Productive generalization - Timothy Gowers

[Music] good evening everyone welcome to the Science Museum I'm Roger hi Phil the science director it's a thrill to be here tonight to introduce the oxford mathematics public lectures they're being webcast from the imax of the museum hello world now mathematics lies at the heart of this museum in our mathematics gallery zaha hadid crystallized abstract mathematical thinking into beautiful physical forms it's central to the cryptography and our top-secret exhibition to the AI and our driverless exhibition if you go to our science city gallery which is brand new you'll see Newton's Principia and the story of the most powerful way that we have to understand the universe which is of course a scientific method mathematics also underpins the scanners the crystallography the epidemiology and more in our massive medicine galleries which opened last week now in a lecture given 60 years ago Eugene Wigner described the unreasonable effectiveness of mathematics mathematics is indeed a universal language it's a means to understand everything that we do it's a tool for in for increasing our thinking power tonight we're going to celebrate its awesome importance with our guest of honor fields medalist Timothy gaurs and with the help of Hanna Frey of UCL who's giving the RI Christmas lectures she's the author of hello world and also a science museum group trustee but first of all I'd like to introduce the Oxford mathematical Institute's director of External Relations please welcome professor Allen Gauri Ali thank you many thanks Raja you know it's an important event when you have a series of a speaker introducing each other sort when maybe have a few more - so what we've been trying to accomplish over the last six years or with the oxford mathematics lecture but also with extended program is to bring the best of mathematics to the public - Electress series first in Oxford but then in London and now to other parts of the country and I'm I am partically grateful to both the science museum and also to Anna Frye we've been great friends and partners of Oxford mathematics and believe in what we're trying to accomplish and I hope we have many more such collaboration but also to help us in our goal I have to give special thanks to XTX market sponsor of the oxford mathematics public lecture series XTX market a leading quantitative driven electronic market maker with offices in London Singapore and New York and I'm glad to see that many of you came to the event today today it's a great honor for me to introduce Professor Tim Gao the rose board chair of mathematics at Cambridge University and fellow of Trinity College since you will hear more about Professor Gower's life mathematics and opinion I've been given the impossible task to introduce him in a few words for inspiration a naturally turned to short description of historical figure I've been recently at Yale University and you refine right at the center of the University beautiful graveyards and you'll find the grave of the great Lausanne saga one of the greatest physicists of all time and simply on if it's gravestone is described by Nobel laureate etc so it could be appropriate to describe professor gory Gower's as field medalists etc and be done with it but it would be a gross mischaracterization of his other accomplishment influence not only on mathematics directly but on the way mathematician work and organized he's also been a passion passionate about explaining mathematics to the public and as the unique record of filling both one of the shortest introduction of mathematics with Oxford University Press and an encyclopedic one maybe the large the longest introduction with Princeton University Press if you follow him on social media as I do you will also discover true true humanists passionate about mathematics but also caring about all social and political aspect of today's world so for my second attempt and knowing that professor goers as a particular interest in music I thought about another short posthumous description which seems to be a good fit this one is about the 15th century English composer John Dunstable who was described as mathematician musician and whatnot tonight we will first hear about the essence of mathematics which is about generalization abstraction then professor gamez would be in conversation with Anna Frye and my hope of course is that a teaspoon we will learn all about professor Gauss etc and whatnots so without further ado please help me welcome professor goers to now Forrester thank you very much I'll just start by saying that this is the largest slide I've ever had the pleasure of presenting I also want to say that I don't think I can quite in half an hour it's what I have live up to the promise of the title of this talk but I can do something towards it because really to say exactly why I believed it will never run out of questions is it's quite a long and complicated task but this is just one I suppose that the emphasis is on one reason because there are plenty of reasons but I suppose the fundamental reason is that mathematics has a sort of Hydra like quality that when you answer one question it somehow gets ten other questions and one of the things one of the sort of big etting mechanisms is generalization and that's what I'm focusing on this evening so before I get underway there are two types of generalization that one could be talking about one is generalizing ideas of mathematics all definitions and the other is generalizing statements and theorems and lemmas and things like that so I'm mainly going to be focusing on generalizing statements but I will say a little bit about generalizing concepts because I think it would be a shame not to have a little bit about that because that is also very very important so let's just dive right in the first two of these concepts are things that maybe I don't know how much mathematical experience people have on average in this room but if you've done maths at school then I would expect it at some stage you've come across something like X to the 3 over 2 and you may remember if you came to understand that and if you didn't table I'm about to explain it sort of feeling that how can X to the 3 over 2 make any sense because you can't take a number one and a half times and sort of multiply those together that just doesn't make any sense so how do we make sense of this concept similarly what about e to the Z if said complex so raising to a complex power that sort even madder than raising to what to the power of one and a half I mean what would it mean to take one plus I lots of E and multiply those together that's even more sort of nonsensical and just to sort of get even worse what could it possibly mean for a shape to have a dimension that wasn't an integer or come back to that in a minute so let's just quickly go through those and then I'll move on to generalizing statements to the main thing I want to talk about so the way that we decide and in each case it's slightly different the process that we use to generalize the concept so if you want to generalize powers to non integer powers one thing we do is just focus on this rule here that X to the M plus N equals x to the M times X to the N although many school children will say that it's X to the M plus X to the N because they always believe that in mathematicians terms all functions are linear but they're not so there's a you're on has the stress at X to the M plus N is X to the M times X to the N or not XDM plastics again and here's the reason or illustrated with an example if I take X to the 5 that's just x times X times X times X times X and if I just split that up as 3 lots of x times 2 lots of X I see today so 3 times X to the 2 once you've seen that example it's pretty obvious at X so the M plus M is always X to the M times X to the N because it's just M plus n X's is the same as M X s times n X's well I don't really mean end EDX as I mean n lots of X multiplied together now the point of the way we then use that is we say well that's a rule and there isn't really a an objective meaning to X to the 3 over 2 it's up to us to choose the most sensible meaning we can and the way to choose the most sensible one is to choose the one that preserves this rule so let's just assume that we've got that rule that would tell us that X cubed would have to be X to the 3 over 2 times X to the 3 over 2 because 3 over 2 plus 3 over 2 equals 3 so if we want that rule to be true when m and n are both equal to 3 over 2 we need this equation to hold but that tells us that X to the 3 has to be the square root of x cubed or if you're feeling very alert you'll object that X cube might be negative or there are two square roots and so on but the convention is we'll assume that the number on the X is positive and we'll take the positive square root that's just convenience rather than some objective reality but it is very convenient okay so that's given us a very good answer to what X to the 3 over 2 is then you can build on that and work out what X to any fraction should be and then you can use other arguments to gift and for from fractions so irrational numbers and you can carry on and on let's move on but that's and that it takes us so far but it doesn't take us to something like this what would eat at a root 2 plus 3i be and again if you've done on a level a mess you'll know how we how we sort this out if you haven't I'm going to ask you to take something on trust so the trick here is to think of e to the square root of 2 plus 3i not as raising e to the power the square root of 2 plus 3i but to reformulate it and the way we reformulate it is to use the following formula which you need to have done some calculus sort of reasonably advanced kind to be able to justify this formula so if you haven't seen this this is the thing I'm asking you to take on trust it is the case that for all real numbers X e to the X is 1 plus X plus x squared plus X s squared over 2 factorial plus X cubed over 3 factorial and so on and this dot dot means you this sum gets closer and closer to some to some number and that is the number that we take us the definition of e to the X sound stories take us the definition of the sum of the series and that turns out always to equal e to the X when X is real we know how to make sense of e to the X when X is real but the great thing about the definition on the right is that now it's much easier to make it to make sense for complex numbers because all we need on the right is addition dividing by integers multiplication and a limiting process and all of those addition application dividing by an integer and taking limits all of those make very good sense for complex numbers they generalize straightforwardly so the left-hand side does not generalize straightforwardly but the right-hand side generalizes very straightforwardly so we take the right-hand side as the definition when we have a complex number okay so now let's move on to fractional dimension so this this collapsed chair is supposed to represent the sort of existential despair that you might feel when trying to conceive of a shape that has sort of two and a half degrees of freedom or something like that so we know there's a one-dimensional shape is one where you have sort of one degree of freedom in two dimensions you have sort of length and breadth or something and then for three dimensions you add height for what would two-and-a-half dimensions be it's not at all clear it seems to say the very notion of a degree of freedom has to be a whole number and indeed it does so we have to do the same trick finding some other way of thinking about dimension that will allow us to generalize it so let's see what we do so the idea is to focus in two and three dimensions on area and volume but since the area is very much a two-dimensional concept involving this very much a three-dimensional concept and I want to have something that isn't an integer I don't want to think of I don't want to call these area and volume I want to call them amount of stuff so another familiar fact is that if you take a two-dimensional shape and you expand it by a factor two in every direction you get a new shape so it has four times the amount of stuff as the original shape so you can see it very clearly hey you even fit one of these squares four times into the big square and what's four what's the significance of the four it is that it is 1 times 2 so it's 2 times 2 ie 2 squared if you write that it'll be 2 very little 2 on top and that 2 is telling us that we're a two dimensional shape and similarly in three dimensions if I double every single direction of a cube I get a new bigger cube into which I can fit eight copy of the original cube and 8 is 2 cubed the facts instead expand by a factor of 3 in every direction I get 27 I get a shape into which I've hit 27 cubes and 27 is 3 cubed and again it's the fact that I'm saying cube which is to the power 3 it's that 3 the one that comes after the words to the power that is telling me what the dimension is in that case so now let me show you a shape or I won't actually show you the shape I will show you a process that eventually leads to the shape the process is this you start with a line segment that goes from this point to this point and you divide it into 3 equal parts and you replace the middle part by any the other 2 sides of an equilateral triangle and then you take each one of the sides of this new shape so that that's not shown but it would be 1 2 3 4 like that here I take each of the 4 parts and do the same process I divide this segment here into 3 replace the middle bit by the other 2 sides of an equilateral triangle and here I've done the same and done the same and on the same and now I've taken each of the 16 segments that make up this shape and replace those by 4 little bits that make that kind of zigzagging shape and then each one of those I replace and if it I'm not showing you is that each one of these little segments are replaced by one of those and then each one of those segments or replace I'll go on infinitely long which is why I can't really show it on one slide and there will be a limiting shape and that limiting sleep shape is called the coughs no flake and I think it's fairly clear from the picture that the coughs snowflake is made out of four copies of itself shrunk down so if I look at this shape here because it's produced by basically exactly the same process as the entire snowflake it is just a little coarse snowflake and that's another little Koch's snowflake that's another little no and that's another little course snowflake so this shape here it's a fractal shape it's made out of four copies of itself but those four copies are a bit smaller let's focus on it the other way around so let's don't look at this part here and say if I were to expand that shape here by a factor three in every direction what would I get well the distance from here to here will expand by a factor of three so I would actually get this shape here so if I expand this bit here by a factor three I get the same shape so the amount of stuff goes up by four so expanded it by a factor three to get this shape here but this shape is made out of four copies of this shape so just let me say that once more I've got a small shape I expand it by a factor three and I get four copies of the original shape so where would a cube when I expanded it by a factor three I got 27 copies of the original view here I've got four 27 was three cubed so what do I need to ask I need to ask for is three to the what fortunately we thought a little bit about raising two non integer powers and think for it turns out to be three raised to the power the logarithm of four to the base three so log maybe that's a number between one and two so this is a shape it doesn't really matter what exactly what the dimension is if you don't like logs but it is definitely between three and four because there's between one and two the dimension because it's more than three to the one and it's smaller than 3 to the 2 3 to the minus 3 3 to the 2 is 9 4 is between 3 & 9 so the dimension of this shape if you use that concept of dimension it's natural to say is but it's between 1 & 2 you might say well that's a little bit you know we can talk about areas of all sorts of shapes they don't have to be squares whereas this is a very special shape with this self similarity property well it turns out that we can build on these ideas and make sense of dimension of lots of other shapes and I've now realized that I've been using slightly more time than I intended which is always the way I want to get on to the main topic so I said I wanted to talk about a beautiful resort called Vandiver dance theorem and talk about generalizations of that one of the reason I chose this is that it had a lot of connections with my own research over the years so Vandiver dance theorem we start by doing the following we take a large number I've taken 50 here not all that large we take a large number and we assign a color to all the numbers up to that number so I've designed either red black or blue to all the numbers up to 50 and now for no terribly obvious reason but you know we're mathematicians we like to do these things I've got a hunt for arithmetic progressions and I want my marinated progressions to consist of numbers that have the same color so what's an arithmetic progression it's something like 4 7 10 13 or something where you have some numbers and to get from each number to the next number you have a jump which in that case was 3 and the jump is the same each time so another one would be say 16 22 28 34 40 46 something where the jump was 6 in that case so if we look around can we see any arithmetic progressions I sort of as you can see 2 4 6 black one but it stops okay can we find any of length four well here's one I've underlined so we go from 2 to 13 to 24 to 35 so the step there was 11 we call that monochromatic but it's just got one color that's a and 35 it stops there I couldn't get to 46 because that was blue not black so anybody feel like finding me a blue one of length 4 thank you 3 7 11 15 and that also has length 4 and it also can't go any further and I looked reasonably hard I don't think there's one of length 5 that's where everything is blue slightly more challenging and red one I think I won't to give you very long for this because I need to press on but just feel you have got a chance and 3 2 1 here it is 23 31 39 and 47 so adds 8 each time what does band of evidence theorem say for this sort of silly game of finding arithmetic progressions whether but everything's the same color it says this is not a formal statement of the theorem obviously but it's it says that basically however you color the numbers up to a number so it says if I take an N that's large enough and I have some palette of colors let's say I've got 25 colors and you want to find you want better be an arithmetic progression of length 79 and as long as n is large enough and however I color the numbers from 1 to N we use 25 colors there will definitely be an arithmetic progression of length 79 where all the numbers in that hair attractive progression have the same color it's just no way of avoiding that happening that when as long as n is large enough saying it's quite an important qualification but the point is that it it does exist some n with that property right so now let's get generalizing I hope you understood that so you color the integers from 1 to N you can't stop there being an arithmetic progression of with only one color so the first generalization I want to talk about is called a density version which starts with the following question you might sort of the one thing that mathematicians do don't just sort of say oh great we've got a theorem now let's go home they stopped asked themselves as I said earlier on more questions so one of the questions you might ask here is well we know there must be one of the colors at least must contain long arithmetic progressions but can be somehow identify which color that is or two is the best we can say that there must be some color in tanta those anime very very surprisingly satisfactory answers to this question which is a result called summary this theorem again this is not a formal statement but let me try to say what this means so when we have a color one of the colors that we use when we're coloring numbers from 1 to N we define its density to be the number of times you use that color divided by n so if you used say your coloring numbers from 1 to 100 and you use the color red 30 times we'd say the density of red was not 0.3 so what some radius theorem says is if you tell me a density like 1 percent and you tell me a length of arithmetic progression like a million then there will be an N with the property that any color that uses one percent of the numbers from 1 to n must contain an arithmetic progression of like 4 million so he actually is not important it's a color it just says any bunch of integers from 1 to n as long as there's at least you've got at least one but 1% of all the integers from 1 to N and n was massively large will have to contain an arithmetic progression of length 4 million so same radius theorem immediately implies found divergence theorem because if you color the numbers from 1 to N with let's say a hundred colors at least one of those colors must be used at least one percent of the time that's a famous principle called the pigeonhole principle in mathematics or if you could or similar if you had four colors at least one of the colors must be used 25% at a time so by semi radius theorem that colour itself will contain an arithmetic progression right let us move on now so I'm going to generalize in a completely different way this is a theorem again not stated formally called the Hales Jewett theorem let me move on to a picture and see if I can explain roughly what the Hales do-it theorem says so when I was young I used to have a game called four pegs and as a picture of the box or other a box that were selling on eBay of the same game so this is three-dimensional noughts and crosses except it's not quite because it's four by four by four instead of three by three by three so lions have length four in this game turns out if you have three by three by three it's a very easy win for the first player but four by four by four ten is a rather good game I think it's if you play optimally it's a draw but it's pretty hard to play optimally and people often win if they're not super expert of the game so a lion here will be something like here's a simple example of the line and they're slightly more complicated example might be that point at that point at that point in that point or yet more complicated one would be one that's sort of diagonal in every possible respect so say at that point that point at that point at that point and what the hell's to it theorem says is however long the lines are so here therefore and however many colors you have say a hundred as long as the dimension is not high enough there must be a line that consists of points of only one color it's considered what have even higher dimensional noughts and crosses might be like so this one we can at least visualize this is an example of four-dimensional noughts and crosses but because I can't do a slide in four dimensions or not even to a slide in three dimensions of a four dimensional thing that I'm projected on to two dimensions what we have to do is find some way of representing it so the way I've represented it I think you can sort of see I've just as a three dimensional noughts and crosses board consists of a bunch of two-dimensional ones placed sort of next to each other and they don't have to one on top of the other we could just place them next to each other and it would be just the same game it would be a little bit harder to visualize what a line is so here I've taken a two-dimensional one and then I've put those three together that way for three dimensional board and then those three together make another three dimensional what types of prefer to visualize it vertically actually that looks sort of like a three dimensional board next to another one next to another one and that makes a four dimensional board and then we can here as here are some examples of lines let's just do the most diagonal one possible we go that point there that point that point there but another one might be say that point that point that point you get the general idea at that point at that point at that point doesn't so gain weight so the hails do it through them again it says if you if you have a sufficiently high dimensional board then have a wide it is and have many colors you you have you cannot avoid light and so the consequence for the game says even if you have a game where you play it with sort of eight players let's say if the dimension is large enough it can't end in a draw because at least one of it once you've once the board is filled you've colored the points with eight different colors and one of those colors has a line somebody must have made a line first so the game couldn't have ended in a draw as a paper the paper of Hale's do it talks about multi-dimensional tic-tac-toe tic-tac-toe being the American for noughts and crosses right let's move on yet another completely different generalization this time what we're doing is if we see let's think of how we might represent an a general arithmetic progression of lengths for we might say you pick a point a number a and you pick a difference D and then the automated progression consists of the numbers a a plus D a plus 2d and they've lost 3d so the the expressions D 2d and 3d are very simple examples of polynomials polynomial functions of D but they're actually polynomials of degree one or linear polynomials of D so d 2 d 3 d they're very simple expressions here I've just replaced those by some rather more complicated ones d squared D cubed and D plus d squared plus D to the 4 and what the polynomial version of Vandiver theorem says is that I could choose any bunch of polynomials like that as long as they don't have constant terms that's a technical reason but the theorems not true if you have constant terms and you can find so however I can also pick a large enough integer and I color the numbers from 1 to that integer I'll be able to find patterns like this let me just illustrate it because I think it will make it clearer so let's go back to it this is exactly the same coloring of the integers I had before and I promise see that wasn't sort of faked I did have a little search and eventually I found an example of a pair of numbers a and D such that all of a a plus d squared a plus D cubed and and they first equals d squared plus T to the 4th belong to the set and here it is I had a equals 7 and D equals 2 so 11 is 7 plus 2 squared 15 is 7 plus 2 cubed and 29 is 7 plus 2 plus 2 squared plus 2 to the fourth I'll check that 2 to the 4th is 16 2 squared is 4 2 is 2 add them together you get 22 add that to 7 you get 29 so I wasn't cheating right so I stress I'm this is just an example but the theorem itself says that whatever I'd chosen that looks a bit like D so I could have chosen sort of 2 d squared plus D D to the fifth 25 D to the 17 plus D to the 13 and a few more 4 levels as well and it would still have been true as long as I color enough integers I will get a pattern of the desired kind right so let's just recap here are our methods that we've got for generalizing we go from a coloring theorem to a density theorem so that means of instead of coloring we just say any old set that's reasonably dense must contain the configuration we're looking for right from integers to high dimensional grid sold noughts and crosses boards I went from arithmetic progressions to more general polynomial patterns those were the three ways that we came up and so they're each completely different they're so different than what we might call orthogonal to each other and using the word orthogonal in a slightly fanciful way but one that mathematicians often do because they are orthogonal we can start asking well what about if we try and combine some of these generalization methods could we actually get even more generalizations so there are eight possible generalizations if you've got three methods if you include what mathematicians would call the trivial generalization where you don't generalize so if you insist on at least at some generalization that goes down to seven because there are seven ways of choosing at least some of these so I'm going to think about what happens if I combine one and two if I combine one and three and if I combine two and three so combining one and three is fairly straightforward the Hales jewett theorem said if you color a sufficiently high dimensional noughts and crosses board you must get a line in one color so the density version says if I just take one percent of or some fixed percentage of a sufficiently high dimensional noughts and crosses board there must be a line in that so if I fill up let's say one percent of the points in a very very high dimensional noughts and crosses board I can't avoid making a line somewhere that would be what the density theorem says the coloring for theorem would say I have to color every single point and then one of the colors will contain a line the density version just says any sufficiently dense color will have to contain a line that's but that turned out to be a much harder result than the hails to it theorem itself due to two mathematicians called Hillel Furstenberg and yet Saint Katznelson so let's try one and three so now I'm thinking about going from coloring to density and from arithmetic progressions to polynomial patterns well then again it's fairly straightforward to see how the generalization ought to work we ought to say instead of coloring we'll just take a sufficiently dense set of the numbers from 1 to N and hope to find a polynomial pattern inside that dense set and that turns out to be a true result as well the theorem that you can do it with coloring was due to two mathematicians called Vitali Ferguson and sessional Lightman there's an interesting story they had turned out that before they did the coloring version they had a sort of machinery which said developed building on the work of actually Hillel Furstenberg they knew that once they had the coloring version they would be able to generalize the density version no it's not obvious how you can but and they're gonna turn out the big they knew how to do that so that was true and the last thing I want to talk about no sorry the third of these combining two methods of generalization it's something that's rather less obvious now here I'm going to be a little bit of a challenge to say what's going on here so you think that this is a two dimensional grid that I've colored black and red but you saw also see I put numbers naught 1 & 2 in there something a bit strange is going on so that's not supposed to be a 2-dimensional noughts and crosses board this is supposed to represent one point that lives inside a sixty four dimensional noughts and crosses board so if you think about it just a straightforward regular noughts and crosses board we might have a coordinate system that went naught naught naught 1 naught 2 1 1 so 1 naught 1 1 1 2 2 naught 2 1 2 2 in 64 dimensions so what are the most of the points of that board they are pairs of numbers and each number in the pair is either naught 1 or 2 yeah I just got 64 of those 64 numbers each one of which is a naught 1 or a 2 so simple that's just a point in 64 dimensional space so why have I colored some coordinates red in some black well just cuz I felt like it I have to think it on trust at this point here remember this is just one point this is the coordinates I happened to arrange the coordinates in an 8x8 grid this point is green okay so this is I'm just declaring it to be green it's not some mathematical reality that you've missed I'm declaring this point for green all right so I want to tell you what a line looks like in this 64 dimensional space and it's a very special sort of line here it goes so why the reason I highlighted those points in red is because they're about to change there we are you see when they're all notes and they're all ones they're all twos so if you've got a point where there's a whole bunch of notes in their coordinates and another one where there's a whole bunch of ones and those same coordinates and another point with a whole bunch of twos in those same coordinates that's going to form a line in 64 dimensional space this is just a natural generalization of what happens in two dimensions and three dimensions if you think about it for a while but there's something else about this line that makes it particularly special which is that the points that do the changing lie in they have both their coordinates in a special set that set is to set one two four seven so that either in there but both the row in the column it admits here we've got the first first row second row fourth row seventh row first row second row fourth row seven so first column second column fourth column seven to column and each point that's colored red lies in a row and a column from one two four seven then those are the points that changed so it's not just any old bunch of points that change and it's the fact that this set here is called a this set of all points where the row and the column lie and set one two four seven is what you might call a Cartesian square so that word square there is important if you look at in particular so focus on the number of points that are doing the changing there'll be 4 times 4 makes 16 which is a square number it turns out that from this so what about what is this this is the statement that however I color all the possible 3 to the 64 points of which this is just 1 with some small number of colors I will not be able to avoid getting one of these special lines a special line where of the coordinates that vary are arranged in a nice pattern like that all of one color actually I lied to so 64 won't be enough it'll be N squared for some very large n but again I want to illustrate it with a it's just one slide now if you didn't completely follow that because I think it is challenging it was so challenging for me when I first came across this theorem to work out what on earth was going on please believe me that this is a generalization that from it you can quite straightforwardly deduce for example the statement that from this sort of case here that if you color the integers from 1 to N and n is very large you can find an arithmetic progression of length 3 where the common difference is a perfect square it turns out to be a fairly easy exercise to get from this statement to that statement and then this is of one special case which generalizes order polynomial thing now getting to the density version is always straightforward just replace is green or is some particular color by belongs to some dense set so here what I'm saying is if you choose one percent of all the possible points that you can get here you must find a line of this special form that lives inside that set so that's the density version of that's generalizing all three different directions that's a density version of the polynomial Hales Jewett theorem but there's one difference between this result and all the other results that I've talked about which is this is not a result it is an open problem and I wanted to get to that because that's something that I am working on right now with one of my research students I wanted to sort of show that just by this process of generalization you can get to the frontiers of mathematics I will stop there thank you very much name so we're gonna have about 25 30 minutes of questions and then throw it over to you so if you have questions that you want to ask professor galas and please do you think of them as we go along and but I think for me what I wanted to kick off with was just asking you about those kind of problems what is it about them that intrigues you I guess how did you choose them as you were your area of study in a sense I didn't actually because when I started out I was doing a completely different area of mathematics so um but I had a friend who worked on summer ADEs theorem and related results and I always found that interesting and at some point I thought I potentially saw a way of coming up with a new proof of some radius theorem which quite a long way down line I eventually did and when that happened the whole sort of focus of my research changed into something called additive combinatorics which concerns problems that are somewhat similar to the ones that I've just been talking about already and but maybe that's not exactly the question you're asking so maybe you're asking these looks like sort of amusing puzzles but what why why concern oneself it's a huge effort to solve any of those puzzles why go to that effort just for one of these puzzles and my answer to that is that they're not mere puzzles so if you want to solve one of them so one of these results that I've just been talking about doesn't have that many typically won't have that many direct applications even to other parts of mathematics let alone outside mathematics but the techniques that you're forced to develop in order to solve one of these problems will very often have much wider applicability and implications for other parts of mathematics very so one of the great mathematicians in combinatorics which is sort of wider area that I belong to Paul Erdos was a master at asking problems that had this kind of quality of seeming like amusing puzzles but they just encapsulated some difficulty he didn't just throw out any old question and when you started thinking about them you realized that there was some difficulty that if you could solve that it would really be in a much better position than you were before for a lot of other things and not just the sort of headline problem that he asked so it's got that flavor a little bit I think is that motivate motivating factor for you then the fact that yes the things that you're discovering along the way will be useful or does is its usefulness even enter into your your motivation well again useful it can mean useful to engineers or it can mean useful to other pure mathematicians I would be thrilled if something that I did was useful to engineers but it's not my primary goal and if something is useful to other mathematicians and I feel as though it's helping to develop the subject and then that's uh maybe more important to me I'm also immediately important i say immediately important it's because if you do then develop the whole discipline it's just inevitable that the applications outside mathematics follow it's just that it may not be my precise theorem that my I still feel I'm contributing to a a big endeavor and then bits of that endeavor something that almost randomly chosen somehow turn out to be very helpful to people outside mathematics how early on did you know that you wanted to be a mathematician not very early I was always one of my favorite subjects at school but it wasn't my wouldn't say it was always by far and away my favorite subject it was and the other thing that makes me say not I heard he was that until when I was an undergraduate and maybe even later than that I didn't really or maybe one of towards the end of my undergraduate timer perhaps not even in times did so they made a full course part three Cambridge's part three course did I have the slightest conception of what research in mathematics would be like I think probably many people if you're not a mathematician you're tempted to ask maybe have even asked if you've ever met a mathematician and sort of what could research in mathematics actually be like and if you're unkind you say is it multiplying larger and larger so then you have to explain that no mass is not just being sort of a bad pocket calculators more to it than that maybe I've conveyed something of the sort of worlds that mathematicians inherit or at least some corner of mathematics so and then another another sort of thing that makes you maybe not want to say I want to be a mathematician is that when you're starting something very intimidating about the very notion of an open problem in mathematics because you're surrounded by people have these reputation for being incredibly clever and so on do you think well there are all those people out there incredibly clever research mathematicians and this is an open problem how am I gonna come and solve that problem there is an answer to that and the answer is not you have brains bursting out of ebbets the mathematics just as I've been saying we don't run out of interesting problems so we get more and more problems and the more you think about it the more questions you ask some more problems get generated so actually there are quite a lot of problems around that have not been thought about by all the world's experts for the last 20 years and so if you don't try to solve the Riemann hypothesis for your PhD then there's a chance of making some progress and then you said the idea as you try and work on problems that are maybe not so central and then increase your level of ambition gradually when did you work that out there when can you remember when you stopped being intimidated I suppose it was when I first made any progress at all well then maybe that's a slight so the first thing I did was just a tiny little tweak to someone's existing argument but improved the answer that improved the bounds that came out of their proof now I gave a talk on that in a rather sort of informal setting and there was a mathematician there who I won't name who at some point sort of walked out of the talk I think to go smoke a cigarette and then we came back for that father gave an indication of his level of interest but I later on found a much more something with the same problem that got not just an improved result but the best possible result that you could get by a much more complicated argument and so at that point I really learn with gains they're very nice I definitely shown that I could do research at that point that's probably in the middle of my first my second year of PhD and that was the thing that ii do announcer yes something I knew I thought something so talk to me then about that process of research so that you know you were saying there but you know being a mathematician on the day say tell us a little bit about your process when you are approaching a new problem how do you do it how do you how do you tackle a challenge part of the answer actually is choosing the problem in the first place to make sure that it just feels like the kind of thing that might conceivably be amenable to the sorts of things that I have in my own mathematical tool box that I've become very often just misjudge that completely and the problem is much harder and not something that I can do so that's another thing to bear in mind so I think if you have the right level of ambition when you're doing research you should be ready for most problems with very much most problems with nine out of ten problems that you try just not to get all that far because you're looking for the one in ten where something kind of gives but how do you find that the the thing that I think is most central is the process that I alluded to of the talk which is you don't try and take route 1 to the solution so if you it's something that quite a lot of people take a while to learn because when you'll set exercises at school and university they're carefully tailored so that either you use the standard method and you just use that method and you get to the solution or perhaps there's one little trick and once you've spotted the trick it's plain sailing from that moment on but a real live open research problem is not like that at all so if you try and use that sort of idea just try and hit it with standard methods or just look for that one little idea after which it'll be easy you won't get very far very occasionally there are problems that surprisingly some little trick works but that's very much the exception so somebody had to ask yourself when you're doing any sort of research is when you have an idea about how to tackle it the first question you want to ask yourself is well if that idea worked why is it that this is still an open problem and if you've got a reasonable answer that some reasonable story to tell that this idea is a little bit left-field in some way or it involves mixing using ideas from a totally different area of mathematics that would not be familiar to the experts in this area method main or something like that some story to tell then probably your approach will fail although it can be valuable to see why the approach fails and then try to think of something more sophisticated I've lost track of what your original question was now I was coming to the answer well it's you're praising it's been you're tackling any program oh yes so the process was you don't take a direct route but you ask yourself other questions so one of the questions that you can ask if you're say you're trying to prove some statement is to see whether you can prove a more general statement so hence generalization and the reason that can work is that sometimes when you generalize a statement it actually paradoxically although you're proving something stronger it's easier to do why is it easier it's because you're trying to prove something that's strong and more general you don't have as much room for manoeuvre to fit like being if you're in a worse position in chess it can be easier to decide what move to make because you're more or less forced the only way of avoiding checkmater to do such-and-such if you make your problems harder it can make them easier for me so that's one method but another method is exact opposite which sort of look at a very special case of what you're doing say with some radius theorem you could look at progressions of length three that turned out to be very fruitful thing to do and build up from a special case until you hope that you get to the point where you sort of spot a pattern and what you're doing in the special cases and then can generalize to get back to the thing you originally try to answer sometimes another methods another question you can ask is well I don't see how to solve this but let me invent a different problem in a slightly different context there appears to resemble this problem involve quite a lot of the same difficulties and see whether I can just solve this somewhat similar problem and get some insights from that but and what someone's always looking for is a new problem that should be easier than the problem you're trying to solve but should help you to solve the problem you're trying to solve if you can solve that one and that's a process that can sort of split up and split up so if I have this is a problem I'm trying to solve this is starting from nothing say I invent this new problem that might be helpful for this farm it might be easier but I can't see how this one helps this one and I can't see how to answer this one so I'll find another one in between and another one in between here and I so try and get a path from zero to full solution of the problem so this process of sort of splitting up a kind of top-down approach is very important and involves a lot of so very much not what one does is sort of say I wonder what the first line of my eventually proved absolutely not like that but I mean that process never-to-be involves a lot of dead ends along the way absolutely do you think it's important to how do you shield yourself against disobey you against being discouraged when you meet dead ends I think it's quite hard to do when you're just starting out because you really really want to have got that first theorem first solution to an open problem but once you have that ideally maybe two or three times and then you sort of know that it can be done and you also know that it won't happen immediately you have to be patient and you have to be prepared to fail several times before you succeed so exam sort of get used to it as being part of the job it's just part of the process of proving theorems it's not necessary the most difficult part either I mean another difficult thing is when you're trying to do something and then somebody else does it you beat you to it or even worse you work hard on a problem and discover that it was done 10 years ago has that happened to you not in a sort of major way so I haven't sort of spent six months on something that's been my big project purposes one exception actually there was something that I worked on but I was very silly too in that particular case to think that it though I was discovering all these things that's what discovered a various things thinking I was just discovering it for myself and then I discovered there's a whole field and all the concepts or well-known but even that was that wasn't a major disappointment to me really because I was nowhere near this is the very nice student I was trying to solve and the solution to that theorem was incredibly clever and I don't think I would have found it myself and if you do rediscover things that's actually very beneficial because you understand those things in a some deep way that you don't quite get if you just passively read a textbook so um I gained quite a lot even even from the sort disappointments one does eventually gain even though it was a little bit crushing for a while has there been an occasion where you've been picked to the post in trying to be something well they've been several occasions where people have proved things that I would very much like to have through but not so many occasions where I I've really been working hard and that's the thing I was absolutely working on at that moment and then someone did it it's more common that I've thought really hard about something and worked pretty hard on it and then sort of sort of maybe I'll try something else it's still one of my favorite problems but you have to move around because otherwise you'd only get sort of stuck not solving a problem five years and that's our Andrew Wiles not a good example doesn't always work out that's not a good model I should say it's a very good example to follow and so even then if someone solves a problem that I have worked on hard in the past and really liked him sort of thought I might well want to return to that can be it also stab of disappointment but that's again something that one has to just take on a chin and keep going I am curious now which theorems would you have liked to sold well the one that I was talking about first of all where I sort of develops the basics only sort of beginnings of the service I was I thought I would have a try at the P versus NP problem this was a long time ago and I saw underestimated how much work had gone into that problem and there's a restriction of the problem to do with a contact called monitoring circuit complexity so if you sort of restrict what a computer can do you know trying to show that it then will not be able to solve certain problems efficiently and so I formulated that problem and thought about it quite a lot and then it turned out us that term sample Sasha raspberr off had years before proved exactly what I was trying to prove and won prizes for us and but that would be you know that would be absolutely great theorem to have my name if I could choose why but of course I mean you're not sure of prizes yourself and not complaining um do you remember when you found out about the Fields Medal that it comes as a surprise yes and no so I it was not a surprise that I was being considered for it because I got all sorts of mysterious messages you know saying could you please send me a CV by yesterday and Melissa but I can't tell you why and that's one thing and it was the right time of in the cycle for the decision to be for the committee for making its deliberations and so I didn't want to so I sort of felt presumptuous to assume that's what it was but it wasn't much else it could be so anyway and then when it actually when I did find out it was quite surprising because I was summoned to the office of my then head of department with someone else and so that couldn't really have been what it was about but it was and we both got fields medals and at the same time yes did it did you muck does that mean that you were in some sense sharing the glory well no but I mean the bit they gave awards of course more each time but it not within within Cambridge or something I suppose it but I think that in a way I think it that wasn't the problem I think they made more of a fuss because it was more sort of unusual to have two in the same institution at the same time so I think it actually works to my benefit really how much do those those prizes mean to you [Music] was made a massive difference my life in a way because in a way that it shouldn't have I would say if the world are a really adjust place because it's not the case that there are each time you know for people who tower above everyone else it's more like you know the committee that makes the decision has a difficult decision and just eventually has to sort of settle for four four people so that's one thing that's it's not some sort of difference in kind between someone who gets one and someone who just misses out or indeed someone who doesn't even just miss out but does something amazing a few years later or something like that and I in the longer since it happened the more remarkable examples of more remarkable things I see other mathematicians doing and the more I sort of see my actually rather small place in the vast body of fast corpus of mathematics but I think what it gives me what has given me it's just a sort of unfair leg up in life really that I just get I get lots of invitations interesting invitations to things people sort of like for the introduction here so you know we've got a fields medalists giving a talk and so on and this sort of it's quite nice but but at the same time in the end I don't think it's I think I've sort of felt after getting it that I had to try to some extent to sort of pretend I hadn't got it so as not to sort of relax too much and just not another thing that could have been a mistake I think would have been to to say well now I've got that I've what's how can you go up from there well I'll have to solve solve one of the clay millennium problems or something like that or sort of solve some really massive problem that's bigger than an had done before and I think if you take that attitude the chances of success are very small and you just have to say no just pretend it never happened just keep on working on the things that interest you and don't be good things happy medium things do you think it kind of liberated you in a way though in some ways yes so I know that so they freed me up to do one or two things that I think I might have thought twice about otherwise so one of them was editing the Princeton Companion to mathematics which mentioned earlier on which I would say took up roughly half my working time for about five years which was a sacrifice but I'm not sure I would have felt I could possibly make if I hadn't somehow just made any sort of reputation I needed to make and it was a project that I believed in but it's must not a research product and also I didn't have to worry too much about people saying oh he's sort of gone soft he's doing popularization now not mathematics or something like that and the other thing which was in the last last ten years I've spent quite a lot of time although certainly not all my time but quite a lot of time thinking about automatic theorem proving that is trying to get computers to find proofs of theorems which is something that really interests me but again it sort of doesn't really count as mathematics so it's something that I think I wouldn't have felt that I could afford the time to spend on if I was continually sort of thinking about how my CV was doing and those sorts of things that liberated me totally so after a while I thought I better get back to mass just to sort of show that I really I can still prove it so I had about three years where I was not doing that much mass at all stay a bit but less I spending less time on mass and I was on automatic so improving but that's changed now and I've thought asked five years or so it's been concentrating more on mass but automatic there improving so my interest on the side do you think those prizes are generally a good thing for the mass community I think they're mixed I do think that one good thing about the Fields Medal is that because it's happens at a young age and because it doesn't come with a massive amount of money in a way that a Nobel Prize does comes with a really quite small amount of money and so that means that the sort of pain of not getting it I think it's much reduced because for two reasons one is you don't have this sort of I missed out on half a million pounds or something which I think would be quite bad and the other is that even after you're it's too late because you're about to turn 40 you certainly haven't stopped being able to do mass and there's still plenty of chance to do to prove amazing theorems and get an amazing reputation and maybe win other prizes that are aimed at older people if you're interested in prizes so I think there may be a sort of some degree of unhealthiness about it but but I think in my in many people's cases it it some benefit as well if people have sort of got a sort of dream maybe that I think it probably in my case when I sort of began to think maybe I would be a chance it did sort of probably make me work quite a lot harder than I would have done for a while my dad had some beneficial effect in that direction and you've also been a really big advocate of sort of collaborative mathematics and thinking of the projects that you've that you've done recently do you think there's ever a contradiction between that sort of working together on a particular theorem or proof and the sort of thirst that might be required for a prize or sort of an individual hunt for glory if you're talking about a huge collaboration then of which you're just one very small part which was the thing that ya set up at one stage then I think not many people would be satisfied with just participating even if they participate it very fruitfully in huge collaborations and for that reason it's not that method of doing things hasn't taken over what's one of the reasons hasn't taken over how we do mathematics but mathematics is now very very collaborative and I would say the majority of papers are two or three authors and that's for the simple reason that two or three people can often work a lot more efficiently than twice as efficiently or three times as efficiently as one person for obvious reasons you can specialize one person might be good at one aspect of finding proofs with another person good at another one plus might throw out lots of ideas another person might be good at hitting them down again somebody might have expertise in one area somebody might have expertise in another area and I think people just find it's easier to come up with severe sort of say it's to author papers to be easier to write fifty two or three papers and twenty five one or so papers and another thing is that I think if you are one of two authors you probably get much more than half the credit for being a sole author of a paper I think that's most people most mathematicians I think would agree with you either I think you probably get 75% of the phrase or something unless you've I says some very strong suspicion that you weren't really the person who had the ideas or something like that but that's another good thing about mathematics compared with other subjects so we have also strictly an alphabetical order rather than having complicated orderings that you have in some sciences where the first person did this and the last name daughter did that and so on so forth we avoid all that in order to exactly to sort of avoid that kind of talk about who really did this and the result of that I think is that it strongly encourages people to work in small collaborations and what I quite like about this is that you're sort of presenting a very rational kind of calculated argument towards small collaborations and this I think as I regularly read your blog I don't know how many people here also do but this is something you're sort of known for right taking if a rational approach to decisions that might not necessarily see mathematical and verbs if I'm worried that one can sort of kid oneself you put a little veneer of rationality just to sort of justify one's prejudices but math does give you the tools for recognizing that when that happens just so they're not always used to full is it true that you once decided whether or not to have a medical procedure basically the risks it is because it was a I have had problems actually David just recently seem to be starting up again which is annoying of arrhythmia in my heart called atrial fibrillation and there's something there's a procedure that can be done to correct that where they stick a wire up a vein to go in your leg and the kumano goes up into your heart and it sounds for a fairly unpleasantly it'll burn the side of your heart but the reason for that is that it breaks an electrical connection which is messing things up if you don't want to have and causing the rhythm to go off and it's a pretty safe operation but when you read about the risks it all sounds quite frightening and the particulars of one in a thousand mortality risk so I thought kind of really says I really want to have this one in a thousand risk of dying then so I looked into it a bit more and of course it depends a lot on how old you are and various other things and also the benefits the operation depend on so I said on how old you are and eventually I decided to go ahead because I had the idea of looking up what the mortality risk was of just being alive for two months fifty it turned out that an operation concentrated about a month's risk into one operation so my risk of sort of falling under a bus or whatever it was for over a month would be about one in a thousand Diedra was then I thought but I'm not terribly scared of the next month so I shouldn't be all that scared as that was a level of rationality somehow I know it all went fine and yeah but is this something that you find yourself doing a lot so I mean I noticed on your blog you often with politics especially you did these sort of very involved posts where you're you seem to be almost breaking argument downs triumph yourself or something one way or the other actually I would say that there I'm fairly sure what my view is in advance first I just want to sort of check that there is some rational justification for it so I wrote a post about why apologies have gotten D voters in the audience but why we should remain in the EU and which had sort of games theoretic aspect to it I wrote something about why one should vote for all the alternative vote in 2011 referendum you can see these posts are very successful and I wrote something recently about tactical voting in the European elections but going back to your question about rationality I do sort of sometimes take things a little bit further than maybe some people would so for example if I'm in the kitchen clearing up I sort of this don't have a mathematician to have this thought but the thought I'm talking about is say there's some stuff on the table and you've got to transfer it to the side of the sink and there's some other stuff in the sink needs to go in the cupboard and that sort of thing it's a mistake to say that first I'll do all the table to the sink stuff then I'll do all the sink to the cupboard stuff because that wastes the journey where you could've when you're walking back to the table you could have been carrying something to the thing so I do have those kinds of sources I promise I have a life does it drive you a family man I try to keep quiet about what I'm doing look on the rare occasions when I've suggested anybody else in the family that more sort of ergonomic weighs down very quickly I'm going to come to the audience in a second but just one last question that wants to ask you today do you think that being a mathematician is something that that requires an attitude that can't be Tour's do you think it's a gift my own view for which I don't really have enough evidence to say that definitely correct but it's a sort of conviction is that you need to be good up to a certain level but you don't have to be genius level that you can as long as you're pretty good at maths and very very keen to make progress but I think the keenness is you know I think if you have a choice between sacrificing a couple of points of your IQ but gaining 10% of enthusiasm go for the infamous it may take you slightly longer to have some ideas but you'll have some in the end there's not when you're solving a problem it's a slow process it doesn't really matter how quick you are there's a long term thing it's not to say that if you are super quick and have those kind of ideas and people can't work out where they came from of course that can help to some extent but there are plenty people who are like that who don't make a huge success I think the one example I can think of somebody who was a prodigy and it's by any account a genius who has also made a huge success I would be Terence Tao but if you're not Terence Tao just don't there are lots of non Terence Taos who've done very good things as well so what advice if you could go back in time and speak to your seventeen-year-old self what advice would you give yourself I think I would say I would go back to what I was talking about earlier that it looks I would say don't be intimidated by all this by all the cleverness that you see around as long as you keep at it learn what you have to learn what try to do plenty of problems including problems that have been solved already but just get practice at problem-solving so practice practice practice keep your enthusiasm and you can make room for yourself exactly okay we'll come to the audience then so does anyone want to who wants to kick us off with questions okay we have let's go we'll go first here one yes please yeah the lady there that'd be perfect and then immediately go behind and then we'll come thank you for a very engaging talk and interview afterwards and my question is to do with how you choose the problem so once you've chosen you know subject area you're interested how do you go about choosing the right problems to pursue and then once you're pursuing them how do you choose when to stop when they if they seem harder than you thought I think roughly speaking I would part of it I can't have a very satisfactory round so I just look at a problem and I sort of feel in that looks interesting and maybe I'd have a chance of solving it but supposing I got beyond that stage and I actually started thinking about it and I might decide to abandon it after an hour or two or I might decide to continue so what would decide me there I think it would be going back to this process of inventing other questions if I found I could come up with promising related questions that was perhaps related to things that I did know how to solve and had a chance of being easier then my enthusiasm would continue and as long as that sort of as I had the feeling I was making progress in my understanding I'd be tempted to carry on but there comes a time when solving a problem sometimes when you just sort of get this feeling a bit bogged down you've found lots of questions and they keep not be able to answer some more you've done calculations that get really complicated and you're not quite sure whether it's worth pursuing those calculations if you can't quite see where they're going to go even if you succeed that may be but then in the other direction maybe you're you do some complicated calculation and and suddenly something simplifies dramatically anything Wow on to something here then you sort of will carry on and see whether you get anywhere so I think that's what it is I have my process of doing research and as long as I feel it's moving forwards then I'll carry on and if I feel it's getting sort of I'm getting to the point where it's not getting forward fast then I'd be better off working on something else as you may be picking up I'm quite I am quite calculating about the process of what to work on at any one time and I think you have to be to be successful because that mattes because there's a little twin risks of giving things up too easily and not solving anything or not giving things up easily enough and not solving anything and who's got to find a sweet spot in between the two there's another questioner just behind you there you get better the only my friend perfect think any of the Millennium problems remaining or any problem in general are truly unsolvable or lumberville I don't really know I of the Millennium problems I think the one that people talk about as possibly being unsolvable is a P equals NP problem my own feeling about that is that it's not because not for a good reason but there's a way of formulating it that makes it look very combinatorial very sort of like it doesn't mention computers at all doesn't sound like a problem to do with logic so it doesn't feel like the kind of problem that would be insoluble but that's not a good reason because there are much more sophisticated reasons for thinking that it might be that by people who have come fully aware of what I've just said about it having this comet oreal reformulation so as for the other ones I really have no idea to live I say the Riemann hypothesis I'd be very surprised if that or undecidable but it always comes as a surprise when a problem that's not sort of set up to be undecidable is undecidable so without these surprises to happen the question just here so we'll go there and then we'll come to you a bit I thought that there just a you know he will come to you in a second yeah thank you very much it's very it's an honor to listen to you my first question what is the most widely accepted mathematical proposition that could possibly turn out to be wrong and second is just like a quick question if given a chance to be some other mathematician who would you like to be and why else of the first is I really don't know at all I could save sort of generally maybe I would mention someone I know called Kevin Buzzard who's quite worried that there might be significant parts of number theory which is an area which rather unlike my area it's rather a lot big hierarchy of this this theorem uses this theorem which uses this theorem and it's Dereham and this theorem which uses this and then you get and so that if you actually were to chase the proof of this theorem at the top through all the different branches of the tree you'd have to read 2,000 pages of very difficult stuff some of which have been published and some of which has only sort of known to a few experts and so on there's just a slight anxiety that some of the results of number theory but which one like I don't know might be wrong in quite serious ways and he takes that so seriously if you set up a whole program of machine verification of proofs in number theory who would I like to be and it sort of think like that somehow speedlight saying what other person would I write spiel I don't really have it all but I'm sort of give up so much else if I had mainly angry someone else Hey okay I think maybe I was gonna say you get sort of attached to your own theorems as well if you - part of the quid pro quo of being another mathematicians I'd have to say goodbye to all those things that I've spent a long time on well I've got time for I think maybe two more quick questions yes they will give you their fascinating talk can I just ask you a bit more about the role of computers in pure mathematics research you've mentioned one or two specific problems and the role of collaboration but in terms of fundamental change in the way pure mathematics is done Jim Harris research is done as it has the computer what what does the computer have given us and what do you think it will give us in the future depends what you call fundamental but it's given us quite a few things so it's given us an ability to check lots of cases which has helped for example proving the four-color theorem it's also got a very nice role in helping to when you're doing research and you invent questions sometimes the questions will be things that where the answer can be checked on a computer because not all problems are amenable to that but sometimes they are so sometimes you might think if this were true that would be really helpful for this but is it true and instead of if it's the right sort of question you might be able to feed it into a computer and get computer to see and then it might after a while say here's a complicated counter example which would have taken you a week to find and the computers found it in five minutes or something so that can speed things up another use of the computer that's rather nice is if this is a rather specific thing but I think it could be a special case of something more general which is that if you're working on a problem and out comes comes the first few terms of some sequence as you generate some integers what you can do these days is put that into something called the online encyclopedia of integer sequences to see whether that sequence is something that's come up in some other context and very often it has in fact almost always it has and when it has it will sometimes just give you a formula for the sequence which would have been extremely hard to find it'll show you that what you're thinking about might be connected with something you hadn't thought of and so on and so forth so computers can be very helpful in just speeding up the natural processes of research going back to your going to the question of where things might go in the future I personally think that computers will eventually take over and just do everything for us but that's a minority view most people think that that humans have this sort of intuition that never be replicated on a computer that is absolutely essential to the process of doing math i I don't believe that it'll take a while but but I think and before that happens I think computers will be able to do two sort of easier parts of mathematics and will that'll be very helpful as well because then again if you ask a question and the computer can answer easy questions and if that question is easy or even if it's not easy for you but easy for an expert in the right branch of math and the computer might then be able to help answer it that could be a huge speed-up I think there will at some stage be a sort of golden age of mathematics in which computers are incredibly helpful but don't quite spoil all the fun but it may be short though yeah did the bit okay last question last a quick question and then we'll do Beth thank you thank you for speaking with us tonight a few years ago on your blog you wrote about teaching mathematics to non mathematicians and as a teacher of a-level maths who's here with some a level students I'm wondering if you have any advice for how we should be teaching mathematics to students who are about to go off to university to study mathematics or other quantitative fields so not teaching mathematics for non mathematicians but teaching it to mathematicians well I don't know how practical this advice is because I know that I feel the sort of weary of offering advice to school you should do this because I know that that if you're actually out doing a chalk face with a big class and a very busy time table and so on you can't just sort of do anything that suggests it but I do think that as a general I feel very lucky that I myself at school was taught by people who didn't really I'd say my last two years didn't worry too much about the a-level syllabus but just sort of taught maths and at the end of the two years you found that actually you had covered material but we did lots of things that weren't really to do with a double syllabus and we would give problems that were much more challenging than you'd find on an a-level it wasn't necessary to do those problems to a level but it was really a very very good experience so it may not be a practical thing to do that in every school especially now when people do for a levels and but I think if space can be found for just the more that can be taught where you're not sort of focusing on the next a level module absolutely maybe there aren't modules anymore but that used to be a whenever modules it was particularly bad because I was always an exam just around the corner but the more it's practical not to focus on exams the better but as I said repeat that I understand it's not necessarily easy to do that I think that's a very good point for us to leave it on playing outside of what you have to do and trying to find the joy in it all and thank you very much everyone for coming and it remains to think a certain together thanks very much [Applause] you

Channel: Oxford Mathematics

Views: 26,455

Keywords: Oxford Mathematics Public Lectures, Generalization, Tim Gowers, Hannah Fry, Science and Maths, Maths Lecture

Id: 95SD9eNNdaE

Length: 88min 31sec (5311 seconds)

Published: Thu Nov 21 2019