500 Years of Mathematics: Are We Living In A New Golden Age?

my title is very ambitiously 500 years of mathematics why have I chosen this title we celebrate this year the birth of Thomas Gresham in 1519 and so I gave myself the task to think what has happened mathematically since Thomas Gresham was born and what will happen in the next 500 years so no small task okay so when I started to think about this I realized that Thomas Gresham couldn't have been born at a more interesting time mathematically because his birth or 1519 in very many respects marks the division between what I would call ancient or classical mathematics the mathematics of the Greeks and then the Arabs and modern mathematics the mathematics that we celebrate today and I illustrate this on the left with a proof of Pythagoras theorem using geometry which is how Euclid did it and on the right we have calculus and a bit of Statistics there as well if you spot it showing the mathematics that has happened since so what I want to do is take you through 500 years of maths now it is impossible to do justice to 500 years of maths in one hour or indeed a hundred hours so what I've decided to do is to look at a few things which happens at around about the time of Thomas Gresham and show you how they have gone on to dominates modern mathematical thinking and I apologize for all the bits of maths that I've left out in the process I put a few more in my transcript but as I said I've only got one hour to do an awful lot so 1519 really does mark the division between classical and modern maths it's where mathematicians got confidence that they could do something which the Greeks could not do and that forms the start of my talk and will be a theme which goes through all the rest of the talk so what is this great thing what is this great thing well it's the solution of the cubic equation so let me tell you a bit about this so here we have the quadratic equation ax squared plus BX plus C I should have put equal zero apologies for that and that equation has been around since the time of the Babylonians and the reason it was of interest to the Babylonians is that it's an equation to do with area and Babylonians needed to calculate areas in order to work out how much tax to pay okay so that equation the quadratic equation has been around for a very very long time and it was found that you could solve it and here is the solution to the quadratic equation I imagine many of you are familiar with this and that solution was sort of known to the Babylonians sorta known to the Greeks and in its modern form was derived we think in India so that is the solution to the quadratic equation but underneath it we have the cubic equation now the Babylonians knew about the cubic equation because it relates to volumes and they actually had tables of solutions which they calculated approximately so this equation but nobody knew how to solve it exactly and this question was open up to the birth of Thomas Gresham so what happened now well what happens was in the 1520's so the decade just after Gretchen was born a guy called Scipione del Ferro solved the cubic equation and as I said this was the first thing that had been done which was new since the Greeks well stuff had been done by the Arabs as well but they weren't actually that familiar with it as far as they're concerned this was new so Scipione looked not at the general cubic but at this particular example ax cubed + CX equals D which as I say is an equation in volumes and any cubic equation can by relatively simple transformation be put into this form and I thought well this is my last lecture of the series let's do some real mathematics yes good so let's see how Scipione solved it so Scipione said look let's think of there being two other numbers U and V so the difference is D and the product is C cubed over 3 now it's possible if you take a V then that's this divided by u you bring that over there and you find that u satisfies a quadratic equation and once you've found you you can find V so these two numbers can be found by solving a quadratic equation and we know how to do this so then I how skip you only thought about this there we are there you come up with those two numbers and then what you do is you say well let X be the difference between the cube roots of those two numbers and then if you cube it and they knew the formula for how to cube so you cube this expression if you cube you to the third you get you if you cube minus V to the third you get minus three and you get this stuff in the middle u minus V we've already sent is D and you can pull out a factor three UV to the third that is ex from up there that CC so you get X cubed equals D minus CX da da you have solved the cubic so providing you can find cube roots and that's kind of expected you can solve the cubic and there we are so that was a major achievement and like all major achievements even in mathematics it led to a punch up okay so let's have a look at the punch up so this guy here called Tartaglia also known as the stammerer came up independently with a solution to the cubic and in fact a more general case of the cubic but he was very suspicious that someone else would steal his ideas so he expressed his solution in the form of sort of an anagram and a poem you know as things were done in those days and he told this guy card on o who is one of the other mathematicians at the time but he swore card our notes of absolute secrecy but then contorno became familiar with the other solution by da Pharaoh and he thought well he should give Tartaglia credit so he wrote a book in which he explained tog conject Tartaglia solution and Tagle was not pleased about this and there was a big punch up as a result but the bottom line is that the cubic have been solved and it had been solved in the time aggression and that was the first thing that had been done which was new since the time of the Greeks and that gave everybody a lot of confidence that they could do new stuff so what I want to do now is tell you what the solution the cubic led to just that thing on its own and it led to an enormous amount of mathematics just on its own right so you've solved the cubic equation what's next well obviously the next type of equation you might want to solve is one instead of a cube there with a fourth power the quartic equation and fairly quickly it was found that you could solve the quartic I won't put up the solution it's a bit technical but it can be done so the next question was well can we solve the quintic equation there it is so this equation doesn't have a direct link to volumes like a cubic it's more of a kind of thing of mathematical interest although equations like this are now used all the time in computer graphics to represent surfaces so if you see something like an animated movie the chances are the shapes in that will be satisfying an equation something like this and people tried very very hard to solve this equation using the same sort of tricks and ideas that have been used to solve the cubic and the quadratic and the kortek and no one could solve it and after a while people began to think well maybe there isn't a way of doing it and árbol who is a norwegian mathematician who as you can see was extremely young at the time was the first person to show rigorously that the quintic equation could not be solved by extracting cube roots fourth roots and fifth roots it was an incredibly major piece of work our book did lots of other stuff and he would have done much more stuff had he not died young but now one of the main prizes for mathematics is named after him the arbol cries close following closely onto our bull was Galois here we are who also died extremely young arbol died I think he died blood poisoning Galois was shot in a duel and at the age of 19 Galois also managed to show that the quintic equation did not have a solution and whilst our balls proof though correct was very technical and rather specialized Galois develops a branch of mathematics now called Galois theory after him from which the solution the cubic was just one of the consequences it have many many other consequences Gallo is generally regarded as one of the all-time great geniuses of mathematics unfortunately he was also revolutionary and came to a very sticky end as I have pointed out what else did Galois manage to show as a result of solving generating Galois theory well he resolved two problems which had been around since the time of the Greeks one of which was could you trisect the angle using ruler and compasses and he showed that that was not possible also another one from the Greeks was could you construct a cube which has volume twice the original which is equivalent to solving the cubic equation X cubed equals two that's another linking with the cubic and Galois was able to show that that could not be done with ruler and compass and that was all done using his theory which he developed at the age of 19 no pressure okay what else came out a Galois theory when I say Galois developed this theory to solve the quintic equation solve those two problems at the same time and Galois theory or part of it the theory of modular forms is now used all the time in such devices as mobile phones and CD players to encode information onto these devices in such a way that it is error-free and all of that follows from his theory and none of that would have been really thought of if you haven't tried to solve the quintic equation so we see how this kind of theme ripples down the ages Galois as well as developing Galois theory formulated a lot of the mathematics of symmetry he was able to oh that the quintic equation didn't have a solution by showing that if it did there'll be certain symmetries between the roots and those symmetries were just not present in certain equations and to study the symmetry of these equations he developed that branch of mathematics we now call group theory so group theory is the theory behind symmetry it's the theory behind how things combine together to produce other things and the sort of structures that you get again developed to help solve the quintic group theory is incredibly relevant to studying symmetry in objects like octahedra icosahedra and so on and so on and that's very very relevant to modern chemistry so group theory is important in studying crystal structures group theory has everything to do with how objects are rearranged when you transform them and when I went to Cambridge back in 1979 as an undergraduate the Rubik's Cube adjust come out and we found as mass additions we became extremely exciting people because we could use group theory to solve Rubik's cubes and that made us kind of cool for a while so there we are and unfortunately group theory well not unfortunately I have to confess guys I'm a morris dancer and you can use group theory to help design morris dances okay so I'm in there somewhere and it has many many applications and as I say this is 500 years of mathematics I want to celebrate all the great achievements in the last 500 years and whilst I was at Cambridge in 1982 a-amazing theorem was proved by John Conway and various others which managed to classify or every single group that you could have or in particular the finite simple groups that's sort of tied up the entire subject of group theory all finished the basic classification of them and that theory is regarded as one of the big achievements of the last 500 years of mathematics and boy did we have a party in 1982 I was nothing to do with the theory but the alcohol was widely spread around on that day so that's one sort of thing that solving the cubic led to it led to solving the quintic led to group theory which led to many many other things but it led to something I think of even more important and that's the understanding of the importance of complex numbers so going back to the quadratic equation here we have it there's the quadratic equation and anyone that's ever tried to solve the quadratic equation and I imagine many of you have at some point in your life had to solve the quadratic equation we'll have noticed that you have this square root term here which has b squared minus 4ac and if 4ac is bigger than B squared you have a 9s term in here and as we all know Utah take the square root of a negative number well at least you can't take the square root and end up with another real number so the Babylonians knew about this the Greeks knew about this and all they basically said was well if this is negative you don't he doesn't have a solution so we weren't worried okay you can solve it but you have to do something which mathematicians love doing which is if you can't solve something you invent something else which is the solution and then you've solved it okay which isn't quite as stupid as it sounds but mathematicians resolve this by saying let there be a new number I which has the property that when it's squared gives minus 1 and we can call that the imaginary number and then we can go and look at what properties that has even if it doesn't really exist lots of maths is like that and suddenly you find it's all very useful so this was develops really so you could solve prints like this so give an example if you take the quadratic equation x squared plus 4x plus 5 which we're going to meet again later in the talk so remember it guys this has solution X is minus 2 plus I am minus 2 minus sign okay so so what well as I say the the attitude of the Greeks and the early earth mathematicians was that if the quadratic equation had a negative sign onto the square root then it doesn't have a solution and there's one branch of understanding which says if something can't be solved and doesn't have a solution it's not worth solving but then the people that were interested in the cubic so Cardno again and another master Titian called con beli notice that if you wanted to solve the cubic and you use this strategy which I explained earlier where you look at these two numbers U and V which satisfy a quadratic equation then sometimes there are examples of cubic equations for which the two numbers U and V satisfied a quadratic equation which could not be solved however if you solved it in terms of these funny complex numbers and then looked for the solution X of the cubic in terms of them the number you got actually was a real number in other words by going through the complex whelmed of solving an equation in complex numbers you could actually find a real answer to a real problem and that could have made complex numbers respectable so even though they couldn't actually make any sense of what the number I meant it turned out to be very very handy in computations and that made them respectable and people started studying them very seriously and without complex numbers much of modern mathematics science and shirring would not be possible so here we are another major part in mathematics which came out as solving that problem which arose at the time of Thomas Gresham okay so those are complex numbers so just to kind of finish polynomial equations off here is the great Gauss Cal Friedrich Gauss who in many people's opinion was the finest mathematician of all time and he proved a result called the fundamental theorem of algebra and the fundamental theorem of algebra said that any polynomial equation that you can write down can always be solved in terms of complex numbers in other words they once you've got the complex numbers you can solve all polynomial equations and that was a very very important result again one of the other big theorems in mathematics he didn't show you how you could solve it but he did show that he had a solution so that's a start okay later on we're going to come across an equation where we don't actually know whether it has a solution or not which is a bit more worrying so there's old Gauss one of the greatest Massa tuitions at all time here is another one of the greatest mass additions of all time this is Euler who sort of came before Gauss in after Newton one of my favorite mathematicians he had a great sense of humor and we're still publishing vast amounts of mathematics fifty years after he died he had so many notebooks and he lived to such a long age that people were still writing up his stuff long after he died and oilers had huge number results I think they had some poll recently for the five best formulas of mathematics and three of them were due to him okay quite incredible I'll come back to this at the end of my series next year but possibly his most important achievement was that he found a link between these complex numbers that we're looking at to solve polynomial equations and trigonometric functions now I want to kind of dwell on this for a minute the Greeks knew all about trig trig is triangles its properties of triangles they knew about the function sine and cosine which are the opposite and adjacent sides of a right-angled triangle okay so these are functions which have been known about and studied in the context of geometry polynomial equations were quite different they were the realm of algebra they were how numbers combine with each other and what Oyler found was a link between geometry professor geometry and algebra we don't have a professor of algebra but we don't need one because they are the same subject and Euler found out why okay so he found this link okay so the next slides a little bit technical but we need to get through it oiler came up with this number now we know it as Euler's number or e which is this limits it's 1 plus 1 over N to the N and this number comes up very naturally in the study of growth and in particular compound interest which is how he was thinking about it and it has a value it's 2.71828 1 8 to 8 or it's 1 plus 1 over 1 factorial 1 over 2 factorial so on so this is Euler's number it's one of the great numbers of mathematics it's up there with PI as one of the great numbers and that is the graph of e to the X as a function of X so Isla studied this number in the context of how things grow and compound interest and then you thought help what happens if you let X in the formula e to the X be a complex number okay we've got these things which are crazy already let's do something even crazier and this is what he found he found that if theta was a natural normal number real number and I is a square root of minus 1 that e to the I theta is cos theta I sine theta an incredibly important form one of the most important formulas both in the theoretical nature of mathematics and also the practical applications in maths and there we have a link between I which comes out of polynomial equations II which is this function to do with growth and there the trigonometric functions sitting there looking at us and you can from this find that cos theta is et I theta plus eat and minus I theta over 2 sine theta is the difference provided by 2i and if you put theta is PI into this expression you get e to the I PI cos minus 1 or if you prefer e to the I PI plus 1 equals 0 and that formula wins hands down as the best formula mathematics ever produced ok every poll puts this top I do my I do a poll my students each year and I have to deliberately exclude this formula because otherwise they will come up with it okay so that's Oilers identity which he proved in the 1800s 1700 so the eighteenth-century linking polynomial equations trigonometric functions and this number E and we'll come back to this in a minute as well but I just want to sort of finish this to the first section of the talk which is largely about algebra and polynomials with just the kind of big problem in algebra which was only sold very recently which is Fermat's Last Theorem so I think many make many of you may have heard of this here we have Fermat and in 1637 which isn't very much long after Gresham College itself was founded so Gresham College was in the end of the sixteenth century Fermat's in the margin of a book wrote here's the equation a to the M plus B to the N equals C to the N I have found a truly marvelous proof that this can never be solved if n is greater than 2 and a B and C are integers now to give firm that credit the case of N equals four was proved by him and we know that and there's lots of evidence for this the cubic case going back to the cubic equation keep the theme going was done by our friend Euler ok significantly harder firm that did the case N equals four by factorizing you couldn't factorize the case N equals three but we managed to find a way around that but no one managed to find a general proof during the time of oiler and certainly not one which would fit in to the margin or anything closer to the margin and 450 years later not quite 500 but not far off this was at last solved and the efforts to prove it led to lots of new mathematics so algebraic number theory modularity theory lots of notoriety there was a prize for it which lots of people try to win and didn't and there's an entire Star Trek episode based around the Fermat's Last Theorem which would you wish to watch it and it was finally proved in 1995 on the left by Andrew Wiles who was working in Princeton at the time and everyone has heard of Andrew but we must also forget that it was a collaboration with and this guy richard taylor and they work together to finally prove Fermat's Last Theorem and that's 1995 so 450 odd years after Gresham and that again is one of the big achievements in the last 500 years I just like to say basking reflected glory but when I was at Cambridge I was vice president of the Cambridge Mass Society otherwise known as the are comedians and the president was Richard Taylor okay so I have to put him in that okay so that's algebra and that came out of the cubic equation which was bought off just after Gresham was born but something else happened just after Gresham was born and that led to another branch of mathematics and another area of understanding and that was the the publication by Copernicus of the heliocentric theory of the planets so up in time till the time of Thomas Gresham the basic theory about the planets was that they all went around the earth and the earth was stationary and the Sun went around the earth as well the Ptolemaic theory and it was only just after Gresham was born that Copernicus turned all this round and published his theory which was that the earth went round the Sun and the planets were around the Sun on circles interesting fact in 1519 Copernicus formulated theorem in economics which has since then become Gresham's law and it is the same Gresham I checked so Thomas Gresham produced this law in economics which i think is something like bad money drives out good or something along those lines and not an economist it's sometimes called the Gretchen Copernicus law and and Copernicus got it in 1519 and even though Thomas Gresham was incredibly gifted I don't think he did it in 1519 himself okay so hard on the heels of Copernicus Kepler in 1610 using a lot of data produced by Tycho Brahe Hey whoo Kepler was Brahe his students took those data combines it with Copernicus's we're thinking about the planets and came up with the Kepler's three laws of kinematics the first that the planets went round the Sun and ellipse the second that planets swept out equal areas in equal time that's called conservation of angular momentum now and the third was square of the period of motion was the cube proportional to the cube of the size of the ellipse these extremely important laws say again just after Gresham was founded why they important because they perfectly described planetary motion really very accurately and from that description you could predict forward very very accurately and it made everyone realize that the Copernican view of the world was the correct view of the world okay so these were laws of kinematics again just after the time of Thomas Gresham so these are laws of kinematics which are descriptions of how planets move and shortly after that we get the laws of dynamics which are the explanation of why they move in that way so two two big heroes of this for me Galileo who was born in 1564 so again not long after Gresham was born who was born on the 15th of February which is my birthday I'm slightly younger than him and he died in 1642 and just to keep the theme going Newton was then born in 1643 and lived through 217 27 and between the two of them they formulated the basic laws and mechanics which have acted as the bedrock for much of physics ever since one of the laws that Newton this.e the Principia by the way was published in 1692 and one of the laws that Newton published in the pre-compute Principia was the law of universal gravitation which is that if you have two bodies x and y at positions x and y then the force between them was proportional to the mass product to the masses of those two bodies and divided by the distance between the two bodies squared so that's Newton's law of gravitation and what Newton did was he took that law of gravitation and he used that to deduce plus three laws of motion again an incredible achievement so these laws that Kepler had formulated by observation Newton was able to show mathematically had to be true provided that you had this law acting okay so it's really interesting the way that Newton published this Newton published his results in the Principia in latin and as I said Kepler's Gresham's birth marks the division between modern and classical mathematics and Newton decided to formulate his explanation in classical terms so not only did he publish in Latin but he put all of his results in the language of geometry but Newton didn't derive them using geometry what Newton did was he derived them using his own method of calculus so he got them with calculus and that's how we do them with students nowadays but he put express them in the language of geometry so calculus I'm now going to say something really really arrogant or bold or whatever so this is my opinion in my opinion the invention of calculus is the singular most important bit of mathematics which has been done in the fight last 500 years I'm not saying there's being a ton of other stuff which is important but in my opinion that's the big one and here's even more awkward I think it has the single most important creation of the human mind I'm happy to argue back with you later I don't think many people would totally disagree with the first one I mean and whatever you were your thoughts the invention of calculus is a decisive moment in the development of mankind okay so what is calculus well calculus is basically the study of how things change that's all it is and the two concepts which Newton introduced were one which was the derivative if you draw a graph of a function f then the derivative is the slope of the graph if F is the position of an object then the slope of the graph is the speed of the object the other concept is the integral which if you have a graph it's the area under the graph so equally if I draw a graph of the speed of an object then the area under the graph is the distance travelled by that object so those are the two fundamental concepts of calculus which Newton kind of came up with just in his bit of technical stuff he's if you can define the derivative is the limit of the difference between two things divided by so if you take something evaluated at this number H take off the other one and divide by action let H ten to zero that gives your derivative the inverse is the integral and here's an example the cubic again there's its derivative as its integral so Newton worked all this out and then he used the calculus to as I say find the equations of motion planets from first principles again this led to a punch up why did this lead to a punch up well Newton kind of claimed credit for the calculus and the British mathematicians all rallied behind him to claim credit but arguably this guy also deserves credit for it this is live knits who would treat another true genius a true polymath living in Hanover came up with many of the same ideas and a far better notation than Newton and indeed the way we think of calculus and write it down nowadays is largely following liveness however many other ideas for calculus were floating around the time in Europe and even in India there's lots of evidence that the indian mathematicians were thinking the same way ideas which were again bubbling up around about the same time as Thomas Gresham now I've told you about Euler and his discovery of the link between complex numbers and trigonometric functions Euler did something very important in calculus Leiden it's a Newton were both thinking in terms of functions changing in time all of them went on to extend it to look at the way things and change in time and space and introduced the idea of what's called partial differential equations which are descriptions of the way things change in time and space and these rather ferocious things which oil are derived are the euler equations and if u is the velocity of the air that's how it changes in time this how it changes in space that's the pressure and to rough approximation those are the equations of the weather and that's what we saw every day let me solve the weather I'll give you some more extended versions of these towards the end of the talk when we talk about where maths is going in the next 500 years so here we are partial differential equations these are descriptions of how things change in time and space and again controversial opinion by me ordinary that's equations which depend on time only and partial which ones which depend on time and space are one of the most powerful tools that we have for studying problems in physics engineering and biology they are the fundamental way of describing pretty well all physical and chemical and biological processes Newton came up with the basic idea behind them and six that was in the 1600s and four hundred years later we still know damn all about them okay this is how mathematics is sometimes I thought my job is to find out a bit more about them let me tell you what we do know about them this equation here which looks a little bit like a quadratic and those reasons for that is called a linear second order differential equation and this sort of equation here arises all the time when you study vibrations in mechanical and physical so systems and Euler worked out that you could solve this by substituting in a solution X which involves his famous number e raced the power lambda t if you substitute this into here and do some manipulations you find that lambda satisfies a quadratic equation remember where we came in beginning of this lecture quadratic equations we know how to solve these the solution of that is this it involves these scary numbers I boil it didn't worry about that he said okay that's what the solution looks like but I know that these are related to sine and cosine and if you substitute in this formula you get that solution is this and here's an example of there's an equation for Vibrations you go through all this complex numbers stuff and you end up with a solution which is perfectly physical okay so that's rather good and this describes damped harmonic motion which looks like that and if I do this okay all that the microphones wobble around for a bit come to rest perfectly described by that if you have more complex differential equations ones involved in third derivatives you get back to the solution the cubic again so it comes up every one if you want to do something more complex this is the equation here for the motion of the pendulum those of you who went to my talk on chaos will have seen this equation and this is one of the many equations which we cannot solve there are two ways of solving this one way is to say let the solution be a function and we'll call that function a name and now we've solved it sort of true the alternative is to solve it on a computer but again in terms of kind of the evolution of mathematics to great discoveries in the 19th and then early 20th century allow us to really make big inroads one was punk ray who found that by combining differential equations with geometry that's a professorship of geometry you may not be able to solve them exactly but you can draw a picture to show what's going on and then in the early 20th century a mean erta arguably the greatest female mathematician of all time arguably one of the greatest mathematicians all time combined differential equations with the group theory that gal wad come up with and found fantastic links and that's led to enormous results in physics ok so that's where Thomas Gresham Copernicus and all that's they Don - that's the end of my kind of second section so I want to talk about now some of the other things that have happened since Gresham and one is the what we call linear algebra and matrices so this is a very unglamorous everyone knows about Fermat's Last Theorem they've made movies about this no one's yet made a movie about matrix theory but I think it's possibly one of the most useful bits of maths with many applications including Google and even brexit I wrote an article for The Times about this series was a question that see if we can solve it there's my daughter and this is her Facebook page and I was 32 and my daughter was ball our combined age is now 86 how old are we anyone solve that so this is a sort of puzzle again people were looking at at the top seven knots in sorry what was you sir 27 yes she is 27 years old how do we do that by the way fantastic so the way to do it is we do some algebra again a bit like the cubic that's my age I'll call myself X my daughter's why she was I was 13 when she was born which means I'm 32 years older than she is so X minus y is 32 our combined age is a 6 so X plus y is 86 and the direct way of solving this is you add these two together the Y's cancel and you get 2x is 118 and you find out from that that I am 59 ha earns if you subtract you find that my daughter is 27 so you're absolutely right so that's a trick there's lots of books on recreational maths which involves similar kind of stuff this doesn't generalize very well and if you want to solve more kind of challenging problems the maths that you use was developed in the 19th century by a guy called Cayley in Cambridge and that he thought well the way to kind of solve these problems is to think about the pseudo structure and he invented the thing called the matrix so s is a matrix here 1 minus 1 1 1 and laws for how you use these things and the law for X minus y is 32 next plus wise 100 that should be 118 apologies looks like this expressed in matrix form now why is this important well this sort of way of writing things down in terms of matrices is very very useful in two two dimensions or in three dimensions to look at geometrical transformations and this is the basis of much geometry and now computer graphics and if you do it in four dimensions it's the basis of special and general relativity so it's very very powerful concept all coming out of the need to solve things like that if you want to solve our equation what you do is you construct a thing called the inverse of the matrix of a times X Y is a B you sort of divide by the matrix X Y Z inverse a B for our problem a inverse can be calculated to be that and that leads to this solution which is the one we got by adding them together and dividing by two while subtracting and dividing by two but why should we bother who cares how old my daughter is when I care about my daughter its but who in general cares well solving equations like this with many many unknowns is actually rather important to modern society here are some of the applications you can read them for yourself for solving many many unknowns but certainly my own area of weather forecasting and the general area of the internet or rely on doing that sort of matrix inversion we couldn't invert matrices we couldn't basically have technology the first algorithm to do this was come up with by Gauss in the early 19th century we still use this although it's a bit slow lorne algorithms like conjugate gradients and multigrid again I'm afraid this is sort of my day job are what's used all the time to solve many of our day-to-day problems so when you book an airline ticket on an airline that is booked by solving this sort of thing when you look something up on Google that's done by so many sort of equations so I've used the words here algorithm which needs me neatly into my final kind of section on what I think is important which is algorithms and this will come as back bring us back to the cubic so one of the main ways that the mathematics that I do impacts or anyone does impact on your lives in a day to day basis is through algorithms I've read you mentioned Google the Internet itself all these things that make your mobile phone work Amazon even dating websites are driven by mathematical algorithms and the first algorithms were developed guess what to solve polynomial equations the same equations that we started the talk with and the Babylonians actually came up with an algorithm for solving the quadratic equation and there we have an example x squared equals 2 and they're kind of perspective was well if you want to solve this if X is a solution divide 2 by its take the mean that's probably better and if you start out with one as a guess then 2 divided by 1 is 2 1 plus 2 is 3 3 over 2 is 3 over 2 that's our next guess substitute that in you get this and in and in and very very quickly you end up with a solution a quote an approximate but still extremely accurate solution or the quadratic equation and that's the solution to a few more decimal places okay so that was our first algorithm developed to solve quadratic equations and it was extended very quickly to solve the cubic equation there's a cubic equation using a method that Newton came up with so if you want to solve the general equation f of X equals 0 xn is a guess to the solution you can get a better guess xn plus 1 by saying xn minus F divided by its derivative this is called Newton's method that's one of the first really important algorithms and that is used all the time in modern computers to solve problems certainly in such subjects as weather forecasting if you take the cubic there we are then there's Newton's iteration for solving this and that works very very effectively to help you find the solutions of the cubic so these are algorithms which we've developed to solve polynomial equations now algorithms are everywhere I've just listed a few you want to solve a differential equation you use them run the cutter method if you want to work out for a series in other words do anything to do with signal processing in other words mobile phones you use the fast Fourier transform if you've got lots of data coming in you want to update your knowledge of a system use the common filter again that's used in mobile phones and if you want to find a website you use the Google page rank algorithm based on linear algebra these are just four algorithms of the men either to use but I would say those are probably four of the most important and these are all course all implemented on computers and it wasn't for various math additions here are perhaps the four most important Babbage Lovelace cheering and von Neumann we wouldn't have computers and therefore we wouldn't have the modern world so the invention of the computer is again a hugely important thing that maths has done for us in the last 500 years when I wrote this lecture I posed the question given we've done all this are we living in a mathematical golden age and I would strongly argue that the answer to that is yes I think we are I find it incredibly exciting to be around as a mathematician at the moment I have three reasons for believing this my first reason is just using my eyes so lots of long-standing mathematical problems are being solved and that includes Fermat's Last Theorem one great conjecture and lots of new exciting problems are being posed so that's one reason a second reason is that the fusion now between mathematics and computers and all the data that we've got allows mathticious to be really creative and deal with major problems of huge complexity that were completely out of reach of people in the past and thirdly the number of applications that we see for maths is just growing exponentially there seems to be no limits for it I'm at a workshop at the moment in not English I'll go back to that off this talk one of the problems that's come up is using maths to help design ornamental flower arrangements you know you may laugh but how many of you have given flowers to someone you know very important and the number of applications in math is just going poor doink but as I said in my last lecture we had exciting times but we do need to encourage people to go into maths or issues maths in their work so I think we are in a golden age and I very much hope that Waterstone okay so where are we heading next well good question at the beginning of the 20th century Hilbert's who mind the greatest maths tuitions of all time posed 23 problems and at the Paris Convention of mathematics and suggest that those 23 problems would drive mathematics the next 100 years and to a large extent he got it right okay so these are called Hilbert's problems and lovely quote who must not be glad to lift the veil behind which the future lies hidden to cast the glass and yet advances of our science and all the secrets development during future centuries and basically got it right as I said most of his problems have now been sold but not quite all so everyone thought wow beginning the 21st century where's maths heading next and an outfit called the clients to published a similar list of challenges not 23 this time only 7 which had been called the 7 Millennium problems which are designed again to stimulate the progress of maths in the next hundred years or maybe beyond one query hypothesis which was the first of these has actually been done already so 6 to go but here are the six remaining problems which are supposed to drive us forward into the next 500 years ok I'm not going to go through all of them I've only got the 5 minutes left but it gives me I'm just going to go very very quick whizzed through three of the problems one of them the first one is the Riemann hypothesis this object here is called the Riemann zeta-function it's a sum of 1 over N to the Z and where Z is the number that oil off remember him found a link between this and the primes so the sum is equal or products the primes he used that to show that the sum of the reciprocals of the primes was infinite which allowed him to deduce that the nth prime number P n was approximately n Times log n so he got a lot out of that and 100 years later a Riemann there's Riemann in 1859 again one of the great masters of all time studied the Riemann zeta-function in more detail and said well you could tell far far more about the primes and if something is true and what he he said would be true was if the the what we call the zeroes of the zeta function all have the property that the real part of them is a half okay what do we know about this well not a lot millions and millions of zeroes have been computed every single one has real plantar half there's currently no evidence at all that none of the rest of them do but no one knows books have been written on this plays have been done on this films have been done on this but no one knows the answer okay so that's that one here's a simpler question you can find the sum of the reciprocals of the cubes and you'd be doing well second one this is these things here called the Navi of Stokes equations they're an extension of Euler's equations for whether these are basically the equations for the whether these equations we solve every day to predict the weather however we have a certain amount of faith in solving them because we don't even know if they got a solution okay tricky the problem is do these have a solution where everyone's trying to think very hard about this it's very closely linked to the phenomenon of turbulence which da Vinci studied which is where you get lots of structure on small scales an interesting rough DaVinci died in 1519 have you heard that date somewhere before so there's a nice continuity huge amount of work into trying to solve these things but we still have no idea whether they got a solution and just look at all the applications for that so this is an error I work in myself and I'd love to know the answer and the final big question is called the P versus NP problem which is the big question of computer science here's the problem let's suppose the size of something we want to solve is n4 that's maybe the number of digits if I multiply two numbers with n digits it takes n square operations to do that if I want to factorize a matrix with n and n by a matrix that takes n cube operations this is what we call polynomial time and that's generally regarded as easy if I want to factorize a number with n digits it takes two to the N over two operations which are called exponential time this is phenomenally hard and hard means useful if you're trying to build computer security around it if I want to solve a jigsaw which has n pieces it's if I take one piece at random and they take another one and take all possible combinations it takes n factorial time which is basically hard to do it but I can look at a jigsaw and say yeah that's right so it's very quick to check and that's what we call NP and the big question is if you can check something quickly can you solve it quickly that's called the P versus NP if we could so that peak was MP we could solve everything the whole world could be done including economics cryptography and pseudocode I've been after years of searching we don't know we have not a clue and it remains a huge challenge and I personally think of that as my equivalent as the cubic as the big challenge for the next 500 years and I would be amazed in 500 years whether we solved it but hey what do I know so it kind of concludes we started with the cubic who at the time of Thomas Gretchen's birth would have known where that would lead to and yet it's led to the modern world I've given you those clay problems who knows what the solution of those will lead to but much more interestingly there's lots of questions out there we don't know even what the question is and they're going to be the most interesting of all and there's bound to be vast numbers of these and hopefully when someone in 500 years time gives a talk they'll be talking about those and marking the Year 2019 as a division between their ancient maths and then modern maths thank you very much [Applause] you

Channel: Gresham College

Views: 18,622

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Keywords: Gresham, Gresham College, Education, Lecture, Public, London, Debate, Academia, Knowledge, Mathematics, maths, Geometry, Chris Budd, IMI, thomas gresham, algebra, Quadratic equation, galois theory, group theory, euler, riemann hypothesis, navier-stokes, runge-kutta, fast fourier transform, complex numbers

Id: kAAhmHLN6ac

Length: 60min 5sec (3605 seconds)

Published: Thu May 09 2019