I recently took part in a Reddit AMA (Ask Me Anything) on the subject of Mathematics. Here’s what we discussed…

tomrocksmaths

Hi all! I’ll be here throughout the day answering any questions you have about maths – whether related to disease modelling or not – and I’ll try my very best to explain things as simply as possible. This should be fun!!

M3CCA8

I have a question about the Banach-Tarksi paradox. The paradox is that you could make duplicates of equal size to the original by separating the data sets by direction. Basically that the sum of all parts is greater than the whole. However i was shown a 3-dimensional visual and in that context i don’t believe the paradox would hold true because of the nature of up/down/left/right requires a frame of reference. There by ruining the methodology that allows the paradox to occur. I was wondering if you could explain whether or not I’m correct in my thinking.

CarryThe2

Banach Tarski absolutely holds with a frame of reference, that is after all, all that co ordinates systems are!

The gist of banach tarski is thst certain infinite chains of transformations don’t preserve volume, it has no basis in the real world because things aren’t infinitely divisable in reality.

M3CCA8

Thank you

tomrocksmaths

I would agree very much with the idea that infinite chains of transformations aren’t possible in the real world. To get a feel for how weird infinity is, just take the simple example of 0/0. It is equal to absolutely anything you want by just taking the limit in a different way. This is not too far from the kind of ideas that are present with Banach Tarski (or at least how I think about them).

BlazingTrail42

Hey! I’ve recently obtained a place at Oxford to study maths & philosophy, what would you say the best prep would be that I could do in quarantine?

tomrocksmaths

Congratulations!! I hope to see you around next year. I can’t speak for Philosophy, but for Maths we have a set of problems that some colleges send to incoming students here: https://www.maths.ox.ac.uk/study-here/undergraduate-study/practice-problems I’d start there and see how you get on.

BlazingTrail42

Fapitalismm

What’s your favourite maths-based fun fact?

tomrocksmaths

I’d have to say the following formula for pi. Take all of the prime numbers and for each one write it as a fraction: prime/(nearest multiple of 4). If you multiply all of these fractions together you get pi/4. I have no idea why…

CarryThe2

Is there some instinctive reason why pi/4 seems to come up more often than pi?

175gr

Sounds like an Euler product. I’m not sure exactly where pi/4 comes from, but that looks like the central value of a Dirichlet L-function.

If I were asked to prove the pi/4 part, I would probably look at the analytic class number formula.

I like this math fact! The tip of the iceberg for a lot of cool number theory stuff.

Fapitalismm

That’s… ok, I did not know that.

Fun fact indeed! 🙂

aeiouicup

Fav mathematician?

tomrocksmaths

Not very original but has to be Euler. That man was a machine. There isn’t a single course I teach (and I teach everything in first and second year undergraduate maths) that doesn’t have a theorem named after him.

tomrocksmaths

Also not really a mathematician per se, but shout out to Richard Feynman just for the charisma. Interesting people make whatever they are talking about more interesting too.

JKTKops

Euler for results, but Galois for story.

CarryThe2

Galois would absolutely be as prominent as Euler if he’d lived longer.

JKTKops

Agreed. I wonder sometimes how much more we would know.

NerdComplex

What advice would you like to give to a 16 year old interested in mathematics?

tomrocksmaths

Practice, practice, practice. Being able to do algebra without making mistakes, being able to recall definitions from memory, being able to work through a wide variety of different style problems – these are all essential for any mathematician. Beyond this, just keep working hard at school and watch videos on channels such as Numberphile, 3blue1brown etc. so that you are exposed to a wide range of topics. Don’t worry if you don’t follow everything – you’re not expected to – that will all come in time.

NerdComplex

Neat! I also maintain a journal filled with goodies from numberphile,Mathigon and futilitycloset (awfully cool website) to keep me engaged when I’m bored. Thank you so much Dr.Tom! Folks like you really help keep the spirit of curiosity alive!

RhynoD

And /u/tomrocksmaths‘ own channel, of course!

Personally, I am also a fan of standupmaths, vihart, PBS Infinite Series, and Crash Course.

tomrocksmaths

I second all of these suggestions. Also singingbanana and mathologer.

NerdComplex

Of course Tom makes great content,but I’m still in high school and don’t understand that level of fluid mechanics;however his recent ventures like the strings and loops in pi and his math essay competition have been of great amusement to me.

CrashAndSideburns

What role do you think proofs play in the teaching of mathematics? Do all things taught need to be rigorously proven, or will an intuitive understanding of a principle, which forgoes rigour, suffice? Alternatively, is there a necessary balance to be struck between explanations in terms of rigorous proofs and of intuitive understanding? If so, what do you think that balance is?

Really interested to hear your thoughts on this one.

tomrocksmaths

Great question – and one I think to which you will get a different answer depending on whether you ask a ‘pure’ or ‘applied’ mathematician, though ultimately there is a balance.

As an applied mathematician, the intuition often comes first. I’m trying to solve a practical problem relating to fluid mechanics, and so I have some physical intution as to what should happen. For example, for my work on river outflows I expect that when there is an increase in the amount of water leaving a river, the coastal current that forms should be deeper. More water in the system means a ‘bigger’ current. But to prove this mathematically you have to start with a complicated set of governing equations (here Navier-Stokes) and then solve them after making various assumptions relative to the problem at hand. The answer is that yes the current depth increases with the volume of water leaving the river (to the power of 1/2 in fact) and so my intuition was indeed correct… in this instance.

I think there’s a lot to be said for first gaining an intuition for problems by playing around with them – visualisations for example are a great tool here – but ultimately you need a proof to be sure the results are correct mathematically. Just learning proofs without first having any grasp as to WHY certain results should be true isn’t very helpful in my opinion. You want to convince yourself that something is true – be it through intuition, examples, visualisation etc. – and then look for a proof. This is particularly true in research level mathematics where most of the time you don’t actually know if a proof should exist in the first place!

Xefoxmusic

Hey Tom, I had the pleasure of working with you in Shanghai 2 years ago, it’s great to see your continued and well-deserved success!

I know you’re particularly keen on Navier-Stokes, so I won’t ask your favourite Millennium Prize Problem. Instead – which Millennium Prize Problem do you think will be solved next?

tomrocksmaths

Hey – great to hear from you! Hmmm… this is a tricky question. I know of recent progress in most of them, but the one that got me most excited was a lecture by Sir Andrew Wiles 2 years ago when he talked about the Birch and Swinnerton-Dyer Conjecture. Whilst he didn’t admit to be working on it himself, he certainly knew a LOT about what was happening in the field. Given it’s close relationship with elliptic curves and his proof of Fermat’s Last Theorem I wouldn’t bet against him solving that in the near future!

NerdComplex

Do you know of any websites like Mathigon? They’re really killing it!

tomrocksmaths

Mathigon is awesome – shout out to the amazing Philipp for creating it! In terms of similar sites, I honestly don’t know. I tend to watch YouTube videos for my online math content, with the occasional textbook thrown in for good measure.

NerdComplex

What’s your way to celebrate math/nerd culture?

tomrocksmaths

Erm tattoos??? I have 58 currently, 10 of which are maths-themed and about 20 related to video games/cartoons.

NerdComplex

Whoaa can I look?

tomrocksmaths

I’ve just entered a photography exhibition at the BRIDGES Math Art Conference where I hope to display some photographs of them – if the exhibit is accepted I’ll post pics for everyone online 🙂

Splatpope

i’d upvote -1/12 times if I could

n00neperfect

Which field of mathematics, equation haunted you mostly, make you confused or seems unanswered now and then? Thanks

tomrocksmaths

If I’ve understood your question correctly (and I’ve interpreted it as which areas of math do I find most difficult) them for me personally it would be anything that is truly abstract, in the sense that I cannot visualise what is happening. When trying to understand a problem the first thing I do is draw a diagram/picture – and I encourage all of my students to do the same. This even works with abstract areas such as Linear Algebra through mapping diagrams etc. But for some areas – and I’m mainly looking at you higher dimensional topology – visualisation is no longer possible and this is where I sometimes struggle to really understand what is going on. It’s not that everything needs to be visualised to understand it, just that for me, it’s a very powerful tool to help my understanding.

n00neperfect

If I’ve understood your question correctly (and I’ve interpreted it as which areas of math do I find most difficult)

yeah right that’s what i mean….. sure the higher dimensions visualizations will be challenging and uncertain. Thanks for response. 🙂

tomrocksmaths

Well that would be an entire 8-lecture course that I teach to my second year undergraduate students… The key thing with Laplace Transforms are they are a tool to turn ODEs (generally hard to solve, particularly if non-linear) into algebraic equations (much easier to solve). The solution to the original ODE then comes from inverting the Laplace Transform. In other words ODE –> LT –> solve algebra –> invert LT –> ODE solution. It’s a VERY powerful tool.

Thank you very much, that really helps me since I’m a first year undergraduate engineering student coming to grips with Laplace transformations

andbm

What is your opinion on the typically platonic way math is taught, i.e. “math exists in the ideal world, not in the real world”? As a scientist, are you opposed to this dualism?

tomrocksmaths

I’d say that’s simply not true. Sure, there are lots of things being studied now that are very abstract and ideological, but i have no doubt they will have uses in the future. Remember that Einstein’s work on relativity had no practical use for over 100 years until the invention of GPS. Now it’s essential for pretty much any location system, as without relativistic corrections it would only be accurate to 6 miles!

andbm

Thank you for your answer. So you would consider that the applicability of math shows that it exists in reality, e.g. that the perfect circle is not only an abstract object but a real object?

It’s mostly a philosophical question. I’m asking from the perspective of theoretical physics, where math is very obviously useful in describing the real world, but the question is whether mathematical concepts, such as the circle, should be considered “outside” the physical reality, as I would argue it is usually taught, or “inside” the physical reality, which would be the consequence of rejecting dualism.

tomrocksmaths

Ah, I get you now… I suppose you could argue that a ‘perfect circle’ will never exist in reality, just like a ‘perfect right-angle’ will never exist (reference to Navier-Stokes video), as no matter what scale you go down to there will be imperfections. In that case then yes I can see why you might treat such things as only existing in the ‘ideal world’. I interviewed the mathematician and author Prof. Ian Stewart about using maths to communicate with aliens and we touched on some similar concepts – I’ll post the link in case you might find it interesting: https://tomrocksmaths.com/2017/05/18/would-aliens-understand-maths/

GhassanZ

Do you think anyone could develop a mind for numbers? Or would most of us work only to make math tolerable?

tomrocksmaths

Oh no Math is 100% a skill that you can learn – just like any other. And just like any other skill, there may well be a limit to what you can achieve. For example not everyone can play the guitar like Tom Morello, but we can all get to a competent level and beyond with enough practice and dedication. I’m actually making a video with Mike Boyd (Quicklearn on YouTube) where I’m going to teach him how to pass the Oxford Maths entrance exam. Mike learns new skills every day on his channel (its very entertaining) and we both want to show that Maths can be learnt just like any other skill. We are hoping to film over the summer so look out for it if you’re interested.

anooblol

Thoughts on how machine learning / AI will impact or change the landscape of pure math?

tomrocksmaths

Ooooooo this ones a good’un… There are already ‘proof writing’ programs that exist and are used to prove certain results, BUT its quite controversial amongst mathematicians as to whether or not computer-aided proofs should be allowed. I interviewed Thomas Hales a few years ago after his computer-assisted proof for the Kepler Conjecture (what is the best way to stack spheres in a box) was finally accepted. I think it took almost 10 years to be ratified and there are some that don’t think it should be allowed.

In short, AI will certainly lead to new results being proved, but whether or not the community deems them acceptable remains to be seen. I don’t see any problem with them personally – if the proof is correct who cares how it was obtained.

NerdComplex

What are your views on entrance exam based institutions?I think the spirit of learning is lost when approaching with competition in mind.

tomrocksmaths

A valid point, but I also counter with the idea that we have to differentiate between prospective students somehow and perhaps unfortunately testing seems to be the best method we have to do that right now. I would also add that as someone who is involved with maths admissions at Oxford, the test score itself is not the be all and end all of an application. We interview people with very low scores if they show potential elsewhere in their application and there are several examples of students being offered places with test scores way below the average mark. We really do consider the application as a while package, not just the test score. (Though this of course is only valid for Oxford where we interview)

NerdComplex

How did you get to where you are today,Dr.Tom?

tomrocksmaths

I guess all I can do is tell my story… I loved maths at school, worked hard, and got good grades. I applied to Oxford and made sure that I prepared well for the test and interview by doing lots of practice questions. Fortunately they offered me a place. In my second year I took a Fluids course and loved it, so began taking more and more options in fluids. When I decided I wanted to do a PhD I asked my tutor for advice and he told me to go to Cambridge. I looked up some of the projects and working in a lab sounded awesome so I met with Paul Linden who then became my supervisor. After my PhD I worked as a science journalist at the BBC for 12 months which is where I learned how to communicate complex ideas effectively. I then set up my own channel and was offered a teaching position at Oxford to go alongside my outreach work.

babahond

What careers might a math enthusiast hope to get into?

tomrocksmaths

Basically anything you want. I’ve been to countless careers talks where employers just talk about how great mathematicians are to employ. Maths teaches you how to solve problems and this is one of the most valuable skills in almost any industry.

babahond

Thanks for the motivation!

6footstogie

the CIO of the company I work for has a masters in math. a few dbas have b.s. in math. many opportunities out there for you.

babahond

I got hope. Thanks brother.

Choib0i

I’m interested in sports betting and the math that goes into developing models people use to try to find an edge. Where could i learn the statistics skills necessary to begin working on my own model?

tomrocksmaths

You want to start with Probability and Statistics. All predictive models start with probability theory as they will need to incorporate some element of randomness into them. Once you have a good understanding of probability then the statistical models will make much more sense.

bishhhhhhhh

Hi Tom,

For someone who was really poor in math in highschool and actively avoided it in university, but now finds themselves in a finance job dealing in an investments industry, where would you recommend getting started?

I truly believed I didn’t have the mind for math all those years but I think my brain has matured and my ability to concentrate has increased, and I’m looking to get my feet wet again. Both in general and specifically more financial mathematics. Any small pointers are greatly appreciated. 🙂

tomrocksmaths

Hey – I’m afraid this is a story I hear a lot. Math is often taught poorly at schools so if you don’t take to it instantly you very quickly lose interest and then you are too far behind to be able to catch up and everything seems impossible… but it’s great to hear that you want to give it a second go. As far as I can remember (I last did financial math in the third year of my undergraduate degree), most of the equations are based around derivatives of one kind or another, which means you need to go back and learn calculus. Calculus is basically the tool we use to calculate ‘rate of change’ and is essential to almost every model that we use today. I’d suggest starting with 3blue1brown’s excellent ‘Essence of Calculus’ series here: https://www.youtube.com/watch?v=WUvTyaaNkzM and see what piques your interest!

SimplySaturday

So I had had a comment here, but accidentally deleted it when I went to edit my comment, in case you think you’re seeing duplicate. Just some general questions on disease modelling.

Since SIR simplifies the complexities of the disease to a conveniently analyzed/numerically solved form, there are definitely more advanced models to capture real world effects e.g. spatial inhomogeneoties/distributions, time-varying recovery/infection time, age related effects, reinfection, complicance, and stochastic nature of the disease. With a limited, but growing set of data, but with complicated population distributions and government mandated social intervention, in your mathematical experience, which of these real world effects are

(1) the most difficult (choose an uncertainty metric I guess) to model/verify with today’s data and what about our data makes this endeavor so hard,

(2) the most crucial (choose a metric I guess) for capturing timescales of breaching specific critical thresholds (e.g. hospital overload, etc.), and how much “better” (again, whatever metric) is it compared to say how the store-brand (SIR) model does

As a follow-up, since most models are usually only good at onset or in the lategame, how much does the existence of endemic diseases (flu, malaria,…) affect applicability of our model when most of the population is recovering? And since most individual data is scarce, how does substituting general population data (like we have) quantitatively affect most of the modern models that probably focus on individual-ish responses?

Gimbeam

I’m so glad you asked this question because my mind was wandering in the same direction but couldn’t find the right people words to ask the question correctly

tomrocksmaths

(1). Spatial distribution can be modelled reasonably well (by adding diffusion to the systems in the form of spatial derivatives) as can incubation time (we actually do this on one of the problem sheets for the Oxford Math Biology course). Re-infection can also be added – you just have some proportion of your infectives, I, go back and join the susceptibles, S, rather than them all going to the removed category, R. I’m not too familiar with the stochastic nature of disease idea, but that seems by far the most difficult to me. Any method to introduce randomness into a model tends to cause things to get very complicated very quickly.

(2). Timescales are tricky, because they are based on the numbers of infectives, I, reaching some critical threshold which is very hard to define. Let’s say for example, the UK has 12,000 beds in intensive care across the country (I think that was the number mentioned by the BBC recently), then we would want to know when the number of infectives will reach 12,000 at a given point in time. This is similar to calculating I-max in the SIR model I discuss in my video, but crucially is entirely dependent on the value of the contact ratio, q. This value is what is SO difficult to determine, because not only is it affected by the disease itself (which we don’t yet know very much about), but also government interventions and whether people follow them, social norms eg. certain cultures share meals whilst others eat alone, immune system response etc. There are almost too many variables to be able to estimate q accurately, but we have to start somewhere.

(3). I’m not sure even the most up-to-date models can focus on individual responses, everything is applied to an ‘average’ person or specific group of people (eg. age 70+, those with long-term health conditions etc.). I think the key thing with all models is to remember the assumptions upon which they are based. They are only tools to aid with decision-making even when they are used properly – when they are taken out of context they are incredibly dangerous. See Black-Scholes and the 2008 financial crash for a perfect example of the misuse of a model.

Spamakin

Any advice for a budding mathematician who isn’t sure if they want to go into industry or academia? I know picking up programming is an invaluable skill but anything else I should know?

tomrocksmaths

Well, as someone who was in the exact same position during my undergraduate degree I can tell you what I did… First, I worked hard and made sure my grades were good enough that I could apply for PhD’s at top universities where I might want to study. Second, I spent my summer vacations doing internships in various different industries to get a taste of what it might be like to work there. I worked for a bank, an insurance company, a consultancy firm, and a non-academic research institution. Turns out I don’t like having a boss and working hours suck, so academia won. But I only knew that because I tried a few different things – hope that helps!

Spamakin

Do you think a major in applied math + a CS minor would be good for going into industry (in any field)? Also any tips for advertising yourself as a math major when applying for internships at like financial institutions or other such industries when you’re competing with people who are majoring in those specific fields?

tomrocksmaths

Math + CS sounds great to me. Also see below response: https://www.reddit.com/r/explainlikeimfive/comments/fn05i0/ama_mathematics/fl7amch?utm_source=share&utm_medium=web2x

oZOV1_L802B

I have a question about the SIR model you presented in your latest video. In Relation to S_0 is it generally assumed to be the entire population say of a given country minus those already infected / immune? Or does social distancing measures in effect decrease S_0 or does it only decrease the transition rate? Or some combination? Thank you I’m trying to understand the dynamics of the system. (Great Video btw)

tomrocksmaths

Glad you enjoyed the video 🙂

S0 is the initial number of susceptibles in a population. So you can think of this as the entire population minus those who cannot catch the disease for whatever reason. This could be immunity (very unlikely for COVID-19), complete isolation (eg. living on a rmeote island) or some other reason. The key thing is that S0 is fixed at the beginning of the model. This doesn’t necessarily have to be the exact moment the disease first appeared, you can starting point at which to initiate the model, but once it is running then S0 cannot be changed. It is a constant value, but one that is already fixed. The social distancing measures will decrease the transmission rate (labelled r in my model). Lowering r, means the contact ratio q = r / a is also lowered and as we see in the video this means fewer infections at any given point in time, and fewer infections overall.

oZOV1_L802B

Thanks for the response and that makes sense. When I was trying to fit the model to data in China I was getting some problems I thought because of the low ratio from infected to total population being somewhere around 80,000/1.4 billion. So I thought decreasing s_0 would account for that but also a low r would also account for very few infections in relation to the entire population at equilibrium. Thanks for the response!

kathryn_grace

Hi Dr. Crawford, Due to COVID-19 all of my classes have been moved online, including calc II. Unfortunately, I don’t do very well in math /science classes without lecture-style instruction. All my professor is doing is putting his notes online, and my brain needs more than that. Are there any resources you would recommend?

tomrocksmaths

Hey! For calculus you can’t get any better than 3blue1brown’s ‘Essence of Calculus’ series: https://www.youtube.com/watch?v=WUvTyaaNkzM to get a feel for the kind of intuition you need to solve calculus problems. Beyond that, it’s tricky as each university will have its own specific syllabus and therefore lectures will cover different content, or at the very least present it in different ways. Ultimately I think you have to stick to the notes provided by your professor and perhaps try to get inspired by some other topics from online videos that will help to increase you interest in the subject, and hopefully make it easier to study.

Macluawn

What’s a short proof that 1+1=2?

Assuming decimal, for simplicity.

tomrocksmaths

Haha I feel like the word ‘short’ is doing a lot of work here… It depends on what we are allowed to assume in terms of axioms etc. In the Oxford first year undergrad Analysis course we teach a proof that the square root of 2 exists which takes a lot longer and a lot more detail than you might imagine… and that’s about 5 or 6 lectures after starting with the basic axioms for the real numbers defined as a field (if you’re interested I made a video on these axioms here: https://youtu.be/9Efsz2hIpxE). So, in short, I would need to know where we are starting from!

Macluawn

In the Oxford first year undergrad Analysis course we teach a proof that the square root of 2 exists

Are video lectures available online to  non students?

tomrocksmaths

Video lectures I’m afraid not. Though this proof is on my list of upcoming video ideas so watch this space.

raindiioannis

How is Euclid’s 5th postulate not self-evident?

tomrocksmaths

Very glad you asked this because it allows me to point you in the direction of an amazing video one of my students made explaining what happens when you don’t assume it: https://www.youtube.com/watch?v=2dUVCswcYF8

raindiioannis

This is great. Thank you. I’m quite bad at maths even though I do enjoy learning more about it and what all of these processes you learn to do in school are supposed to point towards. I got offered a place for the Linguistics program, so if I see you around Oxford in October I’ll thank you personally.

tomrocksmaths

Amazing – all things being well see you in October!

anais9000

I have a question about Fermat’s Last Theorem. Let me phrase it in terms of the cubic. Say we have a cube and two (imaginary) smaller cubes in the upper left and lower right corners. We expand these two cubes at any rate until they touch, then keep expanding them (they pass through each other) until the volume of the cube of their intersection equals the volume of the interior of the large cube NOT occupied by either of the smaller two cubes. Then obviously the sum of the volumes of the two smaller cubes equals the volume of the large cube. So essentially FLT states that these cubes cannot be measured in any unit such that the lengths will be commensurate. My question is, have many of the failed attempts at “simple” proofs attacked the problem from the angle of incommensurate measure, ie, is this an idiot question, not an insight?

tomrocksmaths

This is a really nice way to think about the problem – and one I’ve never heard before so thanks for sharing. My first question would be how we would know the volume that was left inside the original largest cube? As we increase the volumes of the two smaller ones, they begin to overlap, yes, but I’m not entirely convinced that it would be easy to know what that overlap was and whether it was the same as the volume left in the largest enclosing cube? Because as we increase the size of the two smaller cubes, then the overlapping volume increases, and the remaining space decreases. It seems there might be too many things changing for it to be an easy calculation? Also, how do we control the rate of increase of the size of the smaller cubes? To cover all possibilities you would need to fix the volume of one cube and then gradually increase the size of the other until all possible sizes had been exhausted. You’d then need to re-do this for a slightly different volume of the original first small cube and continue in this way, giving far too many possibilities to check.

I could of course be completely wrong with anything I’m saying above – just my first thoughts on what is an awesome way to think about the problem 🙂

EliteKill

Hi Tom! I’m an undergraduate in Physics/Mathematics, and came across your videos with Numberphile on Navier-Stokes following an experiment I performed. The equations fascinated me, especially after I found out they were a Millennium Problem. What similar fields/problems would you recommend studying to someone like me, who is heavily interested in Navier-Stokes?

And another question – my dream is to do a Masters in Oxford, what advice could you give towards a goal like that? What are the special achievements that get people accepted from overseas?

tomrocksmaths

If you’re interested in solving the Millennium Problem then I’d suggest studying courses on PDEs, but if you are just really interested in fluids then Fluid Dynamics based courses are the way to go. Ideally, I’d say do both!

For postgraduate study there aren’t any special considerations beyond needing a very good undergraduate degree with excellent references from your professors. If possible experience of research or other ways to demonstrate your interest in the subject beyond university courses would be very helpful. The key thing though is to get excellent grades in your undergraduate degree.

wendalls

Is mathematics a human invention, or did we discover it?

tomrocksmaths

Great question – and again let me point you to an interview one of my students recorded with Professor Adrian Moore at the Oxford Philosophy department: https://tomrocksmaths.com/2018/10/31/take-me-to-your-chalkboard/

wendalls

Fantastic- thank you

physicsfreak

Hey Tom! Thanks for doing this. I dropped out of a math degree about 10 years ago, but still maintain a healthy curiosity for the subject. Feel free to cherry pick any of my questions!

Possibly a stupidly esoteric question:

I’m wondering if you might know a sort of top-level overview of what the Langlands program is trying to achieve? I never got much past projective / affine geometry but still find myself ending wikipedia-math-binges on Grothendieks(sp?) work and wondering how in the heck anyone manages to learn it. Is Langlands akin to the classification of finite simple groups?

And just because I think it’d be fun:

Do you have a favorite text / author?

Who are the big players in your field? (past or present)

What are you most excited to learn next?

tomrocksmaths

Hey, you’re welcome – some thoughts below.

Langlands Program – I’ll be completely honest and say that despite interviewing Langlands himself about this (https://www.youtube.com/playlist?list=PLMCRxGutHqfmgC5AjHSYYawu6jRSJFE4W) I’m still not entirely sure I have my head around it. From a simple perspective its all about connections between harmonic analysis and number theory – previously thought to be unrelated areas. Rather than repeating what others had said, I found the following article from Alex Bellos at the Guardian very helpful: https://www.theguardian.com/science/alexs-adventures-in-numberland/2018/mar/20/abel-prize-2018-robert-langlands-wins-for-unified-theory-of-maths

My favourite ‘pop math’ book is The Millennium Problems by Keith Devlin – I first read it when I was 16 and am currently re-reading it and loving it more than ever. Otherwise, An Introduction to Fluid Mechanics by George Batchelor is basically the ‘bible’ in my field. It has all of the answers you could ever want and more.

George Batchelor is often described as the ‘father’ of Fluid Mechanics and is the reason the Cambridge Math Department has a lab for conducting experiments. David Acheson in Oxford is also fantastic – he lectured me as an undergraduate and really helped to steer me towards the subject.

I’m teaching a course next term on Special Relativity which I’m super excited about! I remember doing it as an undergraduate but as I haven’t looked at it for a few years, it almost feels like learning it all over again – and the results are still as surprising as ever.

LoveYourLife33

Hi! My preteen & teen kids say some of the math they are learning is useless! Do you have a tip or two, or some obvious fact that would really explain to them why they need to learn complex math, and not just +-* <= > ? Thanks!

tomrocksmaths

Depending on their interests, I’d try discussing math in the context of their favourite hobby. I have a lot of success explaining the math of sport to people – eg. see here for something on the 2-hour marathon: https://tomrocksmaths.com/2017/05/10/breaking-two/ and the math of video games – eg. see here for something on pokemon: https://tomrocksmaths.com/pokemaths/

Finally, you could try my ‘Funbers’ series that I made for BBC Radio here: https://tomrocksmaths.com/funbers-listen/ The idea is to talk about fin/interesting facts about certain numbers, counting up from 0 to 21. We also throw in a few non-whole numbers too to spice things up!

NerdComplex

Who’s your favorite Pokemon and why?

tomrocksmaths

LOVE THIS. I have both Butterfree and Zapdos tattooed on my leg so those two hands down. Zapdoa because it’s so badass and was my favourite Pokemon card I had as a kid (still got it), Butterfree because it is a great Pokemon to train up early and has the powerful combination of sleep powder plus psychic moves.

NerdComplex

That’s love!

octopisces

Okay look, I completely understand that when you multiply a negative number by a positive number the answer is negative. But my question is, why does the tie always go to the negative? Why wouldn’t the number be positive? I understand the rule, but don’t understand the meaning behind it.

I can understand it like this: 3*(-2)= (-2)+(-2)+(-2)=-6

But I don’t understand this: -2*3=???????=-6

tomrocksmaths

I’m not really sure what you are asking because 3*(-2) = (-2)*3 by what we call the ‘commutativity of real numbers’. It is an axiom that we assume is true: for any two real numbers a, b, a*b = b*a. Assuming this is true the statement follows.

Otherwise, you can interpret the second one as 2 lots of 3 and then multiplied by a minus sign. I’m not sure it makes sense to think of ‘minus 2’ lots of a number… Or by my first point we know the order of multiplication is the same so just change it to 3*(-2) and everything works!

helium_ego

How do you feel about the famous -1/12 proof for the sum of all natural numbers? What does it actually mean + how does this solution actually get used in “real” physics?

tomrocksmaths

HA – I was waiting for this one… The -1/12 value comes from something called ‘Analytic Continuation of the Complex Plane’. It is a valuable mathematical tool, but I don’t think anyone intends for it to be used to ‘prove’ that the sum of all natural numbers is -1/12. Without knowing the full set of assumptions under which the theory of Analytic Continuation is valid, I can’t give specifics, but I imagine at least one of them must be invalidated when extending the Riemann Zeta Function to the negative real axis. And as far as I’m aware, the solution isn’t actually used in practice – it’s more of a fun quirk than anything else.

helium_ego

Since you mentioned the Riemann-Zeta function, I’m curious as to 1. Which Millennium Problem do you believe will be solved next and 2. Which one would you most want to have an answer to?

tomrocksmaths

And for 2. as someone who works in Fluid Dynamics I have to say Navier-Stokes! Though it would also be amazing to see Riemann finally solved after all these years…

helium_ego

As a CS guy, I have to say P vs. NP would be great to have an answer to, but I’m afraid we might never get one.

A translation of an article about my work in Spanish newspaper La Razon. You can read the original article here.

Mathematics was, as for so many classmates with little numerical capacity, the coconut of my adolescence. In a twisting mortal with pedagogy, my teacher came to suspend me with a 4.9. I always stayed 0.1 to understand algebra and today I can’t survive without a calculator. I am not proud. I wonder if everything would have gone better with Tom Crawford. This Brit is a professor at Oxford, but he doesn’t wear a herringbone jacket or bottle-butt glasses nor is he older than the polka. Tom is an AC / DC math, the punk kid in the bunch. Unlike the old masters, he does not use the ruler as a throwing weapon but, at most, to measure the meters of cloth that is removed from each lesson. He is a “naked scientist”, not as a nod to precariousness but as a seduction pedagogical strategy. “I want to take the solemnity off the math, make it entertaining,” he says.

That goes through a “look” of a hangover rocker with a given shirt, sucks, piercing, tattoos and hair dye. He calls himself “Tom Rocks Maths.” His profiles on networks and his informative videos, in which he ends up posing in leopard-print briefs, have legions of followers. Will it be the solution to my problems? Be that as it may, Crawford was in Madrid yesterday, for the first time in Spain, to give a talk in his own way about mathematics applied to sport. The event took place at the Student Residence, where in 1923 another weird boy, with more clothes and more hair, Einstein, summed up his theory of relativity in an act presented and translated by Ortega and Gaset. The list of visits to that leading institution is as interesting as that of its well-known students: Lorca, Dalí, Buñuel …

The Residence has long become part of a memorial of what it was, but its teaching program continues far from the spotlight, without neglecting the field of science, which seems to have been overlapped when speaking of the Residence due to talent. creative of the boys of Letters already mentioned. Tom Crawford is the last visit and, although we may feel like a histrion or a secondary actor in “Trainspotting”, we must not forget that this is purely an eminence from Oxford.

Photo: Jesus G. Feria

A translation of my interview with Spanish newspaper ‘El Confidencial’ discussing my approach to presenting maths as the solution to everyday problems. The original interview (in Spanish) with Guillermo Cid can be found here.

## This teacher knows how to shoot the perfect penalty: “The secret is in the numbers”

### Doctor of applied mathematics Tom Crawford has spent years researching and demonstrating how numbers are much more than theory and can be key to our day to day

It is easily seen and is unquestionable. Tom Crawford is not a mathematician, and he knows it perfectly. His image is far and away from those ideas of the typical serious, boring, number-focused expert with squares in all his aspects of life, and it’s not a coincidence. This Englishman, a professor at Oxford University and a doctor from Cambridge University since 2016, is a loose verse in the sector and focuses all his work on proving it . For what? To teach everyone that mathematics is not just theory and paper and that it is present in all aspects of our lives.

With these ideas he has become a famous popularizer in his own country, participating in all kinds of radio programs from stations such as the BBC, and he even has a YouTube channel where he teaches mathematics in a different way. His stage name is Tom Rocks Maths and he is known as ‘the naked mathematician’ because he makes many of his videos without a shirt and even without pants.

This week Crawford is visiting Spain with an event at the Madrid Student Residence where he will talk about one of the aspects that has given him the most success, the relationship between sports and mathematics, and he has been talking with The Confidential on his entire career and, especially, on how the world of sports is intertwined with numbers.

Fan of soccer and of players like N’Golo Kanté or Roberto Firmino, assures that mathematics is leading the human being “to overcome his limits” and that it has been shown that they are a differential point in disciplines such as soccer, but without humans behind it nothing makes sense. “Mathematics is not magic, but a tool that we must know, understand and apply for our benefit.”

Q: Professor of mathematics at the University of Oxford, doctor of applied mathematics, popularizer … Why have you decided to give a talk on the relationship between mathematics and sport?

A: I love doing sports and following it, and I also love math, so I decided to join both fields. My favorite sports are soccer and running, and in those disciplines I focus research and talk. But well , the main thing is that they are very followed and practiced sports and that they have a clear relationship with the world of numbers. Talking about them, it is very easy to demonstrate how ‘mates’ are present in everything and are very relevant to our day to day. It removes the idea that it is only theoretical and that you learn almost by obligation.

Q: Today we have the cases of Eliud Kipchoge or some soccer teams that are clearly committed to technology and science, with mathematics very present, to improve their brands or achieve greater success. Do you think that there will be a limit in which these disciplines can no longer help us and the human being stops breaking records?

A: It is an interesting matter. For example, if we look at the evolution of athletic records in the last 20 years, we see a graph in which there is a constant and very steep drop in marks. Suddenly, in the early 2000s, disciplines such as mathematics began to come into play and the consequence was that records fell at a dizzying rate, also driven by improvements in training, in nutrition, in scientific research, in the professionalization of the industry … That yes, that occurs until a few years ago, and it is that this fall is stopping.

This, in my view, means that we are also reaching a new limit in progression. Come on, it is already difficult to continue breaking current records and you only have to see the case of Kipchoge and the two hours of the marathon. I do not know how far we can continue to improve, although mathematics could end up giving us a prediction, but I do believe that there will be a time when we will not be able to continue breaking more records. I do not know, it is impossible to think that a person can run 42 kilometers in an hour, for example, no matter how much scientific and technological knowledge is used.

Q: In football we see more and more teams and clubs that invest millions in ‘big data’ and other knowledge to improve their performance, is this key for a team as well as in athletics?

A: Yes, I think that investment in these areas can be key to improve a team, to study new players, to see the performance of the squad … Of course, without the intervention of a good human team this is useless . The thing is not only to have large volumes of data and good analysis programs, you need people who know how to interpret that information and can also analyze it and make decisions about what they find.

For me a perfect example is that of N’Golo Kanté. The player, who is now at Chelsea, arrived at Leicester City who ended up winning the English league from a French second division team. They signed him because he had stealing and intercepting statistics well above the average in his league, so much so that he made Leicester scouts look at him. But then the team employees had to go to see if he really was a good player, if he fulfilled what they were looking for, if he fit into his system and things went well. The data can give you clues or help you find the player that fits for a position, but then you must do a personal analysis and check what you are looking for. It is not something magical or perfect.

Another good example that demonstrates this is Roberto Firmino. He is a perfect player for the Liverpool system but that was not seen with the data, let’s say, more often like goals or assists, but with other types of records that are more covered but are very important. Who says what data we should look at is a human being who then uses mathematical tools to find just what he is looking for.

Q: In Spain now the use of ‘big data’ has become very fashionable in the sports environment, can a bubble be generated around all this following the case of ‘Moneyball’?

A: Obviously there is a danger and that is that without the correct human vision, without an analysis that makes sense of data and numbers and knows how to analyze them correctly, databases are only millions of numbers. You need a correct interpretation to give value to what you do, otherwise they are useless.

This type of knowledge is not something magical or perfect. They are super useful tools but without a human team that decides what information is important or how we should look at them, the investment will be useless.

Q: One of your most famous sports-related research talks about shooting the perfect penalty. How does mathematics say that you have to shoot that penalty?

A: Yes, the answer is in the numbers. Obviously there is no place that ensures 100% success, but there are two points in the goal that offer you up to 80%. Where are those points? Well, in the corners, as long as the goalkeeper is in the center of the goal.

Studying the speed of the shots and the capacity of the professional goalkeepers, it can be said that the goalkeeper has half a second to react and move from the moment the player shoots until the ball enters the goal. In that time the goalkeeper can move in an arc that does not occupy the entire goal but leaves the sides and especially the corners free, since it is impossible to physically get there from the center.

You have all that leftover area to mark with great security, but the most interesting thing for me is that if we create a circle between the corner that forms the squad and the semicircle that the goalkeeper can reach, we have the perfect point to shoot drawn on the center of that circle. A point as far from the goalkeeper as from the post as from the crossbar. If you are able to shoot at that point you will have thrown a perfect penalty. I think the measurements are something like 1.7 meters high and 0.65 meters measuring from the stick to the inside. Obviously nothing tells you to score because the goalkeeper can move or guess your intentions, but it is the safest place to score.

Q: Math is usually thought of as boring and difficult, and you try to turn this thinking around with this type of research and topic. Do you think that the idea about mathematics can change with these actions?

A: I think there is still a lot to do. It’s not so much that you don’t know what math is but that people don’t understand or are afraid of math. When you are with friends, you don’t hear anyone say let’s not talk about history because I don’t understand history, but you do hear about mathematics. That is what has to change. It can’t be cool to say that you don’t understand math or don’t like math.

But the worst thing is that many do not believe that mathematics is useful and relevant for life. They believe that everything is theory that stays in class and on paper, and that’s why I decided to change this idea by relating this knowledge to real life. Sport is a great example. People are closely related to sports, and even more so to soccer. If you can show people how numbers are being used or can be used in these fields, the message will come much more than simply talking about formulas or theories. Without going any further, we have already discussed the penalty case.

Q: And does ‘Mathematicians naked’ follow this idea?

A: Yes, well, normally people think mathematics is serious and boring, and almost by accident I thought that taking off my shirt and giving a different image could attract users. I created a YouTube channel to teach math and discovered that many people entered when they saw that there was a guy without a shirt in front of the camera. That was not the initial idea but this is how I have managed to get many people who are not related to mathematics to enter this world.

Many people remember math with bad experiences in class, exams and so on and my videos try to change this and leave at least one good experience to at least lose the fear of math and users see that not only are they not scary but they are very important to your life. In addition, in the videos they see that I have tattoos related to formulas and others and that in itself gives you an idea of ​​something positive, ‘cool’.

Q: In Spain we have a paradox with mathematics because while many students do not like them, they have the highest grade to enter university because they have many job opportunities. Do you think that the ‘boom’ in mathematics in the workplace is good for people to get to know this world better?

A: As a mathematician, I think the more mathematicians there are, the better for everyone. There are many sectors where they are needed and the people who make this career are usually graduates who face problems very well, know how to find solutions and have the ability to analyze all kinds of situations. That is why I think that a ‘boom’ in this sector is good for all of society, but I understand that there may be a double reading for this.

If a lot of people get into a race just for work, they will end up being unhappy and have no passion to do their daily work. If the only motivation that leads you to study a career and dedicate yourself to a profession is that there is work, it is very likely that the bad days with cold, with a lot of work, with personal problems or little desire to work end up leaving everything.

Interview with Venezuelan news channel Impacto TDN.

“If you think mathematics is boring, hold on to the chair because we bring you an exclusive interview with a teacher who teaches without clothes!”

I was recently interviewed by Lucia Taboada for La Redada Podcast about my love of maths and how it is used in today’s world to model everything from penalty kicks to the next TV series you watch on Netflix. The interview was translated into Spanish for the actual podcast so I’ve also included the original recording of my answers in English – enjoy!

### Podcast version

La Redada T08 E29: Tom Crawford the Naked Mathematician

### Questions

1. On your YouTube channel, you present science in an entertaining way. Why is maths so unpopular sometimes, maybe students are afraid of maths?
2. How would you define the importance of mathematics in our life?
3. Tom, I’m a huge supporter of a Spanish team called Celta de Vigo. You explain the possibilities using maths to improve the performance of football players. How can Celta de Vigo use this to improve? (unfortunately, we are now in the last positions)
4. Penalty kicks are a science? Can you predict them?
5. Have you been hired by any football team?
6. Do you think football teams should hire math workers?
7. You are a tutor in St John’s College at the University of Oxford where you teach maths to the first and second year undergraduates. Oxford is a traditional university – how are your methods received there?
8. You have some maths tattoos on your body, thats right? Explain them to us?

### English (unedited) version

Image: Residencia de Estudiantes

Following my talk in Madrid in November, I was asked to answer a few questions about the current status of maths teaching based on my experience as a university lecturer. Here are my answers…

### How should mathematics be taught in schools?

Through stories. Teaching through story-telling is an incredibly powerful tool and one that is not used enough in mathematics. For example, when teaching trigonometry, rather than just stating the formulae, why not explain WHY they were needed in the first place – by ancient architects trying to construct monuments, by explorers trying to estimate the height of a distant mountain – these are the reasons that mathematics was developed, and I think that teaching it through these stories will help to engage more students with the subject.

### Are teachers prepared to teach this subject correctly?

I don’t believe the teachers are at fault – they are told to follow a particular curriculum and due to their heavy workload have no time to develop lessons with engagement at the heart of their design. There are of course ways that we can help teachers, by providing examples of ways to make maths content more interesting and engaging. This can be through story-telling or applications to topics of interest to students such as sport and video games. This is what I try to do with ‘Tom Rocks Maths’, for example see my video teaching Archimedes Principle by answering the question ‘how many ping-pong balls would it take to raise the Titanic from the ocean floor?’.

### In your view, how should a math teacher be?

The most important thing is to have passion for the subject. The level of excitement and interest that the teacher demonstrates when presenting a subject will pass on to the students. Just as enthusiasm is infectious, so too is a lack of it. Beyond passion, there is no typical profile of a maths teacher. Anyone can be a mathematician, and it is very important that people don’t feel that they have to conform to a particular stereotype to teach the subject. I have always just been myself, and hopefully as a public figure in mathematics will inspire others to do the same.

### Sometimes, this subject becomes more complicated for some students, not so much because of its difficulty, but because of the way in which they have been taught. What should be done with these students?

The trick is to find a way to explain a topic that resonates with a particular group of students. Let me give you an example from my research: the Navier-Stokes Equations (NSEs). For students who have no real interest in mathematics, I would try to get them to engage by explain the \$1-million prize that can be won by solving these equations. For students who have more interest in real-world applications such as in Engineering or Biology, I would tell them about how the aerodynamics of a vehicle or the delivery of a drug in the bloodstream rely on an understanding of Fluid Mechanics and the NSEs. If the students are fans of sport, I can explain how the equations are used to explain the movement of a tennis ball through the air, or for testing the perfect formation in road cycling. Finally, for students who are already keen mathematicians, I would explain how the equations work in almost every situation, except for a few extreme cases where they result in ‘singularities’, which as a mathematician are the ones you are most interested in understanding. Once you know the interests of your audience, you can present a topic in a way that will help them to engage with the material.

### Can you get to hate math?

It is certainly possible – though of course alien to mathematician such as myself! I think this feeling of ‘hate’ relates back to either the way that you have been taught the subject, or from a lack of understanding. If you did not enjoy your maths lessons at school and harbour ill feelings towards your teacher, then you will begin to develop negative feelings towards the subject. This is not because you dislike the subject, but more because of the way that it was taught to you. Likewise, if you do not understand mathematics then it is very easy to develop a ‘fear’ of the subject, which can quickly turn into hatred due to feelings of inadequacy or stupidity if not addressed. It all comes back to finding a way to approach the subject that fits with the knowledge and experiences that you already have. If you present a problem in an abstract manner of manipulating random numbers to find a given total, then most people will struggle – regardless of their mathematical ability. But the same problem presented in a relatable situation suddenly becomes understandable. Here’s an example:

(a). Using the following numbers make a total of 314: 1, 1, 2, 5, 10, 10, 20, 20, 50, 100, 100, 500.

(b). You go shopping and the total is €3.14. What coins would you use to pay for your items?

They are the same question, but in (a). the problem looks like a maths question, and in (b). it is an everyday situation that people all over the world are used to. Both require the same maths to solve, but even people who ‘hate’ maths could tell you the correct answer to (b). using their own real-life experience.

### Women are at a great disadvantage compared to men when entering a STEM career, why do you think this is happening?

First of all, as a man I am certainly not qualified to answer this question, but I will at least try to provide you with my opinion based on personal experience. At high school level I believe that the difference is less severe (eg. see article here) and even at university there is a slightly higher number of females than males studying science-based subjects. BUT, the issue occurs after this. In graduate degree programmes and beyond there is a definite lack of female researchers, and this is amplified even further at more senior level positions. One explanation could be that academic ‘tenure-track’ positions exist for life, and so many of the men that now hold these positions have done so for the past 30-40 years and were employed when we were doing a much worse job of tackling the gender gap. Now that awareness of these issues has increased, and in general we are doing a much better job at addressing them that we were 30 years ago, hopefully we will begin to see more females in leading positions over the coming years, it will just take a little while for the effect to be seen. I also think that in general there are not enough female role models within many subjects (especially maths) that have reached the pinnacle of their field (through no fault of their own), and as such there is a lack of role models for young female researchers. The achievements of female mathematicians such as Maryam Mirzakhani (2014 Fields Medal) and Karen Uhlenbeck (2019 Abel Prize) should be even more celebrated precisely for this reason.

### Do you think that enough importance is given to mathematics in the educational world?

In the past perhaps not, but attitudes are certainly changing. With the increased role that technology and algorithms play in our lives, people are beginning to realise that we need to better understand these processes to be able to make informed decisions – and maths is the key to doing this. Employers are certainly aware of the invaluable skillset possessed by a mathematician and as a result more and more students are choosing to study the subject at degree level and beyond to improve their competitiveness in the job market. Ultimately, attitudes are changing for the better, but there is still more that can be done.

### In your opinion, what is the best way to teach this subject?

Exactly as I have described in questions 1 and 4. Storytelling is key to making the material as engaging as possible and knowing the interests of your audience allows you to present the subject in a way that will appeal to them most effectively.

### What is the current situation of mathematics research in the university?

I think the main issue facing research mathematics is the relatively recent trend of short-term research outcomes. The majority of funding available to mathematicians requires either continuous publication of new results or outcomes that can readily be used in an applied setting.  The issue of continuous publication means that researchers feel the need to publish a new manuscript every few months, which leads to very small advances at each step, and a wealth of time spent writing and formatting an article instead of conducting actual research. In many cases, the work would be much clearer if published as one piece in its entirety after several years of careful work. The drive for short-term research outcomes means that it is now very difficult to study mathematics just for the sake of it – you have to be able to convince your funding body that your work has real-world applications that will be of benefit to society within the next 5-10 years. To show why this is a disaster for maths research, let’s take the example of Einstein and his work on relativity. Now seen as a one of the most fundamental theories of physics, his work had no practical applications until the invention of GPS 60 years later. In today’s short-term outcomes driven market, it is highly unlikely that Einstein’s work would have been funded.

Photo: Residencia de Estudiantes

#### Tom Crawford talks about how mathematics can help win a football league or the real ability of algorithms to manipulate people’s behaviour.

Tom Crawford (Warrington, United Kingdom, 1989) is presented as an atypical math teacher. He teaches mathematics to first and second year students at the University of Oxford (United Kingdom) and carries out an intense dissemination work in which he tries to approach a discipline that is not usually found among the favourites of young students.

In his attempt to popularise science, he does not hesitate to stay in his underpants , using the striptease as a metaphor for his work deepening the meaning of equations such as Navier-Stokes, unveiling them layer by layer, to make something affordable that can result in principle esoteric.

This week, Crawford visited the Student Residence, in Madrid, where, within the Mathematics in Residence cycle organised by the ICMAT, he offered the conference  Mathematics of sport . In it, he uses sport as an example of a daily activity that can be better understood and practiced using mathematical equations.

Question. You undress or use sports to make mathematics impose less. Why is it necessary to show that mathematics is fun? I don’t see lawyers or judges, who also deal with very complex issues, trying to present the law as something fun.

Answer. I think it’s because people, for whatever reason, happily admit that they don’t like math, it’s socially acceptable. If you tell someone that you are a lawyer, their default answer is not going to be “I don’t like the law,” and that does happen with math. And it shouldn’t be like that. Everyone should have a basic understanding of math, but many people don’t have it. For me, that is why I want to emphasize that mathematics is fun and accessible. It doesn’t have to be something very hard or something that was taught badly in school.

Q. Do you think mathematics is taught especially badly in school, worse than other subjects?

A. Mathematics has a hard time competing with other subjects in the sense of teaching them through stories. When you learn something, if they can teach you through stories, it is something very powerful, which serves to catch people. And that is easier with literature or history.

A very simple example of how to add stories to mathematics would be trigonometry. The properties of the triangles you learn in high school. If you think about how these functions were discovered or invented, why we invented the sine, the cosine and the tangent, it was the ancient architects who tried to build buildings, churches, pyramids and created those intellectual tools. This is how trigonometry should be taught to me. Imagine they are in ancient Rome and you have to build a concrete building. How would you do it with the technologies available at that time? This prompts you to think about angles and distances and that is where trigonometry is useful and what it was invented for.

Q. A little more than a century ago, in a country like Spain, more than half of the population was illiterate. Do you think it would be possible and desirable to get a large majority of people to be able to handle basic mathematical tools?

A. It is completely possible and I would say that we are already doing it. It depends on what you consider a basic level of mathematics. Most people can, for example, looking at a clock know that the needles return to the same place every 12 hours, it is modular arithmetic, something you don’t study until you get to college. Even being able to calculate changes when they give you a ticket is to do mental arithmetic. Or calculate when you have to leave home if it takes 35 minutes to the station and the train leaves at 12.45. There are many things you do without thinking, but that involve mathematical calculations. So it depends on what you consider a desirable level of mathematics, but a large part of the population already has some capacity to use them.

“You can question whether trying to influence voters is good or bad”

Q. He also talks about the possibilities of mathematics to improve the performance of athletes. There is a movie like Money Ball , which talks about the experience of a baseball coach who uses mathematical analysis to lead a small team to compete against the big ones in the league with much less budget. Do you use math a lot in elite sport?

A. As far as I know, it is an important part of the scout systems of large teams. Today, these scouts, in addition to the classic analysis of a player’s performance, strengths and weaknesses, include teams of mathematicians and data scientists. As in Moneyball , your job is to analyse large amounts of data and detect marginal gains to take advantage of. That works well in baseball, because you have many controllable factors: The pitching of the pitcher, the batter, the race to the base. It is very formulable and they are repetitive behaviours. In football it is more difficult to find those marginal gains because it is less controllable.

The best example I can think of in football is Leicester City, which won the Premiere League in 2016. A big surprise. They had climbed to the first few years before and suddenly they win. In that victory, N’Golo Kanté was very important. He was the star of the season and won the player of the year award. He had been signed by a French second division team because the scout network had identified him among all the defensive midfielders in Europe at any level. As a defensive camper centre, one of your jobs is to stop the attacks of opponents. You can measure this in tickets, but one of the best ways to do this is through interceptions, which has to do with the player’s ability to read a game. It is something very difficult to assess with a number, quite subjective. But interceptions suggest that you are very often in the right place. And from that point of view, their number of interceptions was much higher statistically than the rest of midfielders. If the average of all midfielders in Europe is two, but most of the players are between 1.9 and 2.1 and Kanté is at 3, we see that it is an atypical case. It was not just a statistical analysis, because the human element is valued, but it was a factor to hire him.

Q. Can mathematics tell us what is the limit of human performance in sport? There have already been examples in the past, such as Roger Bannister’s, which went down four minutes on the mile when almost everyone said it was impossible, in which the predictions were completely wrong. Can these limits be accurately identified using mathematics?

R.If you look at the men’s marathon record during the last century, the marks descend, but not at a constant pace. You can estimate, for example, that every 10 years, 10 minutes are trimmed at the beginning, but then, in the 1940s and 1950s, the curve begins to flatten out and already in the 1990s it seems completely flat. So if we had sat here 30 years ago, when the record was around two hours and five minutes, we could have thought we would never run below two hours, because even if it keeps going down, the pace is getting slower. But in recent years, there has been much progress in long-distance races, such as new shoes that can provide 4% more energy. In addition, there is a professionalisation that allows you to train all day and not have a job besides running.

“I could predict with some confidence that the human limit for the marathon would be about an hour and 55 minutes”

So these are new factors that modify our calculations. In the future, in 30 years, new improvements may appear, but it is certain that we will not run a marathon in less than an hour. Given what has happened in the past, I think I could predict with some confidence that the human limit for the marathon would be about an hour and 55 minutes.

Q. Some people, when talking about the possibilities of mathematics to bring humans to the limit of perfection, may think that sports will become more boring, because there will be less and less space for the unexpected.

A. I think that also has to do with the human psychological trait that is nostalgia. But sport evolves and there is always a human factor. If the study allows you to perfect the place where it is better to throw a penalty, the goalkeepers can also work with that information. And then, there are some players who do not shoot at that supposedly perfect space, such as Eden Hazard, of Real Madrid, who when he threw the penalties for Chelsea waited until the last moment to decide where he threw it, a method that goes against what he says The mathematical model. In the end there are many variables in sports.

Q. Can mathematics help us better understand human groups? Does that technology have the potential to improve living together or to make it worse?

A. With all the data available, there are huge technology companies that can make profiles of people. Knowing that you are white, American, that you earn so much money and live in such a state, they can try to predict what you like or what you do and influence your vote in one direction. But this technology could also be used for good and you can also question whether trying to influence voters is good or bad. I think that ultimately we depend on the big companies that have control over these data so that they assume their moral responsibility and use the data well.

In any case, I think that most of the mathematicians working in this field would say that the idea of ​​using mathematical data, algorithms and models to try to predict people’s behaviour is incredibly new and we don’t know exactly what we are doing. Algorithms may be a part of the decision making process, but not the only criteria for making a decision.

You can read the original article on El Pais here.

Author of the best-selling book ‘Fermat’s Last Theorem’, Simon Singh tells the story of Andrew Wiles and his proof of the world’s most famous maths problem…

If you don’t know the story, this is a great place to start to gain an insight into the world of research mathematics – and make sure you read the book it’s fantastic!

Produced by Dr Tom Crawford, with thanks to Simon Singh, the Isaac Newton Institute, the Heidelberg Laureate Forum Foundation, Klaus Barner, New York Times, and Rex Arbogast.

During my recent trip to Madrid to speak at the Residencia de Estudiantes, I was interviewed by national newspaper ‘El Mundo’ about my talk on the ‘Maths of Sport’ and my mission to popularise maths. The original interview can be found here.

Known as Tom Rocks Maths, the Oxford University scientist transforms boring formulas into fascinating models that he applies to sport to improve records and reduce errors.

Football World Cup, 2018. 1-1 on the scoreboard. Spain plays its pass to the quarterfinals in the penalty round against Russia. Koke has failed one. Cheryshev is ready to throw. He scores. It’s up to Aspas. Expectation. Whistle. Launch and … Akinfeev stops it. We miss the game.

Could it have been avoided? The answer is Tom Crawford, l’ enfant terrible of numbers, a punkrocker in the court of mathematicians at the University of Oxford. And he explains it with a worn shirt, leather jacket, curled hair, piercing and tattoos. Because Crawford is not a common scientist. He is Tom Rocks Maths, an alternative researcher and communicator who transforms boring formulas into fascinating models that he applies to sports science, his second passion as a marathoner and a follower of Manchester United.

But, since all science is not exact, nor is Crawford a fortune teller, his predictions are based on data, taking into account all possible variables and, above all, on the highest probability of hitting. It is about getting ahead of the facts, of having all the necessary information to reduce errors and improve the records.

The mathematics of sport consist of “building models using data from the past to predict the future. When you don’t have them, you have to go to the field, contact the athletes and gather new information, ”explains Crawford in an interview with EL MUNDO after the talk he gave Tuesday in Madrid during the cycle of conferences ‘Mathematics in the Residence’, organized by ICMAT, the Student Residence and the Deputy Vice Presidency of Scientific Culture of the CSIC.

Win or lose penalties

The countries that best know how to throw penalties are Uruguay, Germany, Argentina and Brazil. Spain is not good, not bad. We are 50% among this list of experts and 50% of the worst, Mexico. We share media with France and Ireland. But how a team is better than another is not a matter of tradition or genetics, but numbers.

The first thing is to know how the players have responded before to the penalties, their statistics of failures and successes. According to Crawford, in the case of the 2018 World Cup, while Iniesta had four hits of five shots, Koke had zero of one and Aspas 16 of 17. The great surprise could have been given by Thiago, a substitute with a full in hits, four of four.

There is also a way to measure their stress responses with glasses that observe the movements of the eye. Footballers who are not immuted by pressure keep their eyes fixed and the most distracted move them. Knowing this in advance could decide the choice of a coach to choose the most focused players on those decisive penalties of a World Cup. “Football clubs now have entire teams of mathematicians and scientists who analyze all this data,” says Crawford.

But math doesn’t end there. In a goal you can make as many measures as your imagination, as Archimedes, to find the radius that indicates the exact area where you should place the ball without being stopped by the goalkeeper. It is called an insurmountable area and it depends, among other things, on the distance the goalkeeper moves in the shortest possible time from his position in the center. It looks like this: r2-2r(a+b+R)+a2+b2-R2. “These calculations are going to help you but they don’t guarantee that the penalty is perfect. In fact, the goalkeeper may also have trained against these formulas, ” Crawford alerts.

Roberto Carlos, the king of the Magnus effect

If penalties are a science, free kicks are not far behind. Defining its trajectory is one of Crawford’s favorite equations when the ball is given effect, as we have allways called it, which also has its scientific name: Magnus effect. “The ball that spins does not go in a straight line, because the rotation moves it to the side,” he said.

In this modality, for Crawford there is a master: Roberto Carlos, king of the Magnus effect in a match against France in 1997. It happened like this: he carefully placed the ball with his hands on the ground. He kicked. The ball passed over the barrier of players, turned in the air to the right, then to the left, hit a stick and entered as a stroke.

“I saw it when i was eight years old and I thought that it was impossible, that it was magic. But years later, using this equation to model Roberto Carlos’s shot, by entering the correct data (the speed of the ball, the distance to the door and the spin that applies to the ball) the formula accurately predicted that movement. It is still amazing. Although now I have an explanation that tells me that it did not break the barriers of physics.”

Marathon in less than two hours

In Tom Crawford’s mind there are not only favorite sport formulas but also graphics. And if there is one that drives him crazy it is the one that calculates, with a curve, when you will be able to run, with conventional methods, a marathon in less than 2 hours, something that it could happen between 2027 and 2035.

The record is owned by Kenyan Eliud Kipchoge, 2:01:39. He obtained it in Berlin in September 2018. In October of this year, the same athlete beat it at 1:59:40, but his feat was not accepted by the International Athletics Federation. “It has not been taken into account because they have broken the rules. As a new official mark, this record below two hours does not count, ”says Crawford.

How they did it? “Creating the perfect race,” says Crawford. And something else: a flat route in a straight line to go through the center of the track; a pair of shoes with carbon fiber that balances and saves 4% of energy; a tape on the leg with lumps (like golf balls) that create streams; a squad of escorts in V to cut the wind (called hares); a car that laser marks the ground so that these satellite corridors maintain the perfect position; a scanner that controls the muscle accumulation of carbohydrates and an enriched diet.

“Where you draw the line between what is due to the human element or an incredible shoe. What is the next? Putting rockets in our soles? ”Crawford wonders. We saw it in swimming a few years ago with high-tech swimsuits that reduce friction with water. They were even questioned for increasing buoyancy. With them, in some competitions 130 records were broken in just two years.

It is clear that mathematics helps to overcome tests and marks. However, in sports there are uncontrollable factors, such as the mental control of athletes, to disarm algorithms. “You can never add that factor to your models. You can never really predict a sport with total certainty. There are many unknown variables, ”reflects the English mathematician.

And, returning to the football game that we lost in 2018, would we have win if we knew the data in depth and having other players thrown the penalties that took us out of the World Cup? According to Crawford, we could have reduced the risk of losing, but this is something we will never know. What is highly certain is that their talks not only reinforce the devotion to sports, but also awake the mathematical vocation of the youngest students.