2018 Abel Prize Laureate Robert Langlands: Theorems v Theories

2018 Abel Prize Laureate Robert Langlands explains his work in the context of theorems versus theories. The second in a series of videos documenting my experience at the 2018 Abel Prize week in Oslo.

With thanks to the Norwegian Academy of Science and Letters for kindly providing me with a scholarship.

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Maths, but not as you know it… (St Hugh’s College Oxford Magazine)

In October 2017, Dr Tom Crawford joined St Hugh’s as a Lecturer in Mathematics. He has since launched his own award-winning outreach programme via his website tomrocksmaths.com and in the process became a household name across Oxford University as the ‘Naked Mathematician’. Here, Tom looks back on the past year…

headshot-cropped

I arrived at St Hugh’s not really knowing what I was getting into to be completely honest. I’d left a stable and very enjoyable job as a science journalist working with the BBC, to take a leap into the unknown and go it alone in the world of maths communication and outreach. The plan was for the Lectureship at St Hugh’s to provide a monthly salary, whilst I attempted to do my best to make everyone love maths as much as I do. A fool’s errand perhaps to some, but one that I now realise I was born to do.

The ‘Naked Mathematician’ idea came out of my time with the Naked Scientists – a production company that specialises in broadcasting science news internationally via the radio and podcasts. The idea of the name was that we were stripping back science to the basics to make it easier to understand – much like Jamie Oliver and his ‘Naked Chef’ persona. Being predominantly a radio programme, it was relatively easy to leave the rest up to the listener’s imagination, but as I transitioned into video I realised that I could no longer hide behind suggestion and implication. If I was going to stick with the ‘Naked’ idea, it would have to be for real.

Naked-Mathematician

Fortunately, the more I thought about it, the more it made sense. Here I was, trying to take on the stereotype of maths as a boring, dreary, serious subject and I thought to myself ‘what’s the best way to make something less serious? Do it in your underwear of course!’ And so, the Naked Mathematician was born.

At the time of writing, the ‘Equations Stripped’ series has received over 100,000 views – that’s 100,000 people who have listened to some maths that they perhaps otherwise wouldn’t have, if it was presented in the usual lecture style. For me that’s a huge victory.

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Of course, not all of my outreach work involves taking my clothes off – I’m not sure I’d be allowed in any schools for one! I also answer questions sent in by the viewers at home. The idea behind this is very simple: people send their questions in to me @tomrocksmaths and I select my favourite three which are then put to a vote on social media. The question with the most votes is the one that I answer in my next video. So far, we’ve had everything from ‘how many ping-pong balls would it take to raise the Titanic from the ocean floor?’ and ‘what is the best way to win at Monopoly?’ to much more mathematical themed questions such as ‘what is the Gamma Function?’ and ‘what are the most basic mathematical axioms?’ (I’ve included a few of the other votes below for you to have a guess at which question you think might have won – answers at the bottom.)

The key idea behind this project is that by allowing the audience to become a part of the process, they will hopefully feel more affinity to the subject, and ultimately take a greater interest in the video and the mathematical content that it contains. I’ve seen numerous examples of students sharing the vote with their friends to try to ensure that their question wins; or sharing the final video proud that they were the one who submitted the winning question. By generating passion, excitement and enthusiasm for the subject of maths, I hope to be able to improve its image in society, and I believe that small victories, such as a student sharing a maths-based post on social media, provide the first steps along the path towards achieving this goal.

Speaking of goals, I have to talk about ‘Maths v Sport’. It is by far the most popular of all of my talks, having featured this past year at the Cambridge Science Festival, the Oxford Maths Festival and the upcoming New Scientist Live event in September. It even resulted in me landing a role as the Daily Mirror’s ‘penalty kick expert’ when I was asked to analyse the England football team’s penalty shootout victory over Colombia in the last 16 of the World Cup! Most of the success of a penalty kick comes down to placement of the shot, with an 80% of a goal when aiming for the ‘unsaveable zone’, compared to only a 50% chance of success when aiming elsewhere.

unsaveable-zone
Image courtesy of Ken Bray

In Maths v Sport I talk about three of my favourite sports – football, running and rowing – and the maths that we can use to analyse them. Can we predict where a free-kick will go before it’s taken? What is the fastest a human being can ever hope to run a marathon? Where is the best place in the world to attempt to break a rowing world record? Maths has all of the answers and some of them might just surprise you…

Another talk that has proved to be very popular is on the topic of ‘Ancient Greek Mathematicians’, which in true Tom Rocks Maths style involves a toga costume. The toga became infamous during the FameLab competition earlier this year, with my victory in the Oxford heats featured in the Oxford Mail. The competition requires scientists to explain a topic in their subject to an audience in a pub, in only 3 minutes. My thinking was that if I tell a pub full of punters that I’m going to talk about maths they won’t want to listen, but if I show up in a toga and start telling stories of deceit and murder from Ancient Greece then maybe I’ll keep their attention! This became the basis of the Ancient Greek Mathematicians talk where I discuss my favourite shapes, tell the story of a mathematician thrown overboard from a ship for being too clever, and explain what caused Archimedes to get so excited that he ran naked through the streets.

toga

This summer has seen the expansion of the Tom Rocks Maths team with the addition of two undergraduate students as part of a summer research project in maths communication and outreach. St John’s undergraduate Kai Laddiman has been discussing machine learning and the problem of P vs NP using his background in computer science, while St Hugh’s maths and philosophy student Joe Double has been talking all things aliens whilst also telling us to play nice! Joe’s article in particular has proven to be real hit and was published by both Oxford Sparks and Science Oxford – well worth a read if you want to know how game theory can be used to help to reduce the problem of deforestation.

Looking forward to next year, I’m very excited to announce that the Funbers series with the BBC will be continuing. Now on its 25th episode, each week I take a look at a different number in more detail than anyone ever really should, to tell you everything you didn’t realise you’ve secretly always wanted to know about it. Highlights so far include Feigenbaum’s Constant and the fastest route into chaos, my favourite number ‘e’ and its link to finance, and the competition for the unluckiest number in the world between 8, 13 and 17.

The past year really has been quite the adventure and I can happily say I’ve enjoyed every minute of it. Everyone at St Hugh’s has been so welcoming and supportive of everything that I’m trying to do to make maths mainstream. I haven’t even mentioned my students who have been really fantastic and always happy to promote my work, and perhaps more importantly to tell me when things aren’t quite working!

OxTALENT

The year ended with a really big surprise (at least to me) when I was selected as a joint-winner in the Outreach and Widening Participation category at the OxTALENT awards for my work with Tom Rocks Maths, and I can honestly say that such recognition would not have been possible without the support I have received from the college. I arrived at St Hugh’s not really knowing what to expect, and I can now say that I’ve found myself a family.

You can find all of Tom’s outreach material on his website tomrocksmaths.com and you can follow all of his activities on social media via TwitterFacebook, YouTube and Instagram.

 

Answers to votes (watch by clicking the links):

  1. What is the probability I have the same PIN as someone else?
  2. How does modular arithmetic work?
  3. What would be the Earth’s gravitational field if it were hollow?
  4. What are grad, div and curl? COMING SOON

What are the chances that two England teammates share a Birthday?

Cast your mind back to the summer of 2018… we saw the warmest ever weather in the UK, Brexit was not yet a complete and utter disaster, and seemingly against all the odds the England football team reached the semi-finals of the World Cup for the first time since 1990. No doubt the team had a huge celebration together afterwards – but it wouldn’t be the first time that two of them have celebrated an occasion at the same time. As well as playing together at the heart of England’s defence, Manchester City duo Kyle Walker and John Stones also share the same birthday! Stones was born on 28th May 1994, making him 24 years old; Walker was born on the same day in 1990, meaning that he is exactly four years older than his teammate. How strange! Or is it…?

John_Stones_2018-06-13_1 Kyle_Walker

On the face of it, it seems quite surprising that in an England squad of just 23 players, two of them happen to share a birthday. However, as we’re about to see, this isn’t a freakish coincidence – maths says that it’s quite likely! What we’re talking about here is commonly known as the birthday problem: if there are a group of people of a certain size, what is the likelihood that at least two of them have the same birthday?

Let’s start by saying that we have a group of N people, and assume that birthdays are equally likely on every day of the year. (There is some evidence to suggest that this isn’t the case for top athletes; some say that they tend to be born early in the school year, such as around September in England. This is because they are slightly older than the other children in the year, and so they have a slight head-start in their physical development. However, we don’t want to make things too complicated, so we’ll ignore that for now.)

The easiest way to think about the problem is to first try to work out what the probability is that none of the N people share a birthday. Suppose our N people walk into a room, that is empty at first, one at a time. When the first person walks in, it’s obvious that they don’t share a birthday with anyone else in the room, because there isn’t anyone else. Therefore, they have the maximum probability of not sharing a birthday with anyone else in the room, which is 1.

Now think of the second person who walks in. The only way that they could share a birthday with someone in the room is if it happens to be exactly the same day as the first person. That means there is a 1 in 365 chance that they do share a birthday, so there is a 364 in 365 chance that they don’t.

Suppose that the first two birthdays don’t match, and then the third person walks in. They now have 2 days that they can’t share a birthday with, so there are 363 possible choices out of 365. Because we assumed that the first two didn’t match, we multiply the probabilities, so now the chance that none of them share a birthday is (364/365) * (363/365).

We can repeat this process until we get to our final person, number N. For example, the fourth person has 3 birthdays that they cannot share, so we multiply by a chance of 362/365; the fifth person has 4 days to avoid, so we include a probability of 361/365… By the time the Nth person walks in, there are N-1 people already in the room, so there are N-1 days that their birthday cannot fall on. This leaves them with 365-(N-1) possibilities out of 365.

To work out the total probability, we multiply all of these terms together which gives the likelihood that none of the N people share a birthday as

1 * (364/365) * (363/365) * (362/365) * … * ((365-(N-1))/365).

You might be thinking that this still looks like quite a big probability that none of them share a birthday, because all of the terms are very close to 1. But, if we try some values of N in a calculator, then it tells a very different story. (The percentages are calculated by finding the probability from the equation above and multiplying by 100.)

When N = 10, we get an 88% chance that none of them share a birthday. However, this drops down to 59% when there are N = 20 people. When we get to N = 23, the number of players in the England squad, the probability reaches just under 50%. That means that, incredibly, the likelihood that at least two of the 23 people share a birthday is just bigger than 50%!

So, in a random group of 23 people, it’s more likely than not that two of them share a birthday! This seems very strange at first; surely you’d need more than 23 people for a shared birthday to be more likely than not?! This is why the problem is commonly known as the birthday paradox – it might be very hard to get your head around, but the maths doesn’t lie!

Perhaps, in order to convince ourselves, we should look at some real-life examples. This is where the World Cup squads come into play: each team is restricted to bringing 23 players to the tournament. (We’ve seen that number before…) If our calculations above are correct, then if we picked any one of the World Cup squads, there would be roughly a 50:50 chance that at least two of the squad members share a birthday, which means that out of all of the squads that went to Russia, we would expect about half of them to have a birthday match. Well, let’s take a look…

Of the 32 teams, which were divided into 8 groups of 4, the following teams have at least one pair of players who share a birthday:

Group A Russia
Group B Iran, Morocco, Portugal, Spain
Group C Australia, France, Peru
Group D Croatia, Nigeria
Group E Brazil, Costa Rica
Group F Germany, South Korea
Group G England
Group H Poland

So, not only is there at least one team in every group with a birthday match, but if we count the total, there are 16 squads with a shared birthday pair – exactly half of the teams! The experimental results have matched up with the mathematical theory to perfection. Hopefully that’s enough to convince you that our calculations were indeed sound!

A slightly different question that you might ask is as follows: if I am in a group with a certain number of people, what are the chances that at least one of them shares my birthday? Is it the same idea? What we have worked out above is the probability that any two people in the room share a birthday (or rather, we worked out the opposite, but we can find the right answer from our working). Note that the pair doesn’t necessarily include you; it’s a lot more likely that it’s some other pair in the group.

In order to work out the answer to this similar sounding question, we work the other way around again, by calculating the probability that none of the N people share my birthday. For each of the N people, there is only one birthday that they cannot have, and that is mine (14th November, in case you were wondering), which means there are 364 out of 365 possibilities for each person. We no longer care whether their birthdays match up; we only care if they match with mine. So each person has a 364/365 chance of not sharing my birthday; and the overall probability is just 364/365 * 364/365 * … * 364/365, N times, which we write as (364/365)N.

Once again, we can plug some values of N into a calculator: N = 10 gives a 97% chance that no-one else has my birthday. For N = 50 the probability is still very high: there is an 87% chance that none of these 50 people have the same birthday as me. N = 100 gives 76%; N = 200 gives 58%; you have to go all the way to N = 253 before the probability dips below 50%, and it becomes more likely than not that at least one person will celebrate their birthday with me.

Applying this idea to all 736 players (32 squads of 23 players) involved in the World Cup, we should expect around 3 of them to have been born on the same day as me – 14th November. And I am very happy to confirm that France’s Samuel Umtiti, Switzerland’s Roman Burki, and Belgium’s Thomas Vermaelen all have what is undoubtedly the best birthday of the year… Two similar problems with two very different solutions!

Thomas_Vermaelen_2018 Samuel_Umtiti_2018AUT_vs._SUI_2015-11-17_(250)

You can check which footballers share a birthday with you at www.famousbirthdays.com/date/monthDD-soccerplayer.html, where you enter the month in words and the day in numbers (no preceding zero required).

Kai Laddiman 

Just give me a second…

On the 30th June 2015 an extra second was added to clocks across the world. Seeing as you now have all of this extra time, here’s everything you need to know about the leap second…

  • The leap second arises because the atomic clocks that we use today are actually more accurate than the earth at time keeping – one million times more accurate to be exact.
  • Changes in the Earth’s orbit are influenced by a number of factors: from an occasional wobble to a gradual slowing of its rotation. This causes the Earth to speed up and slow down unpredictably and is the reason why we need to add leap seconds.
  • A total of 27 leap seconds have been added since 1972 when the idea was first introduced.
  • The last leap second was added at midnight on December 31st 2016, but due to the unpredictability of the Earth’s orbit I can’t actually tell you when the next one will be!
  • Don’t worry though, all electronic devices are updated automatically so long as they’re connected to the internet.

You can listen to the 2015 announcement with the Naked Scientists here.

Carnival of Mathematics 167

Next month (March 2019) I will be hosting the ‘Carnival of Mathematics’ – a monthly blogging round up hosted by a different blog each month and organised by the Aperiodical.

The Carnival of Mathematics accepts any mathematics-related blog posts, YouTube videos or other online content posted during the previous month (February 2019): explanations of serious mathematics, puzzles, writing about mathematics education, mathematical anecdotes, refutations of bad mathematics, applications, reviews, etc. Sufficiently mathematized portions of other disciplines are also acceptable. Links to the previous monthly posts and a FAQ section can be found on the Aperiodical website here.

The deadline to submit your posts will be the 1st March 2019.

Click here to submit an idea!

This incarnation will be the 167th Carnival of Mathematics so here are some fun facts about the number 167…

  • 167 is the only prime number that cannot be expressed as the sum of 7 or fewer cube numbers.
  • 167 is the number of tennis titles won by Martina Navratilova – an all-time record for men or women.
  • 167P/CINEOS is a periodic comet in our solar system.
  • M167 Vulcan is a towed short-range air defence gun.
  • 167 is the London bus route from Ilford to Loughton.

The previous Carnival can be found at Math with Bad Drawings hosted by Ben.

My favourite Carnival is number 146 which featured Tom Rocks Maths for the first time!

Climate Change will increase Turbulence on Flights

We’ve seen many recent extreme weather events – from mudslides in Columbia to flooding in Australia – which scientists say are a consequence of climate change; but it’s not just the weather that is affected. The Earth’s atmosphere is made up of several layers of air which all flow around each other in patterns known as jet streams and an increase in temperature will cause these to speed up. This is bad news for air passengers, including the 1 million people currently airborne at this very instant, because an increase in the speed of the jet streams will cause more turbulence making flying less comfortable and potentially more dangerous. I spoke to atmospheric scientist Paul Williams…

  • Climate change will cause a 59% increase in light turbulence, 94% increase in moderate turbulence, and 140%  increase in severe turbulence.
  • Turbulence is measured on a scale from 1 to 7 where 1 means light turbulence, 3 means moderate, 5 means severe, and 7 means extreme.
  • Light turbulence is a slight strain against the seat belt, moderate turbulence causes unsecured objects to become dislodged and makes walking around difficult, and severe turbulence results in anything that isn’t strapped down being catapulted around the cabin.
  • Turbulence is caused by wind shear – the higher you go up into the atmosphere the windier it gets – and instabilities within these layers of shear generate turbulence.
  • As the atmosphere is heated, the temperature increase causes the jet streams to move faster, creating more wind shear and thus more turbulence.
  • The researchers hope that results such as this will encourage us to think more carefully about our carbon footprint as there are likely many effects of Climate Change that we do not know about.

You can listen to the full interview for the Naked Scientists here.

Size matters when it comes to speed

How fast should an animal be able to move? And why are the biggest animals, which pack more muscle, not the fastest? That’s what Yale scientist Walter Jetz was wondering, so he and his colleagues looked at hundreds of animal species and have come up with a new theory that successfully puts a speed limit on most species…

  • There is a theoretical maximum speed that is expected to increase with body size,  however, in order to actually get to any speed you need to first accelerate, and larger animals take much longer to do so – much like a truck accelerating to 60mph compared to a motorbike or car.
  • Large bodied animals simply do not have sufficient energy to reach their theoretical maximum speed.
  • The general distribution is a ‘hump-shape’ as shown in the plots below. Maximum speed increases with size until we reach a critical mass beyond which the maximum speed reached starts to decrease.

screen shot 2019-01-24 at 10.59.30

  • Data for over 450 species were included in the study, across land, air and water.
  • The study provides insight into evolutionary trade-offs for different species as they evolve to increase their chances of survival.

You can listen to the full interview with the Naked Scientists here.

Image copyright Dawn Key

 

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