Christopher Budd (mathematician)

Christopher John Budd OBE (born 15 February 1960) is a British mathematician known especially for his contribution to non-linear differential equations and their applications in industry. He is currently Professor of Applied Mathematics at the University of Bath, and was Gresham Professor of Geometry from 2016 to 2020.

Quotes

Math Encounters (2017)

Vital Math: How Mathematicians changed the World. Speaker: Chris Budd, OBE. A public presentation series presented by the Simons Foundation and the National Museum of Mathematics. A YouTube Creative Commons License Video Source (posted Jul 8, 2017) @National Museum of Mathematics channel (momath.org)

On the whole, mathematicians don't have a particularly great image.

View #1 about math: Math is completely useless. Another one... Mathematicians are evil, soulless geeks... [A]nother... everyone seems to believe: All mathematicians are mad! ...Unfortunately, they are views quite commonly held.

[S]ome people are... quite frightened of math, or even... suspicious of math.

[T]he shame about all of this is not only is it not true. It's really, really, really not true! ...Math is basically the basis of the modern world. The modern world would simply not exist without mathematics.

The technology that we celebrate today, everything we do, is all heavily based on math. In my pocket I have my smartphone... absolutely stuffed full of mathematics, and lots of mathematicians work in the smart phone industry.

Much of industry has problems which can potentially be formulated and solved using mathematics. ...It's numbers. It's information. It's mathematics.

If there wasn't air around me, I'd die very very quickly... and math is like that for technology. Take the math away, the technology fails, but just like the air around us it's invisible, and lots of people don't know it's there.

I'm going to show a picture of a mathematician... that has changed your life profoundly more than anyone I could possibly think of. ...Whilst I'm a great admirer of Washington and Franklin and... all these wonderful people, I reckon this guy's changed your life even more... This is Maxwell, but... most audiences haven't a clue who this is.

The Royal Institution was... one of the first scientific research establishments in the U.K. ...They reckon about 30 elements were discovered in the Royal Institution. A number of Nobel Prizes came out of the Royal Institution. Humphry Davy worked in the Royal Institution, the Braggs [William Henry and Lawrence]... [etc.,] but possibly the most famous... was... Michael Faraday... [H]e discovered experimentally the link between electricity and magnetism and... essentially invented the electric motor and the dynamo, and those have then been developed by people like Edison and Tesla into the power generation system we so celebrate today.

Faraday... wasn't a mathematician and so he relied on other people to do his math... and Maxwell... took Faraday's experimentally derived results, and... turned them into mathematical equations.
$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}-\mathbf {M} ,\quad \nabla \times \mathbf {H} =-{\frac {\partial \mathbf {D} }{\partial t}}+\mathbf {J} ,$
$\nabla \cdot \mathbf {D} =\rho ,\quad \nabla \cdot \mathbf {B} =0.$
...and these are called Maxwell's equations, and if you go to Edinburgh and you see his statue, on the base of the statue you will see these equations... which link electricity $\mathbf {E}$ with $\mathbf {B}$ and $\mathbf {M}$ , which are magnetism, and $\mathbf {J}$ is current, and $\rho$ is charge, and these are... vector operations, and that's a derivative. ...Maxwell ...took those equations and looked for solutions, and... discovered that there were solutions... where you have an electric wave and a magnetic wave... paired... and traveling together, and he worked out the speed of those waves... exactly the speed of light. ...[I]n those equations he unified electricity, magnetism and optics all in one setup.

But then he did something which mathematicians can do... [i.e.,] what-if experiments. You can say... what if these equations have other solutions, and he found... waves with the same speed as light, but a different frequency and wavelength than light... and we now call them radio waves. ...Maxwell discovered radio by pure mathematics alone. It was later... that... Hertz found them experimentally, and... later... Marconi and others took the theory and turned it into practical means of communication.

But it all started with Maxwell. Let's think where we'd be without radio. We wouldn't have radio... wi-fi... TV... radar... our mobile phones, we wouldn't have microwave cookers. The world would be utterly different without radio! In fact, the modern world simply would not exist without radio waves, and it was Maxwell who discovered them, purely from mathematics, and... very, very few people know who he is. This guy should be on the bank notes... It took us a long time to get his statue up. ...One of the other things he's... noted for is his work in color photography. He essentially invented color photography. He also wrote poetry... He was a... very good guy.
- Ref: "Experiments on Colour, as perceived by the Eye, with remarks on Colour-Blindness" The Scientific Papers of James Clerk Maxwell (1952) pp. 126-154.

Who was the most famous female mathematician? ...Emmy Noether, [was an] excellent... fantastic mathematician, but if I went into the street, who would know Emmy Noether? ...Even more famous than Marie Curie. Films have been made about this woman. Ada Lovelace... famous, but not as famous as this one. I've seen films, books have been written about her. Hugely famous, most children would know her name. I'm going to put her picture up and it's going to surprise you. ...Florence Nightingale's an incredibly famous woman because... she basically founded modern nursing. ...The story ...she was sent to Crimea and... set up hospitals... which saved huge number of lives, and when she went back to England she developed modern nursing and her practice... are used all over the world, and everyone thinks she's a nurse, but... she was a statistician. She was one of the first members of the Royal Statistical Society and was a really good statistician... [T]he way she cured people wasn't so much through medical care. It's through the... more modern approach, which was to try to work out what was causing people to be ill. ...[S]he gathered loads and loads of data on this and... produced graphs of this data... essentially to convey what she was doing to politicians, because politicians then and sadly now, don't know what numbers are... [S]o she did this through graphical information and she developed... rose diagrams which are very like pie charts... [S]o she not only developed medical statistics... she also developed graphical presentation of data, which is universal, and she's incredibly famous, but noone knows she was a mathematician. ...The Royal Statistical Society ...building is called the Nightingale building, after her.

The way maths is used... has... changed enormously, essentially in my own lifetime. ...The whole thing has changed remarkably, largely because of the develoopment of computers.

So... [who needs math?] Thirty years ago [traditional industrial users] typically using mathematics... [were] telecommunications, the aircraft industry [aerospace], power generation, oil, iron and steel, weather forecasting... a big user since the 1920s, security-code breaking... and of course, finance.

But nowadays mathematicians are... using maths in a far far broader capacity. One of the biggest users of mathematics, certainly in the U.S., is the film industry... because if you have something like Shrek... What he is, is a whole load of mathematical formulae inside a computer which some mathematicians come up with... constructed to produce a shape which looks like an ogre, and they've done a great job... So Pixar employs loads of mathematicians... [T]he film industry, a massive user of mathematics!

One of my other talks... is about the math in The Lord of the Rings and all the maths that was used for that, because... they used lots of mathematicians there.

So film, entertainment, graphic design, the retail industry uses lots of mathematicians. When you go and buy something... sadly, they're collecting data... If you go to Amazon and... buy a book, it will say, "People who bought this book also were interested in this!" ...[T]here's a mathematical algorithm that's doing that. ...It's great news for young people, because... there's vast numbers of jobs, if you stick with the maths.

The way you... deal with... a problem from industry is you take all the math that you know... and you try and solve the problems with it. ...[A]fter a while you find that you've run out of math. The math that you've learned isn't enough to... solve the problem. So...you have to... invent new math, and... that new math can be... abstracted and turned into other stuff, and then... used to solve new problems, and... you look at those new problems and... find that you need new math from that... and so it... cascades, with problems generating maths, generating problems, and so on. ...[I]t's a really good virtuous circle, and this is... how math is developed over many years.

Math is also developed by curiosity, and just pure abstract reasoning as well, so there's lots of ways it works.

I'm going to take you through this, looking at the way history of the way math has impacted our civilization. ...I'll try and take you right up to where we are at the moment, and beyond.

Where did math come from? ...Early people counted on their fingers, and numbers basically came from that. ...There's no other reason for choosing ten. Ten isn't a great number for a base. Very few numbers divide into it. If we had 12 fingers we would have been better...

The Babylonians were a little... more advanced. They used the knuckles as well. They counted not only in 10s but in 60s... and that's why we have 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle.

Math was... invented to count things with. What was it then used for? ...Once you have numbers 1,2,3,4,5, and 6... and you want to start using them, you... find they're not useful for everything. You have to invent... more numbers... to include things like 0... invented around the year 0, and negative numbers were invented to deal with things like debt, and fractions were invented... I suppose you've got 3 fields and... 5 children, then each child will inherit 3/5 of a field. So they were invented to deal with that.

What was the first application of numbers? ...[W]e're pretty sure that the first application... was ...the tax man. Why can we be sure... because if you go to museums... you can find Babylonian cuneiform tablets... and the [Egyptian] Rhind papyrus... where there's loads of wonderful maths developed, and it's all to help the tax man.

But they then discovered something else, and... had to extend numbers a bit more. Suppose a farmer has a field and that field grows 100 cabbages. ...[T]he king wants to have a war so they need... 200 cabbages. How much bigger should my field be? ...It needs to have twice the area, and the area... is proportional to the square of the length of the field, so... how much bigger should the length of the field be, and the equation that you have to solve... is
$x^{2}=2$
[W]e know the Babylonians were interested in this problem because... a cuneiform tablet, which I believe is in the British Museum... is... trying to solve this equation.... and... gives the answer. ...[T]hey tried to solve this using fractions and they... couldn't. There was no fraction which equaled the answer... and so they had to invent... what we call an irrational number to give a solution... 1.4142135623730950488... That's to 20 decimal places, and it goes on and on and on. ...[T]hese were numbers called real numbers, and were originally invented for the tax man to work out how to double the area of fields.

The great triumph of maths around... 1690 was the development of calculus... and calculus is now probably the best tool that we have in math to tackle the problems of the real world.

We had a... newspaper competition in the U.K.... to identify the greatest ever invention... and I wrote in ...calculus. ...It didn't win. ...The greatest ever invention was apparently the printing press ...the second was the wheel, and the third was fire... which was misguided because calculus is, without a doubt, the best tool that we have... But of course, I am biased.

[A] lot of maths doesn't develop by solving problems of practical importance. A lot of it... develops purely out of curiosity, of from doing stuff for fun! ...You're doing maths when you do Sudoku, and it's good fun ...[S]olving puzzles ...and having fun is ...an extremely good way of doing math, probably the best way.

[R]ecreational math is a huge... subject... [A] particular favorite of mine... mazes and labyrinths, which were originally recreational. One of the earliest examples... involves... the Minotaur... the product of a one-night-stand between the queen of King Minos and... Zeus, dressed up as a bull... turned into a bull, or whatever they do. The product... was the Minotaur... half man and half bull... [H]e was... ferocious and... lived in the center of a labyrinth underneath the palace of King Minos. ...Theseus... said I will go with the 9 young men and 9 young women and... attempt to kill the Minotaur. So when he went to Crete he was met by one of my heroes... Ariadne... the first female mathematician... recorded in the classical literature. ...[S]he gave him a sword. By the way she fell in love with him. ...[S]he also said ..."I will give you an algorithm for cracking the labyrinth ...and using this algorithm, he went into the labyrinth, found ...[and] killed the Minotaur, got out of the labyrinth and took the young men and... women back to Greece... [O]n the way he stopped off at an island... Naxos where they had a great party... and only after they had sailed off did they realize they'd left Ariadne behind, and she died of a broken heart and turned into a spider... [T]he real hero of the story is the labyrinth. ...[T]his design, although they've found it everywhere in the ancient culture, is universal. There are clear examples of Native American populations in the U.S. having essentially discovered the same design.

Maths is universal.

[I]n the 18th century the idea behind the labyrinth was evolved into... the modern maze, and people... used to build mazes in their large houses... designed to trap the unwary. You'd go into them and... occasionally get lost... People would try to puzzle how to get from the entrance into the center.

Euler... worked out the math of how you could get into the center and... back out again, and that math led him into the theory... of networks, and that all came out of mazes. What do we use networks for now? ...[T]he biggest network in the world is the internet ...and Euler's work on mazes is directly used to help us do the internet. ...[W]hat uses that, Google. Understanding networks, combined with Matrix Theory (due to Cayley) also forms a major part of the algorithms behind Google.

I want to tell you a little... about some of the algorithms that have been used to solve mazes. ...Ariadne ...was a mathematician and a computer scientist, because she came up with [an] algorithm for solving the maze or the labyrinth, and her algorithm was beautifully simple, but remarkably effective. She gave Theseus a ball of thread... and as you go into the labyrinth you unwind it and... to get out, you wind it up again... It worked very well, and that is the basis of a modern computer algorithm called the Flood algorithm for solving the labyrinth.

Another algorithm is, if you... read the book Three Mean in a Boat... they try and solve Hampton Court Maze, and... Harris... says you solve it by always turning left, or... you put your left hand on the hedge and keep it there, and that will actually work... and it will solve a lot of mazes. It won't solve all of them, but it's... a very good algorithm to try. It will always get you out of a maze, even if it won't get you into the center. So always turning left is a good algorithm.

A labyrinth with this of [King Minos] design is... unicursal, which means that you can go in and out without making any decisions... So you didn't need the thread after all, but the other mazes like Hampton Court and Hatfield House, and all the other ones, you have to do a little... more. So a labyrith is something where you don't have to make decisions...

People often say there's a close link between math and music, and mathematicians and musicians, and that's absolutely true. Music uses a lot of math. So some musical notes sound better when you play them together. ...The reason was discovered by... Pythagoras. ...He is very very famous for Pythagoras' theorem, which was invented by the Chinese about 1,000 years before him. ...But he absolutely did do the work on musical notes. ...[H]e measured the length of strings of instruments and he compared the lengths with notes that sounded good together. ...He realized that the octave [C:C] corresponds to two strings, one being twice the length of the other [2:1], C:G ... 3/2 and C:E proportion 5/4, and Pythagoras found an incredible link between musical harmony and fractions.

Pythagoras... took the idea further. He said... let's suppose that we have a sequence of notes with simple fractional relationships. So he came up with notes with these relationships [the Just scale]
[C] 1 : [D] 9/8 : [E] 5/4 : [F] 4/3 : [G] 3/2 : [A] 5/3 : [B] 1/ 5/8 : [C] 2
So Pythagoras invented the scale... the basis of modern Western music...

This scale was used... up to about the 18th century... [when] keyboards were invented. ...It was found that this scale worked brilliantly for one key, but terribly in another key... because the ratios differ... and so the mathematicians developed a different [well tempered] scale where the notes were constantly varying in frequency [in the same proportion, a geometric progression of the semi-tone frequencies], so the ratio is this wonderful number $1.059..=2^{\frac {1}{12}}$ which is the number when multiplied by itself 12 times gives 2, which works well in all keys.

Bach was very familiar with this work going on, and he so liked the well tempered scale that he wrote... The Well-Tempered Clavier where you go through every key on the harpsichord... and it's all based on maths, and it's the maths of geometric progressions... [I]t all goes back to mathematicians working with musicians. We're not evil, souless people at all.

In the 18th century... people traveled... by boat, but the problem... was... that they were finding it very difficult to know where they were. ...[O]ne of the big problems of the 18th century was finding your position at sea. ...We have latitude, which is the angle from the equator, and longitude, which is the angle measured round... from London. ...I've stood on the zero longitude line in Greenwich. ...[T]he first thing they cracked was latitude. So you could work out your latitude... by using... the sextant, which measures angles very accurately... [T]hey realized that if you measured the angle between the sun and the horizon, or between the pole star and the horizon, you could... accurately find your latitude. ...[Y]ou had to use a ton of trigonometry... [i.e.,] more math, in fact trigonometry on the surface of a sphere, which is... spherical trigonometry... [T]rigonometry and the sextant combined... meant that navigators could work out their latitude... how far above the equator they were.

So one way to get to America was that you would sail from England on the same latitude, and when you got to something big that was probably America, or possibly Canada.

Longitude was the big problem... Many mathematicians tried to solve it using math alone, but the solution was a beautiful one, and it's the sort of math I do... a combination of sums on paper, but also a lot of work with a computer, so it's combining technology with formulae. ...[T]he solution ...was due to ...Harrison, who was a clockmaker... [I]t was based on the observation that the earth goes once around every 24 hours and so if you can time when the noonday sun is, and measure that time on a clock which is the same as the clock in Greenwich, for example, then by measuring that time you could... work out how far around the earth you are. You have to do a whole ton of other stuff as well, but that's the basic idea. ...[T]hat was math, but it was also linked up with technology because you needed the clock ...[H]arrison's clock, called H4 ...was the first clock ...accurate enough to make that possible.

[A] ton of math was involved as well. You have to produce tables of the motions of the sun and the stars... These ephemerides... were calculated by computers. ...A computer in those days was a room full of people who computed. ...They were typically youngish people, often students, and it was found ...that women were better at it than men ...[T]he midshipmen on the boat doing navigation had to do long [tedious 22-step mathematical] calculation based on these [tables] to [plot a ship's position] find where they were. But those calculations changed the world.

Why is America such a powerful nation? Because you could reliably sail your ships to Europe and sell all your stuff, and that reliability came from navigation, and that navigation relied on math.

The calculations were tedious. They took a long time. They were also very error prone, and so people thought... can we automate these using machines? So... Babbage, working with Ada Lovelace... a very very fine woman computer programmer, essentially the first ever computer programmer, worked together to design the machines which would automatically calculate these tables [the Ephemerides.] Sadly they never managed to build them properly due to engineering problems, but Babbage's machine has been recreated... at Science Museum. You turn the handle, all the cog wheels go round and it calculates the tables for you. Brilliant! ...[B]abbage's ideas led directly on to the invention of the modern computer, with Turing, von Neumann and people like that, and that all came directly by navigation and the need to do that.

I would argue that mathematicians save hundreds, if not thousands, if not millions of lives, almost every day. ...Medical scanners have revolutionized medicine because you can be scanned and... find out what's wrong... without cutting you open. ...[T]he medical scanner was basically invented by ...Radon ...one of my favorite mathematicians ...[H]e did brilliant maths which is used in medical imaging and saves millions of lives. He's a fantastic mathematician, and he was studying in 1917 a kind of abstract problem. He was looking at shadows. You know, objects cast shadows, and he wanted to know if you know what the shadows were, can you find out what the object was that cast them... [H]e wrote down a formula for taking an object and the shadows it cast, and then with a bit of genius, he worked out another formula saying, if you know what the shadows are, this is what the object is. $f$ is the object, $R$ is the object.
Shadow $\quad R(\rho ,\theta )=\int f(\rho \cos(\theta )-s\sin(\theta ),\rho \sin(\theta )+s\cos(\theta ))ds$
Object $\quad f(x,y)={\frac {1}{(2\pi )^{2}}}\int \limits _{-\infty }^{\infty }\int \limits _{0}^{\pi }\int \limits _{-\infty }^{\infty }e^{ik(x\cos(\theta )+y\sin(\theta )-\rho })R(\rho ,\theta )\left\vert k\right\vert dkd\theta d\rho$

Various things happened in 1917. We had the Russian Revolution, the Great War was going on in Europe, and America came in on the side of the allies... and so a lot of people were being injured, and they had X-rays... Madame Curie was driving... an ambulance X-raying people. ...You could see some detail, but not very much, and it was realized that Radon's [object] formula... if it could be made to work, could turn an X-ray into a really good image that could show you what was going on. ...[T]hey didn't have the technology to do that. You had to wait 50 years for computers to be powerful enough to use that, and a company called EMI with a guy called Cormack developed the computers... the scanner... and that was the first scanning device. ...[T]he scanner that relies totally on Radon's formula, with a lot of other stuff. Cormack got the Nobel Prize, quite rightly, for doing that. So that's all based on math. ...Medical imaging has utterly transformed medicine, hugely reliant on maths. ...[I]t's an area I work in myself.

Sudoku and Killer sudoku are basically the same maths as scanning. [If you solve Killer sudoku you are using the math of tomography.]

This was the first photo of Saturn that was taken by a satellite. It was taken in the early 70s by the Voyager satellite. That was... beamed to earth by a transmitter with 30 watts of power... less than that light bulb. That light bulb, well you can see it, but imagine from Saturn, which is millions of miles away... [T]hey took this picture, turned it into numbers and... turned those numbers into a code, and that code had lots of [redundant] information put into it, so it didn't matter how far it went and how much it was distorted. You could reassemble it at the other end, into the picture... ...[I]t's absolutely wonderful. It's a combination of math, and technology and brilliance.

This technology is used all the time now. If you have a CD player, then you're using that technology. ...[O]n the CD you put lots of extra information to make sure that when you play the CD it doesn't get distorted or destroyed. [Error correcting codes... store the numbers... and other data... such that any errors can not only be detected but corrected. They work by asking extra questions to make the answers as different as possible so we can still tell the right answer even if it has mistakes in it.] This technology... was invented by a marvelous American mathematician working at Bell Labs... Hamming, one of my heroes, a fantastic mathematician, a brillian inventor of... error correcting codes [using... (Galois theory)], which he used to send all the information... essentially he created the mobile phone and the internet technology which could which could send the information around... But... it goes back even further... to a branch of math called Galois theory... invented at the end of the nineteenth century by a guy aged 19, and Galois's stuff is heavily used in modern technology now.

So this is error correcting, which brings us right up to date with the information age, sending huge amounts of information around the world, sending stuff around with Google. Absolutely fantastic.

Mathematicians have changed the world.

What is often called pure math, which is maths developed for interests only, suddenly finds applications in remarkable ways and leads to whole new technologies. Google was invented by two mathematicians who are now very wealthy. Let that be a lesson to us all.

[T]here are incredibly exciting times ahead and the great thing is that there is going to be young people driving us forward, and I can't wait to see where we get to next.

Quotes about Budd

For the last 20+ years, Dr. Budd has run a module at the University of Bath designed to help undergraduate students communicate mathematics to broader audiences while providing mathematics masterclasses to young aspiring mathematicians at the Royal Institution.
- Honoring Dr. Chris Budd SIAM News Blog (April 16, 2021)

[E]ven in his academic work it is clear that Budd enjoys breaking out of the university bubble and getting his hands dirty in the real world, working on problems that he hopes can "benefit society directly".
- Ed Goldsmith, Emily Maw, Sam Porritt, Sean Jamshidi, "In conversation with Chris Budd" Chalkdust magazine (Mar 12, 2018)

External links

Wikipedia

Wikipedia has an article about:

Christopher Budd (mathematician)

Chris Budd OBE, FIMA, C Math, NTF @University of Bath
Christopher John Budd @Mathematics Genealogy Project
Professor Chris Budd Professor of Geometry (2016–2020) @Gresham College

Gresham College
- Mathematics and the Making of the Modern and Future World (Oct, 2016-Apr-2019) 18 Video Lectures
- Mathematics in the 21st Century (Oct, 2019-Apr-2020)

YouTube Videos
- Maths and the Making of the Modern and Future World Playlist of 23 Videos

Quotes

Math Encounters (2017)

Quotes about Budd

See also

External links