salary of (according to HESA) 23,160 pounds (according to AGR figures 26,500 pounds) with rapid increases, or spend a year on a bursary of less than 20,000, and join a profession where after a number of years they reach top of scale at 31,552 pounds? And yes I know I should take into account holidays, etc, but even with shorter holidays the working conditions are better in many professions. Looking at the figures, which do you think is the better...

## Best of xkcd: Maths I

posted by Kevin Houston

I’m currently at a conference so have been working hard on research-y type things rather than thinking up exciting blogs posts. So here are some of my favourite maths strips from...