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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How Plesiosaurs Ruled the Ocean using their Flippers

Plesiosaurs ruled the oceans during the time of the dinosaurs with specially adapted flippers that enabled them to swim faster and with greater efficiency than any other animal. Luke Muscutt studied the 4-flipper arrangement by conducting experiments at the University of Southampton to investigate exactly how it all worked…

With thanks to the UK Fluids Network and the Journal of Fluid Mechanics for supporting this video.

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Abel Prize Laureate 2019: Karen Uhlenbeck

Karen Uhlenbeck was selected by a committee of five mathematicians nominated by the European Mathematical Society and the International Mathematics Union. Her work involves the study of partial differential equations, calculus of variations, gauge theory, topological quantum field theory, and integrable systems. The full citation from the announcement can be found here and a short biography by Jim Al-Khalili here.

“Karen Uhlenbeck receives the Abel Prize 2019 for her fundamental work in geometric analysis and gauge theory, which has dramatically changed the mathematical landscape. Her theories have revolutionised our understanding of minimal surfaces, such as those formed by soap bubbles, and more general minimisation problems in higher dimensions.” – Hans Munthe-Kaas, Chair of the Abel Committee.

Karen’s work covers minimisation problems, such as solving for the shape of a soap bubble acting to minimise its energy under gravity. Here’s a fantastic slow-motion experiment from Ray Goldstein at the University of Cambridge demonstrating the change in the shape of a soap bubble as the two supporting wires are pulled apart.

Karen also works in topological quantum field theory which has very important consequences for physicists, not least in relation to the Yang-Mills Mass Gap Hypothesis – one of the 7 million-dollar Millennium Problems. You can read more about the problem here.

“If I really understand something, I’m bored.” Karen Uhlenbeck

Throughout her career Karen has been very active in the area of mentorship and furthering the cause of women in mathematics. She is the founder of the Institute of Advanced Study Women’s Program, now entering its 25th year, and the Park City Mathematics Institute Summer Session, which places a huge emphasis on interdisciplinary research and collaboration between mathematicians from all areas.

The Abel Prize was established on 1 January 2002 – 200 years after the birth of Niels Henrik Abel. The purpose is to award the Abel Prize for outstanding scientific work in the field of mathematics. The prize amount is 6 million NOK (about 750,000 Euro) and was awarded for the first time on 3 June 2003.

You can read the official announcement from the Norwegian Academy of Science and Letters here.

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STEM for Britain Competition

On Wednesday March 13th I’ll be presenting my research to MP’s at the Houses of Parliament in the final of the STEM for Britain Competition. You can find my research poster on modelling the spread of pollution in the oceans here.

Read coverage of my entry by the Oxford Maths Institute, St Edmund Hall, St Hugh’s College and the Warrington Guardian. The press release from the London Mathematical Society is also copied below.

Dr Tom Crawford, 29, a mathematician at Oxford University hailing from Warrington, is attending Parliament to present his mathematics research to a range of politicians and a panel of expert judges, as part of STEM for BRITAIN on Wednesday 13th March.

Tom’s poster on research about the spread of pollution in the ocean will be judged against dozens of other scientists’ research in the only national competition of its kind.

Tom was shortlisted from hundreds of applicants to appear in Parliament.

On presenting his research in Parliament, he said, “I want to bring maths to as wide an audience as possible and having the opportunity to talk about my work with MP’s – and hopefully show them that maths isn’t as scary as they might think – is fantastic!”

Stephen Metcalfe MP, Chairman of the Parliamentary and Scientific Committee, said: “This annual competition is an important date in the parliamentary calendar because it gives MPs an opportunity to speak to a wide range of the country’s best young researchers.

“These early career engineers, mathematicians and scientists are the architects of our future and STEM for BRITAIN is politicians’ best opportunity to meet them and understand their work.”

Tom’s research has been entered into the mathematical sciences session of the competition, which will end in a gold, silver and bronze prize-giving ceremony.

Judged by leading academics, the gold medalist receives £2,000, while silver and bronze receive £1,250 and £750 respectively.

The Parliamentary and Scientific Committee runs the event in collaboration with the Royal Academy of Engineering, the Royal Society of Chemistry, the Institute of Physics, the Royal Society of Biology, The Physiological Society and the Council for the Mathematical Sciences, with financial support from the Clay Mathematics Institute, United Kingdom Research and Innovation, Warwick Manufacturing Group, Society of Chemical Industry, the Nutrition Society, Institute of Biomedical Science the Heilbronn Institute for Mathematical Research, and the Comino Foundation.

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LMS Holgate Lecturer

Very happy to announce my appointment as a Holgate Lecturer with the London Mathematical Society (LMS). The position means that the LMS are supporting my outreach work for the next 4 years so all you have to do if you want me to come and give a talk/run a workshop at your school is to get in touch here.

You can find out more about the details of the scheme on the LMS website – and make sure you check out the other amazing speakers.

If you’re not already excited about the prospect of Tom Rocks Maths coming to your school then here are some examples sessions to really get you in the mood for some maths!

1. Maths v Sport (Y9 onwards)
How do you take the perfect penalty? What is the limit of human endurance? Where is the best place to attempt a world record? Maths has all of the answers and I’ll be telling you how to use it to be better at sport (results may vary).

2. Maths: it’s all Greek to me! (Y9 onwards)
You’ve probably heard of Pythagoras, Archimedes and Plato, but do you know the sins behind their stories? From murder and deceit to running naked down the street, the Ancient Greek mathematicians were anything but boring. I’ll be telling you all about their mischief – mathematical or otherwise – as I bring the history of maths to life (featuring live experiments and togas).

3. The Millennium Problems (Y10 onwards)
The seven greatest unsolved problems in mathematics, each worth a cool $1 million… In this session I’ll introduce each of the puzzles in turn and try to give you a feel for the maths that you’ll need to know if you’re planning to take one of them on.

4. Navier-Stokes Stripped (Y12 onwards)
The Navier-Stokes equations model the flow of every fluid on Earth, but yet we know very little about them. So little in fact, there is currently a $1 million prize for anyone that can help to improve our understanding of how these fascinating equations work. In this session, I’ll strip back the Navier-Stokes equations layer-by-layer to make them understandable for all… Based on my hit YouTube series ‘Equations Stripped’.

5. How to make everything about maths (Teachers)
Since completing my PhD, I have transitioned from maths researcher to maths communicator with the launch of my outreach programme ‘Tom Rocks Maths’. In this session I will discuss the most successful ways to increase engagement with maths through examples from my work with the BBC, the Naked Scientists, and from my YouTube channel, website and social media pages @tomrocksmaths.

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Carnival of Mathematics 167

You know you’ve made it as a maths communicator when you have the honour of hosting the Carnival of Mathematics (if you have no idea who I am or what I do then check out this interview for St Hugh’s College Oxford). But, before we get to the Carnival proper, as the creator of ‘Funbers’ I can’t help but kick things off with 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 the name of a periodic comet in our solar system
  • The M167 Vulcan is a towed short-range air defence gun
  • 167 is the London bus route from Ilford to Loughton

Now that we all have a new-found appreciation for the number 167, I present to you the 167th Carnival of Mathematics…

Screen Shot 2019-03-06 at 00.15.32

Reddit’s infamous theydidthemath page tackles ‘fake news’ on Instagram with a quite brilliant response to a post claiming that avoiding eating 1 beef burger will save enough water for you to shower for 3.5 years. Whilst the claim is hugely exaggerated we should still probably stop eating beef…

Next up, Singapore Maths Plus take a light-hearted look at the definition of ‘Singapore Math’ on Urban Dictionary – which is apparently the world’s number one online dictionary (sounds like more ‘fake news’ to me).

Math off the grid jumps in ahead of hosting next month’s Carnival to discuss the book ‘Geometry Revisited’ with a re-examination of the sine function as a tool for proving many fundamental geometric results. Scott Farrar also has the sine bug as he encourages us not to reject imprecise sine waves, but instead to consider the circle that they would form (warning contains a fantastic GIF).

soviet_license_plate

John D Cook introduces what is now my new favourite game with his explanation of the ‘Soviet Licence Plate Game’. Have a go at the one to the right – can you make the four numbers 6 0 6 9 into a correct mathematical statement by only adding mathematical symbols such as +, -, *, /, ! etc. ? Send your answers to me @tomrocksmaths on Social Media or using the contact form on my website.

If by this point, you’ve had enough of numbers (which apparently happens to some people?!), then here’s a lovely discussion of ‘numberless word problems’ from Teaching to the beat of a different drummer. If that doesn’t take your fancy, how about some group theory combined with poetry via this ridiculous video of Spike Milligan on The Aperiodical…

If like me you’re still not really sure what you’ve just watched, then let’s get back to more familiar surroundings with some intense factorial manipulation courtesy of bit-player. What happens when you divide instead of multiply in n factorial? The result is truly mind-blowing.

koncnaslika-krog

Finding our way back to applications in the real world, have you ever wondered how the photo effect called ‘Tiny Planets’ works? Well, you’re in luck because Cor Mathematics has done the hard work for us and created some awesome mini-worlds in the process!

Sticking with the real world, Nautilus talks to Computer Scientist Craig Kaplan who discusses how the imperfections of the real world help him to overcome the limitations of mathematics when creating seemingly impossible shapes. They truly are a sight to behold.

With our feet now firmly planted in reality, let’s take a well-known mathematical curiosity – say the Birthday Problem – and apply it to the 23-man squad of the England men’s football team from the 2018 World Cup. Most of you probably know where this one is going, but it’s still fascinating to see it play out with such a nice example from Tom Rocks Maths intern Kai Laddiman.

The fun doesn’t stop there as we head over to Interactive Mathematics to play with space-filling curves, though Mathematical Enchantments take a more pensive approach as they mourn the death of the tenth Heegner Number.

Focusing on mathematicians, Katie Steckles talks all things Emmy Noether over at the Heidelberg Laureate Forum Blog, whilst I had the pleasure of interviewing recent Fields Medal winner Alessio Figalli about what it feels like to win the biggest prize of all…

And for the grand finale, here are some particularly February-themed posts…

The next Carnival of Mathematics will feature mathematical marvels posted online during the month of March, which of course means ‘Pi Day’ and all the madness that follows. Good luck to the next host ‘Math off the grid’ sorting through what will no doubt be an uncountably large number of fantastic submissions!

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What is the blast radius of an atomic bomb?

Picture the scene: you’re a scientist working for the US military in the early 1940’s and you’ve just been tasked with calculating the blast radius of this incredibly powerful new weapon called an ‘atomic bomb’. Apparently, the plan is to use it to attack the enemies of the United States, but you want to make sure that when it goes off any friendly soldiers are a safe distance away. How do you work out the size of the fireball?

One solution might be to do a series of experiments. Set off several bombs of different sizes, weights, strengths and measure the size of the blast to see how each property affects the distance the fireball travels. This is exactly what the US military did (see images below for examples of the data collected).

Picture1 Picture2

These experiments led the scientists to conclude that were three major variables that have an effect on the radius of the explosion. Number 1 – time. The longer the time after the explosion, the further the fireball will have travelled. Number 2 – energy. Perhaps as expected, increasing the energy of the explosion leads to an increased fireball radius. The third and final variable was a little less obvious – air density. For a higher air density the resultant fireball is smaller. If you think of density as how ‘thick’ the air feels, then a higher air density will slow down the fireball faster and therefore cause it to stop at a shorter distance.

Now, the exact relationship between these three variables, time t, energy E, density p, and the radius r of the fireball, was a closely guarded military secret. To be able to accurately predict how a 5% increase in the energy of a bomb will affect the radius of the explosion you need a lot of data. Which ultimately means carrying out a lot of experiments. That is, unless you happen to be a British mathematician named G. I. Taylor…

Taylor worked in the field of fluid mechanics – the study of the motion of liquids, gases and some solids such as ice, which behave like a fluid. On hearing of the destructive and dangerous experiments being conducted in the US, Taylor set out to solve the problem instead using maths. His ingenious approach was to use the method of scaling analysis. For the three variables identified as having an important effect on the blast radius, we have the following units:

Time = [T],       Energy = [M L2 T-2],      Density = [M L-3],

where T represents time in seconds, M represents mass in kilograms and L represents distance in metres. The quantity that we want to work out – the radius of the explosion – also has units of length L in metres. Taylor’s idea was to simply multiply the units of the three variables together in such a way that he obtained an answer with units of length L. Since there is only one way to do this using the three given variables, the answer must tell you exactly how the fireball radius depends on these parameters! It may sound like magic, but let’s give it a go and see how we get on.

To eliminate M, we must divide energy by density (this is the only way to do this):

eqn1.png

Now to eliminate T we must multiply by time squared (again this is the only option without changing the two variables we have already used):

eqn2.png

And finally, taking the whole equation to the power of 1/5 we get an answer with units equal to length L:

eqn3

This gives the final result that can be used to calculate the radius r of the fireball created by an exploding atomic bomb:

eqn4

And that’s it! At the time this equation was deemed top secret by the US military and the fact that Taylor was able to work it out by simply considering the units caused great embarrassment for our friends across the pond.

I love this story because it demonstrates the immense power of the technique of scaling analysis in mathematical modelling and in science in general. Units can often be seen as an afterthought or as a secondary part of a problem but as we’ve seen here they actually contain a lot of very important information that can be used to deduce the solution to an equation without the need to conduct any experiments or perform any in-depth calculations. This is a particularly important skill in higher level study of maths and science at university, as for many problems the equations will be too difficult for you to solve explicitly and you have to rely on techniques such as this to be able to gain some insight into the solution.

If you’re yet to be convinced just how amazing scaling analysis is, check out an article here explaining the use of scaling analysis in my PhD thesis on river outflows into the ocean.

And if that doesn’t do it, then I wish you the best of luck with those atomic bomb experiments…

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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 Twitter, Facebook, 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
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