Fermat's Last Theorem - The Theorem and Its Proof: An Exploration of Issues and Ideas [1993]

when I was about 10 years old I found a book in the public library which talked about this problem and like many other mathematicians that's one of the things that got me excited about mathematics obviously as a ten-year-old I didn't do very much on this problem I kept trying in my teenage years and then I abandoned it when I was a student this problem has a long history was tried by many mathematicians in the nineteenth century but most mathematicians seem to have given up in the 20th century until about eight years ago there was a big breakthrough and someone related bamas Last Theorem or this is the problem I've been working on with another well-known problem in number theory and from that moment when that link was made I've worked on it so it's about seven years now I didn't start with any idea on how to solve this problem so that was a problem and it took me five years to get the first real breakthrough and then I finished it probably about three or four weeks ago there are lots more problems in mathematics there are many unsolved problems I think I'll wait until I've written this problem this solution in a way that satisfies all the experts and then I'll start thinking about my next project the main significance of this problem is SiC is symbolic one doesn't expect any real applications of this kind of result but one never knows primarily it's symbolic because so many mathematicians have tried it it's led to the creation of so much mathematics itself everyone has thought of it as the dream of mathematics to solve such a problem welcome to tonight's special program on from AHS Last Theorem and I see a few seats and a few people coming in so please feel free to take a seat while I make a couple of introductory remarks I'm will Hurst the publisher of the San Francisco Examiner and I'm going to be the moderator for tonight's session and introduce some of the speakers on June 23rd of 1993 something extremely exciting and the mathematical world happened some of us were reading the New York Times innocently that day others heard about it via email or maybe a day later and that was the announcement that a mathematician named Andrew Wiles had proved something that mathematicians had been trying to prove for 350 years he had finally resolved from AHS Last Theorem in the affirmative now tonight's presentation is really aimed for a general audience and so we expect that we may have some students in the audience professionals from other allied fields perhaps some professional mathematicians as well and members of the general public that are interested to find out why the fuss about this theorem what's exciting how did Andrew Wiles do it what does it mean and we're going to try and get at some of those questions and we have some very talented and brilliant speakers who have condensed a lot of complicated mathematics into some simple talks of about 10 minutes each I don't think anybody should be worried if they miss a detail here or there even professional mathematicians go to talks and miss a detail here and there but try and get a hold of the general concepts feel the the flow of it and we're going to have some questions at the end and don't worry there will not be a quiz Thanks ought to go to the mathematical sciences Research Institute who is our primary host this evening msri as it's known is a research institutes and a little bit like the Princeton Institute for Advanced Study located over in the Berkeley Hills they have a Zen monastery like building overlooking the Bay Area and a lot of interesting people come through their mathematical researchers who give talks listen to other people's talks and talk to each other a bill Thurston who is the director of MSR I set out a goal for the institution to do some outreach this was before the announcement of the proof of Fermat's Last Theorem and so this program tonight is an effort to try and have MSRI be the agent for public awareness of mathematics and we hope you'll enjoy the program I guess I should also answer the question why am I here why is will Hearst here and I can only offer two answers one is that about 25 years ago I was a college student in mathematics and I had a high school teacher who told me look all these things that you learn in elementary school arithmetic fractions geometry algebra all of these things are the preparation for you to find out what mathematics really is and although I didn't go on in the field I think sometime in my college education a dawned on me that there was a whole world of mathematics beyond the elementary a world of beauty and logic and a kind of adventure and hopefully the speakers tonight will convey that for you in ways that I cannot but I guess my second reason for being here as a publisher I believe in the public outreach and in providing information to the widest public about not only news but also ideas tonight's speakers are the real experts and they will tell us a little bit about the history the prehistory of Verma's Last Theorem and the part between Verma making his conjecture and it's proof and they will take you through some of the important ideas that are within this proof and they will try to answer the question why is this important what does it connect to if anything in ordinary life and along the way what is math culture like what mathematicians do why do they do it what excites them so before we begin I've been asked and this is the hard part for me to sort of remind the audience what exactly does from Oz Last Theorem say and perhaps the easiest way to try and illustrate this theorem and it has a very simple statement it's the proof which is hard is to recall something from high school algebra and geometry Pythagorean theorem and perhaps you have come across or recall the three four five triangle this is a right triangle with three sides most triangles have three sides it has this side next to the right angle and this side and then it has this hypotenuse and Pythagoras notice that in fact the ancient Babylonians noticed that if you took the length of this side and squared it the length of this side and squared it added those two together you got the length of this side squared and the 3 4 5 triangle illustrates this because 3 squared is 9 4 squared is 16 the sum of 9 and 16 is 25 and that's 5 squared now the interesting problem the problem that makes this a a number theory problem is that these are whole numbers these aren't just sort of arbitrary dimensions and the ancients understood that you had a 3 4 5 triangle and a 5 12 13 triangle and in fact you have an infinite number of triangles with integer sides whole number sides that stand in this relationship of a square plus a square is equal to another square what from ah noticed was that you could not do this if instead of squares you tried it with cubes or with fourth powers and a cube is something times itself times itself again x times X times X and he conjectured he thought in fact he had to prove that it was impossible except in this case of squares and what this theorem says is that no matter where you look no matter how big the numbers you search for you will not find a triple that stands in this relationship not four cubes not four fourth powers not for any power except two people have been trying for 350 years to give a mathematical proof that there are no other solutions except the ones we've talked about now that's the easy part that's the statement of the theorem now comes the fun and the interesting part how do we know it's true and we only found out it was true on June 23rd so it's a relatively new piece of knowledge to begin our first speaker is Robert a sermon and Bob a sermon is the deputy director one of two of MSRI a professor of mathematics at Stanford University his research interests are in the direction of geometry and professor Osmond is going to talk about the Pythagorean theorem the kind of prehistory of Vermont Bob as you heard what I will be talking about was basically the equation a squared plus B squared equals C squared the original case that led Fermo to the thought what about if you go to a cube plus b cubed equals C cubed and fourth powers and so on ad infinitum and the later speakers will start from the third power and tell you what happens the rest of the way but I will just stick to the original equation but what I really will do is start out with a very practical problem which I think many of you may have faced at one or another time and tell you about how one can find a solution and the problem is simply this suppose you have a choice you want to decide which is a better deal would you like one large pizza or a small plus a medium how as usual will make life simple and let's simply assume that the price comes out the same you charge the same amount for the small and medium as they do for the large what we really want to know which one gives us more pizza well this is a very old problem the Greeks thought about that too and some 2000 years ago they came up with a method of solving it which I'll let you know so the next time you face the problem you can do it you bring your pizza knife and you simply cut each of them and it half right down the middle that and then put it out there you cut the medium and place that next to it and finally you cut the large and place all three of them with the straight sides together so they form a triangle and then the trick is this you look at the angle that's opposite the large pizza that's this one down here and you check is it bigger or smaller than a right angle now in this case it's smaller than a right angle and that means you're better off taking the small in the media on the other hand had you had a pizza and you did this and it was an extra-large there the angle is bigger than a right angle and in that case you want the large one rather than the small and a medium now it is just one case which is right in the middle where you carry out this process and lo and behold what you get is a right angle here opposite the large side and that leads you to what we call the Pythagorean theorem and as you all undoubtedly learned in school what the Pythagorean theorem says is that the pizza on the hypotenuse is the sum of the pizzas on the two sides now you may have learned it with other words often they use no words at all they just use this formula a squared plus B squared equals C squared which in fact simply refers to the fact the a squared if the sides a and B are the sides the two small side C is the hypotenuse then what this tells you is that the sum of the squares of the two sides is equal to the square of the hypotenuse now of course when you're talking about pizzas then you want to put a pie in there and so sorry you you all know that that the area of a circle is PI R squared so if you're faced with a six-inch pizza and eight inch pizza and attend eight inch pizza then you see the ten inch pizza would have a radius of five pi times five squared would be the whole pizza you put a half and that's what you have on your hypotenuse you do the same on the two sides and you have to check is it or isn't it true that the one on I pot use adds up you write down the equation and it ends up being three squared plus 4 squared is five squared and as we'll mentioned earlier this is a correct equation it's an example of the Pythagorean theorem I wanted to mention one special case which arises and turns out to be particularly interesting that's where it turns out you have two small pieces of the same sides so that you get what's called an isosceles right triangle perhaps where these are equal hypotenuse and see if you do the Pythagorean theorem you'll find that the ratio of the length of the hypotenuse to the side is just the square root of two now the square root of two is a very interesting number you can prove that it's what's called an irrational number it is not the ratio of two integers to whole numbers it is not a fraction so if you try to make such a triangle with all numbers you'll never get a right angle here so that was an interesting fact that the Greeks discovered also I think that's as much as I want to say about the Pythagorean theorem but I did want to say a little bit more about Pythagoras and other things that he did Tiger is founded a philosophy or religion whatever you want to call it a cult sometimes the Pythagorean Brotherhood of which a basic tenet was that nature which often seems totally irrational and unpredictable at the whims of these gods that are have full of human frailties in fact when you look closer has numerical and mathematical underpinnings and you can often find mathematical reasons for explaining things which on the surface you wouldn't understand at all and one of the examples that was given most strikingly was that of music people had noticed that if you take a say a string and tighten it and pluck it you get a certain note and if you divide it in different ways sometimes the notes you get will sound very consonant or harmonious and sometimes they'll sound discordant and this is a kind of an aesthetic judgment or an emotional reaction and yet what they discovered there was a simple mathematical reason and to illustrate that I'd like to show you an instrument which is a rather simple one known as the mono core looks like this you have one string but you're able to divide it with a kind of movable threat here so that the ratio of the two sides can be divided any way that you like now I just put it here where there's a marker and if it's marked right it's divided so that the ratio of the lengths of the two sides is four to three that's a simple numerical ratio if the Pythagorean theory is right then that should give you two consonant sounding notes so let's try that's one side that's the other and indeed that is an interval which we're familiar with which is called fourth and modern notation but any rate is a consonant sound between those two now if you move a little bit so that the ratio is not simple numerical then it should probably not be so nice sounding let's try that's the kind of thing you'll get in general when you don't have it simple interval now if we move it up further so that you get now it's divided three to two so again one would hope for something simple and again you have a consonant sound harmonious that's actually the interval we call a musical v with the ratio three to two so what I wanted to do would simply ask the question what happens when you combine the Pythagorean theorem use ik with a Pythagorean theorem about right triangles and of course you can guess what you get is a musical triangle now that's not exactly the kind I was referring to but in fact what would I mean by a musical triangle I would mean a right triangle where the sides have lengths that are in simple ratios so that if you constructed such a triangle it would make nice harmonious sounds and in fact I believe there may be one in the wings here comes one there are two musical triangles I would like to thank the employer Exploratorium shop for this as well as all the of our great contributions now first of all let's look at this triangle this one you notice has two equal sides it's an example of what we called an isosceles right triangle and so there's a string simply wrapped around with pulleys to make sure that the tension is the same on all three sides so if we try these two sides you get what's called unison it's the same note because of the same length on the other hand if you try these two sides now I have one side and the hypotenuse what do we get that's an interval that you get when the ratio of the lengths is the square root of two to one you remember I said when you have two sides equal then the ratio of these lengths is irrational not only is it not a simple numerical ratio of three to four or five to three it is not the ratio of any two whole numbers so it should be kind of maximally discordant well that particular chord you might be interested know ever since the Middle Ages was considered the most dissonant that there was it was known as the Diabolus in musica the devil and music and you were told to avoid that at all costs on the other hand we can look for a triangle where we have simple ratio of sides and again in this one this is the one that we've mentioned several times the sides are length three four and five so that the ratios are nice simple one and if we now play for example the three and the four and then we get exactly what we had before musical interval of the four or the four and the five a third and any two of them that you play give you a nice musical sound thank you so the question is is this the only one are there a lot of such musical triangles and the answer is yes there are a lot and a Greek mathematician named Diophantus in fact wrote in a book called arithmetic ax a system of finding all such triangles that is the book that Fermo was reading when he made his famous conjecture and that's what the next speaker will tell you about thank you the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides do not tolerate letting your part is simple dangle so please affect the self same respect for your geometric slides ole Einstein said it when he was getting nowhere give him credit he was heard to declare Eureka square of the hypotenuse of a right angle is equal to the sum of the squares of the two adjacent sides sure as shootin when problems get in your hair be like new who was her to declare yuri gamma square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides now the two Wright brothers before they conquer the air like those other Orville holler look yeah I'm a right triangle is equal to the sum of the squares of the two adjacent side thank you that wonderful performance was by Morris Bob Roe who is a writer composer lyricist of numerous musicals and reviews as well as special entertainment for major corporations our next speaker is Leonor Blum and she is the other deputy director at MSRI she founded the Mills College math and computer science department serving as its head or Co head for 13 years her talk is going to be about the history of efforts to prove from Oz theorem and she will also talk briefly about computer efforts to resolve the theorem by brute force well that's some act to follow and also it's amazing to see all these people here out for mathematics 100 a places the initial birth of modern number theory around 1630 when French law student cara de vermois received a copy of a translation from Greek into Latin of diophantus book after arithmetic oh it was widely thought that diophantus wrote his study of numbers around 250 AD a typical problem taken from Brooke too would be to divide a given square into two squares and Bob talked about that problem by the late 1630s for mass correspondence makes it clear that he absorbed diophantus ideas and that he had a number of creative ideas and results going well beyond Diophantus my job this evening is to explain the meaning and context of one particular statement that firm r wrote in the margin of his copy of diophantus / mas certainly was not writing for posterity but his annotations come down to us from addition of diophantus published by his son his oldest son after FAMAS death by this curious route a casual statement by firma has led to an enormous amount of mathematics including the recent excitement of Wiles proof I like to describe the context of this statement by starting with a whisper ma is a person from all was born in 1601 and studied law in 1631 he was appointed judge in the French town of Toulouse he held this position for his entire life Verma had an excellent classical education and was talented languages in fact he wrote poetry in a number of different languages he was certainly part of the scientific mathematical ferment of the time and worked on many areas of mathematics and indeed Newton later remarked that he was led directly to the differential calculus by fromage method of tangents to put this a little in historical tant timeframe the early 1600s was the time the pilgrims was settling in America most of 4ma scientific mathematical ideas must be inferred from his correspondence in modern parlance he had serious writer's block although the question of publishing his ideas especially a number theory recurs in his correspondence he essentially never published anything I know about tenure for this guy in addition for my travel very little travel or never even leaving Southwest friends with one or two possible exceptions he never had face-to-face meeting with other mathematicians which is quite a remarkable and in contrast that today with so many international meetings in places where mathematicians yet to gather like at the math science Research Institute Berkeley or the Newton Institute in Cambridge where Wiles recently announced his result from our guide in 1665 at which time Samuel the oldest of five children began work on the project of publishing his father's mathematical work for Mars most famous annotation indeed surely the most famous marginal note in all history is the following quote but this is just the folio copy from an edition of the diophantus published by Samuel this is the quote more closely in Latin and since I suspect that some of you may not have a classical education equal to firma I'll be kind enough to give you an English translation it is impossible to separate a cube into two cubes or by quadratic into two by quadratics or in general any power higher than the second into two powers of like degree I have discovered a truly remarkable proof which this margin is too small to contain firm ugh didn't have the benefit of modern algebraic notation but we do so let's try to express the idea that a cube can't be written as a sum of two cubes well what is a cube well it's a number like 8 2 times 2 times 2 that is a product of a number by itself three times for instance a cube number naturally measures the volume of a cube it's not just a coincidence we call raising to the third exponent to the third power cubing so there we have an algebraic statement of Lisa the first thing Sofer mods first assertion was that it was impossible to find three whole numbers a b and c such that a cube a times a times a plus b cubed equals C cube and when we're talking about home numbers we mean positive numbers one two three four not zero suppose we were trying to disprove or show this empirically well we have some props here we could do some experiments right I have a collection of cubes right and so I might be able to take one cube and see if it balances the sum of two cubes we have a scale here and we have various cubes lined up one inch by one inch by one inch I can't see I guess you can so that's one volume well we might imagine that the way this proportion to volume so that's how we would use a balance - so check things out then we have I guess the two by two by two which would be a weight two inches by two inches by two inches so volume eight and so forth so let's see if we might just check out with some examples so these some of these look pretty heavy so what I'm going to start with is one we have a six here right six cubed six times six times six so how much should that be we have some of the audience's that's 216 good and then I'm going to take five cubed that's 5 times 5 times 5 225 and that weighs up there the sum is 341 and let's see suppose we want to prove for a ma so I take a seven seven times seven times seven is what through almost right three so we had the sum there with what 341 and now 343 see whether whoops almost but not quite so I was an experimental mathematician at least I didn't find a counterexample yet so it's close but not exact and from I was saying that no matter how hard one tries it would be impossible to find two cubes of whole number size that balance another cube with a whole number side and for my search more generally the impossibility holds for all larger exponents this is the more general statement and notice to show that form is wrong all you have to do is find a single example of whole numbers a B and C and greater than two that satisfy this equation although many people have tried no one has found a solution n bigger than two and now Andrew well says that no one ever will it's impossible to know what was going on in firm oz mind the only proof affirm are that we know is the proof of this theorem N equals four and by an ingenious argument using a technique called infinite descent it's sort of a technique by showing if you have at one example you can get a lower one and the lower one the low one you show that the equation a to the fourth plus B to the fourth equals C to the fourth is impossible in whole numbers first to the cases N equals 3 and N equals 4 frequently the subsequent correspondence but never returns to the general case more than 100 years later the Swiss mathematician Euler who spent much of his time in our Russia gave a proof of her MA for N equals 3 from our seemingly innocent mark has led to an enormous amount of mathematics although number theory was regarded almost as a recreational endeavor and for my and Euler's time by the early 1800s the deep work of the German mathematician Carl Friedrich Gauss gave number through a respectability and importance that it retains to this day by the middle of the century for Maas statement has become known as for mods Last Theorem because it was the last of her as assertions that remained unsettled not because it was the last one that he ever stated galas perhaps the greatest mathematician of all time felt that the question had little importance in itself but was instead the tip of an iceberg of a far larger mathematical domain in 1860 the French Academy offered a prize for work on firm odds Last Theorem and this led to a flurry of activity that again has continued unabated to this day some of the first general were compromised Last Theorem was done by Sophie Germain Germain was the first to make progress for a large class of ends at one so up to this time it only been shown for particular numbers in particular three and four and Sophie starts looking a more general takes a more general approach and to give a sense of what it was like to be a woman interested in mathematics at the time it is perhaps worth taking a few minutes to tell a little bit about this remarkable women so I'll digress for a minute or two Sophie Germain was born in Paris in 1776 the Europe American independence and was a young girl during the French Revolution for protection she was kept at home starved for mental stimulation young Sophie began to read math books in the family library and this became a passion with her her family vehement Lee disapproved of such studious tendencies and concern for mental well-being took desperate measures they denied her light and heat for her bedroom in order to force her to sleep at night instead of to study but after Sophie's parents were asleep she would wrap herself up in quilts take out a store of hidden candles and work all night after finding her asleep at the desk in the morning with ink frozen and calculations on her slate her family finally relented by the time Sophie Germain was 18 paris was settling back to normal and the occult Polytechnique was founded although women were not accepted Sophie Germain collected lecture notes of various professors and started communicating with mathematicians using the pen name Monsieur Leblanc so they would take her seriously in eight for after reading Gauss's disk was easy honest mr. LeBlanc began a lengthy correspondence with gals it was not until 1807 that Gauss discovered her true identity he remarked that the question of gender mattered little to him and let me read to you a few lines from who has led it to her at that time but how to describe to you my admiration and astonishment at seeing my esteemed correspondent mr. LeBlanc metamorphose himself into this illustrious personage who gives such a brilliant example of what I would find difficult to believe apparently she'd sent him a theorem a taste for the abstract science in general and above all the mysteries of numbers is excessively rare it is not a subject which strikes everyone but when a person of the sex which according to our customs and prejudices must encounter infinitely more difficulties than men to familiarize yourself with these sorting researches succeeds nevertheless insurmountable obstacles then without doubt she must have the noblest courage there was much progress over the years in proving form on I'll put up a table here let me mention and so we could see the progress progressing we see out of the French Academy Sophie for Germain and other things but let me mention the flurry of excitement of the spring of 1847 cautionary note on March 1st LeMay announced the Paris Academy that he could solve for mods Last Theorem for all in by introducing complex numbers to the problem LeMay enthusiastically told the Academy that he could not claim entire credit for this idea since the mathematician leagoo had casually suggested the idea some months before immediately Leoville rose to the floor saying he didn't share Lemay's enthusiasm and declined any credit for himself indeed he suggested any competent mathematician approaching the problem for the first time with a flood of that idea so what but then the eminent mathematician Koshi took the floor indicating he believed Lemmy would succeed pointing up that he himself had presented to the Academy the previous fall an idea what she believed would lead to a resolution but unfortunately he said he had not found time to develop his idea further on March 22nd both LeMay and Co she deposited secret packets with the Academy this was a convention which enabled one to claim priority of ideas without having to reveal them in the following weeks they each published those notices in the Proceedings of the Academy notices which according to some of the books were annoyingly vague on May 24th a letter from the German mathematician Coomer was read to the Academy's proceedings pointing out how their prose had failed but Coomer added by introducing a new kind of complex number and ideal complex number so you get the dynamics of how these mathematics and mathematicians work here but indeed Coomer did develop an entirely new approach laying the foundation in what is now called algebraic number theory in particularly proved the theorem is true if n is a so-called regular prime and he established the VAD a more efficient way to verify the theorem for individual primes those were techniques that were used in the century so moving ahead to the 20th century Vandiver together with graduate students and desk calculators at the University of Texas and that's in the 30s used coomer's idea to verify for mods last theorem for n less than 600 and then there was the advent of computers and here at Berkeley Derek lamer a longtime professor Berkeley was one of the first to use computers to work on mathematics and in this way he was able to confirm for ma o a up to n equals 4,000 and more recently using a network of computers Jo Bueller who is an audience somewhere here verified that firm odds last theorem was true up to N equals four million so we're getting way up there and in fact if you think of it if you try to get a counter example over four million even to show there's a counter example doing the computations would be quite quite difficult one we might imagine a computer proof that could prove for a mod it's not quite impossible to imagine that but the techniques that were used just prove it for one n at a time they do that methods that have been used could never be used to verify the whole theorem all it was but now just five weeks ago to the day in one fell swoop using mathematics and not computers at all Andrew Wiles has proved Fermat's Last Theorem is true for all N thank you there is a delta for every epsilon it's a fact you can always count upon there's a delta for every epsilon and now when again there's also an N but one condition I must give the epsilon must be positive a lonely life all the others live in Noth erm a delta for them now sad how cruel how tragic how pitiful and other adjectives I might remain the matter merits our attention if an epsilon is a hero just because it is greater than zero it must be mighty discouraging to lie to the left of the origin this rank discrimination is not for us we must fight for an enlightened calculus where epsilon zone both minus and plus half Delta's to call there our next speaker is Professor Karl Rubin who was a professor at Ohio State University and he's visiting msri right now Karl was also a PhD student under Andrew Wiles and has done some important work of his own in the theory of elliptic curves and we're getting in now to the sort of meet of how Wiles accomplished this feat and one of the essential ideas a mathematical development that has its own history and was not considered to be linked in any way to from Oz conjecture but a very important part of number theory nonetheless is the theory of elliptic curves and you have to get a little piece of this in order to really understand the large-scale architecture of how this theorem was proved so professor Rubin on elliptic curves okay well let's start with an example of an elliptic curve here we have the equation y squared equals x cubed minus x and then we have the graph of this equation so we have the horizontal x-axis the vertical y-axis and on these axes we plot all the pairs XY that satisfy this equation y squared equals x cubed minus X and you get this curve with those two pieces you are probably familiar with graphing equations if you start with a very simple equation that has X's and Y's but no higher powers then you get a straight line if you have X Squared's and Y Squared's and maybe XY but no higher powers than that then you get what are called conic sections you can get parabolas or circles if you make it just a little bit more complicated as we did here by adding an x cubed this is an elliptic curve it's a little bit more complicated and makes it a little bit more mysterious and a little bit more interesting this particular elliptic curve was studied by fair ma you've already heard about how he was reading in his copy of Diophantus where there are lots of results stated about right triangles Fermo had asked himself are there any right triangles whose sides are whole numbers and for which the area is a perfect square simple question about right triangles fair ma was able to prove that the answer is no there is no such triangle and he proved that by showing that this equation has no solutions with fractions x and y except for the three points up there where y is zero and x is either minus 1 or 0 or plus 1 so elliptic curves which are going to play an important role in the proof of Fermat's Last Theorem their study was really begun by fair MA and we know all all of this and we know he was able to prove this because he did manage to put the proof into the margin of his Diophantus so this is one elliptic curve but there are lots more so if you take any two different integers nonzero integers so two different positive or negative whole numbers call them a and B look at the equation y squared equals x times X minus a times X minus B that's an elliptic curve if you take a 2 B 1 and B 2 B minus 1 and multiply out that product you get Y squared equals x cubed minus X the example we were just looking at there are two types of questions mathematicians would ask about elliptic curves one of them is like Fermo asked if you've got an elliptic curve how can you find all the solutions maybe you want to find all the solutions for x and y are whole numbers or maybe where they're rational numbers like fair minded on the other hand you can ask questions about the whole collection the whole family of different elliptic curves by trying to identify important properties of elliptic and the second problem can help you with the first because identifying the right sort of properties can help you in this question of finding solutions there's one particular property of elliptic curves that plays a crucial role in Wiles proof of pheromones Last Theorem and that's what we want to talk about now this is the property of an elliptic curve called being modular and I'd like to explain what that means when mathematicians have a question which may be too difficult to answer one strategy is to try to replace it with a simpler question or maybe a whole lot of simpler questions in this case we start with the question how do you find all solutions to Y squared equals x cubed minus X or the same thing all x and y for which Y squared minus X cubed minus X is 0 that's the hard question the easier question is take a number like 5 and ask how often is Y squared minus X cubed minus X a multiple of 5 well that the question you can answer with just some arithmetic here we have a table of the values of x going from 0 to 4 the values of y going from 0 to 4 and then in the table we just write down the value of Y squared minus X cubed minus X you notice a couple of zeroes those correspond to actual solutions where that difference is 0 but there are also some numbers that are divisible by 5 but not 0 those are the ones that are sort of answering our question now that let's mark them they're 7 in this picture now you might ask why did I stop at 4 for X and why why didn't I continue well I leave it to you to check that if I were to continue X or continue Y the pattern of which numbers are multiples of 5 would just repeat the numbers themselves would change but this pattern of the seven numbers which are divisible by 5 would remain the same well there's nothing special about 5 you can do this for any number will be interested in prime numbers like 2 3 five seven and so on that have no divisors except for one in themselves take one of these prime numbers call it P make the same sort of table count the number of values that are multiples of P and call that number n sub P this table we made for five shows that n sub 5 has the value 7 well you can compute this for the other Prime's here the table for some of them I've listed the Prime's up to 31 and then I've listed a few larger primes if you start to look at this table you'll notice some patterns one thing you might notice is that often but not always the number in the top row is the same as the number in the bottom row also if you look at it for awhile you'll see that after the first one after the number two all the numbers in the bottom row are 1 less than a multiple of 4 what do these patterns mean well it turns out there's even more structure to these numbers than you might guess in 1814 the mathematician carl friedrich gauss who's considered one of them most eminent mathematicians of all time found a formula a recipe for computing this number n sub p the recipe goes like this 2 is always a little separate so you just keep it separate and you count the n sub 2 is equal to 2 otherwise you look at your prime if it's 1 less than a multiple of 4 then the number n sub P for that prime is just the number itself and if it's one more than a multiple of 4 there's a slightly more complicated formula but if you look at this table here I promise you that I computed the numbers in the bottom table by using Gauss's formula and that's a lot easier than writing out all the values of X up to a million and all the values of Y up to a million and filling in the table well what Gauss's formula tells us is that this sequence of numbers in sub-2 and sub-3 and sub 5 and so on has a very special structure and because of this special structure we say that this lift it curve y squared equals x cubed minus X is modular well you can do the same thing with any elliptic curve remember we have a whole family of elliptic curves what that means is you start with your equation you can build a sequence like this in exactly the same way by making a table counting the number of the visibility's you find so you get a sequence and sub-2 and sub-3 and so on and we define this elliptic curve to be modular we say it has the modular property if the sequence you get this way has the special kind of structure analogous to the structure that Gauss's formula gave for the elliptic curve y squared equals x cubed minus x so to be modular is a special property of elliptic curves which is related to this sequence you get from the elliptic curve well for a sequence like this to have this modular property is really very special very few sequences if you just wrote down a sequence would have this property we know that the sequence coming from y squared equals x cubed minus X has this property but it was very surprising when in 1955 the Japanese mathematician Yutaka taniyama suggested that maybe all elliptic curves are modular he suggested this to a group of his colleagues at a mathematics conference in Japan but no one really knew what to make of this suggestion because no one really knew a reason why this should be true so the suggestion didn't get a lot of people interested for a while taniyama died in 1958 but sometime after that a colleague of his Gaurav shamora who is now a professor at Princeton looked into this some more he thought about what taniyama had suggested he made this conjecture more precise and also made it more believable he came up with some reasons why you might expect it to be true so mathematicians call this a conjecture meaning it's a guess it's something they believed to be true but don't know how to prove this conjecture over the years gained more and more support in the sense that lots of people believed it until the last few years certainly it was universally believed by mathematicians but still no one had any idea how to prove it well if this conjecture that every elliptic curve is modular which goes on to play a crucial role in the proof of dharmas last year our next speaker is Ken ribbit and cannas a professor at Berkeley right across the bay his specialties are number theory and arithmetic algebraic geometry which is a difficult field ken rivets own research has played an important part in the pieces that come together and established from Oz Last Theorem and so he's going to talk about how all these pieces fit together including his piece and tell us a little bit about what it was like to be there on June 23rd in Cambridge when the theorem was announced ken ribbon thanks very much I would like to take you from Yutaka taniyama in 1955 through June 23rd this year possibly a little bit beyond if one asked 20 years ago or even as recently as 1981 what taani honest conjecture had to do with farah naaz Last Theorem the answer would have been nothing there's a man who's very important in the story his name is Gerhard Frye he's a mathematician in Essen who really saw that there was a connection for the first time Frye was in Seattle last week and I'd hope to bring him here unfortunately he's back in Europe I don't even have a photo of him to show you but I do have to show you is a letter that he wrote me in 1981 where he proposed coming to Berkeley for two months to talk about modular elliptic curves and modular curves and Jeremiah's Last Theorem I was very busy at the time I thought maybe I would discourage him from coming but in fact he was welcomed with open arms in Berkeley and we have many discussions about elliptic curves and he wrote down quite a bit in the way of formulas linking elliptic curves and fair Moz Last Theorem and frankly neither of us really saw what he was up to he had the connection between elliptic curves and fair Moz Last Theorem but he didn't see how to relate Tunney honest conjecture to this important assertion of Fermo now if we jump a little further we see that 1985 the situation had changed drastically Frey announced through the mathematical public in fact a very small group of mathematicians that indeed Hani honest conjecture had as a consequence fair laws last theorem and this was the first time that anyone had really exceeded in linking up Carla's last theorem with the tools of modern mathematics the success unfortunately wasn't complete because Frey gave his lecture in a little retreat in Ober Volkov in the Black Forest there were about 20 people present I certainly wasn't among them and the vast majority of the 20 people realized by the end of the lecture that there was a big mistake in what Frey was saying the mistake occurs in his manuscript which was only two and a half pages long um it's a little bit out of focus for you but in fact that's the way most mathematicians saw it there was a very important idea in the manuscript the link between the two but it really wasn't made complete a lot of the people who were present in the Black Forest this for January 1985 came rushing back to Paris and tried to deepen the link that Frey had first seen and there were a number of people who had ideas that made some partial progress in the direction of really setting up would fry it hoped for I was busy teaching calculus in Paris and was hoping that people would get somewhere but it didn't really seem clear how the result was going to come out but again by the summer of that year the situation had changed there was a mathematician in Paris named jean-pierre ser who's certainly one of the greatest living mathematicians he won the Fields Medal which is our version of the Nobel Prize Sarah writing from his mountain home in August during the vacation realized that there was a precise method of linking up pheromones Last Theorem and tani honest conjecture unfortunately this introduced a new layer of complication because in his letter which he actually wrote to a young colleague in Paris Jean Francois mess he explained that one couldn't directly make the length of fry it started but instead what would happen is that you could only prove two little conjectures and if these new conjectures of ser were established then you would know that taani all's conjecture implied fair laws last theorem now the news of such things spreads rapidly among professional mathematicians this was the case because as Nestor was getting this letter he was boarding a plane for California and he came to a workshop which we had at Humboldt State University in Arcata were once again there were something like sixty mathematicians assembled and this caused quite a line of the coin-operated photocopy machine because everybody was very eager to get news of this deepened link that Sarah had set up now one thing I should point out is that this indicates at least it seems to me lustrated of the way mathematicians work during the year we're busy teaching courses and we don't really have time to interact with one another and it's been the pattern that we've had week-long conferences typically when universities are out on vacations so we can live in the dormitories and enjoy dormitory food and we typically have four or five lectures a day which are surrounded by walks in the forest and discussions amongst ourselves and this is a very intense period for people who are unable to see their colleagues during the academic year electronic mail has changed that to some extent because now we can bounce news around the world but it's still very important to speak to people face-to-face and to see exactly what the other person has done now for roughly one year Sarah's conjectures which he called c1 and c2 in the letter were unsettled and the person who actually was able to prove them is myself yours truly I proved that conjecture in 1986 and the way it happened is that at the time when I was proving this conjecture there was an International Congress of mathematicians is one every four years the next one will be next summer in Zurich and it happened to be in Berkeley California there I was sitting with Barry Mazur who's a professor Harvard we were having cappuccinos on the south side and we were talking about these conjectures c1 and c2 and I told Barry that I had more or less proved them but that I still didn't understand to what extent my techniques would work and Barry was the person who said well you're just being silly because in fact what you're doing is proving these conjectures directly and by the time I had licked off the foam from my cup I realized that in fact he was absolutely right and since there were 2,000 mathematicians present in Berkeley I was able to confide in a few of them that this was going to work and then the news just spread and it was somewhat embarrassing because there was a lot of interest in Fermo much more than we realized at the time and people were really pressing me for the details of the proof which were still being washed off the cappuccino cup and what happened is that again by coincidence there was a special year at MSRI which is this mathematical sciences Research Institute it's now sometimes called emissary because it reaches out and tries to convey mathematics outside of professional mathematicians and I gave a lecture one lecture a week for a period of several months at the NSRI to specialists in my field and this was absolutely great because as the proof was taking shape I got a lot of direct criticism from people who really knew all the ins and outs of the mathematical objects that I was working with and by the end of that special year I had an airtight manuscript at least I thought it was airtight but I do want to point out that the process of publishing a proof is a very lengthy one because there referees that intervene and make changes that the author may or may not agree with but nevertheless there's some negotiation which takes place you get a final manuscript it goes to the printer and then there's this concept of a publication delay before it comes out on your local newsstand and so actually four years elapsed between the time when I realized that I was going to prove this in the time that was actually in print well that's the sort of main thing that happened before 1993 but then the next chapter of the story starts so this is Andrew Wiles in a photograph that I took of him quite a number of years ago perhaps it's photographed from 1980 I first met Andrew in 1975 when he was a research student in Cambridge his adviser John Coates is a very close friend of mine and he was busily solving all the problems that Coates was putting to it we certainly became friends and colleagues especially since andrew has done most of his career in the United States after his thesis he became a postdoc at Harvard and after his three years at Harvard he stayed on as a permanent professor in Princeton in Princeton New Jersey I have some other photos to show you I don't know how easy this is to see but this is some variant of the published photo of Andrew writing down the theorem that he can prove taani honest conjecture for the class of elliptic curves that intervene in the story you might have read in the newspaper accounts that many of the mathematicians were taking both notes and photographs during this lecture because we sort of well understood what was going to happen and now in the next photograph you see two gentlemen who may be familiar to you the person on the right is the previous speaker call Rubin and I got myself at the center and that's Andrew he's finishing what turned out to be napa valley Brut for some reason it found its way to Cambridge England on June 23rd one thing that I think was absolutely amazing is that when I went through my Wiles file trying to find evidence of his interest in pheromones Last Theorem I found a postcard that he mailed me I believe in 1980 in which he talks about my demonstration of pheromones Last Theorem this was his way of making a joke but I think it was very much on his mind he explained to British television on June 24th that he had been thinking of pheromones Last Theorem ever since he first encountered it as a statement when he was a boy and when he learned that Fairmont on llaman were connected in 1986 he basic the closet in himself he actually worked in an attic for seven years trying to get a proof of tani honest conjecture and this is really an unparalleled event in the history of mathematics that I know because usually people work in close contact on close contact with their callings they go to conferences and they talk and whilst thought this was so important that he really had to work alone until he knew that he could solve the problem next I just want to tell you of course that Andrew announced that he had salt Santini honest conjecture I guess you know that by now and one thing that I want to stress is that there's a tremendous amount of modern mathematics that goes into his proof you might have gotten the impression from preliminary accounts that he had somehow sat down and written 200 pages which were independent of all the developments that you've heard about tonight but in fact what he did was he took everything that he needed from the most potent techniques in modern number theory including many that really hadn't been devised when he started working the problem in 1986 I just like to tell you who the personalities are involved just so you get some sense of the global sweep on Caruso HIDA is a professor at UCLA but of course he's from Japan he's from Sapporo where a lot of the ideas linking modular forms and elliptic curves first again Barry Mazur I've mentioned before and then there's a topic called the asawa theory Yukichi Misawa who is a close colleague of shamora and of the people whose work that was used by Andrew Wiles of course Carl Rubin when you've just heard from and Ralph Greenberg who's a professor at the University of Washington in Seattle and then there's something called Euler systems Euler is the person who first saw Sarah Maas Last Theorem for exponent 3 and oiler his name occurs in many different places in number theory there's a man named Victor Cole even who's from Moscow who found a new technique in the theory of elliptic curves and he named it oiler systems because he thought that that was very appropriate in the context in which he worked and it was really wild for the first time working with some ideas of Mateus who is another student of John Coates and Cambridge who really bent the Euler systems into the context that was needed my name also occurs in the paper because I proved something involving congruence between modular forms these come up in the proof and then there is another aspect involving modular forms which really relies on work of Robert Langlands who's a professor at the Institute for Advanced Study in Princeton and Gerald Tunnel who's a professor at Rutgers University also in New Jersey well what I'd like to do for you next is just summarize the logic the way the proof goes if you have at your disposal the tools that have surfaced since 1985 if you want to prove fair laws Last Theorem you do it by contradiction you suppose it's false and that means that there really are integers a B and C with a to the N plus B to the n equal to a perfect nth power of course n is some number which is big bigger than 4 million you know that when you start because of the computer calculations I think you just use that it's bigger than thought so you start with this counter example and you do this important thing that Frye wrote down you make the elliptic curve y squared equals x and so on this is just some variant of the elliptic curve that Carl Rubin told you about now it's a little more complicated and Gauss doesn't tell you that it satisfies Tunney honest conjecture but Andrew Wiles does that's what we learned on June 23rd that this elliptic curve satisfies taani honest conjecture on the other hand if you believe what I told people in 1986 you know that it doesn't so it's an impossible situation you have an elliptic curve which has contradictory properties and the only thing that makes this elliptic curve is the supposed solution of pheremones Last Theorem what's wrong there wasn't a solution and that's the way the proof ends now you can look at this from another point of view you can just have the chain of ideas in terms of when they occurred taani alma maters conjecture in 1955 of course before that we could have put pheromone in 1637 one says but there's a lot of intervening mathematics and it didn't fit into the margin of the slide so we just jump ahead to 1955 and then we have taniyama and 1980 excuse me we have fries ideas in 1985 which were fully realized only in 1986 and then there's a big jump of seven years before andrew's lecture in 1993 well in conclusion I'd like to say that this proof of pheremones Last Theorem is a tremendous triumph for modern mathematics it's a tremendous triumph for number theory of course it's a wonderful personal triumph for Andrew Wiles and I'm very happy to have been part of it and it's a pleasure to be here tonight to tell you about it our next speaker is John Conway professor of mathematics at Princeton and Ken ribbon told you a little bit what it was like to be with Andrew Wiles and John tells me he's going to tell us a little bit about what it was like to not be there now that's because John Conway is one of Andrew Wiles closest friends and coworkers and they are in fact working on a book together and John Conway was probably the first person outside of Cambridge to hear about it via email he's also the inventor of the game of life which you may have heard about if you're a computer science person and after John's talk there will be a break 15 minutes 10 minutes or so and then we'll come back for a panel discussion so here is John Conway with his personal history from oz Last Theorem I know one of the first things I want to say is that I'm not really connected with this at all and you'll see why that's true later but I was very interested I think among the people here I'm probably the person who's known Andrew Wiles for the longest time and I'm very very delighted of course that he succeeded but I'm going to give you a very quick history I'm going to go right back to 3,600 years ago because you know it's really very interesting to me I'm very interested in mathematical history this problem really does have a history that far back this clay tablet actually contains essentially a formula it's the formula in diophantus this book that gives you integer solutions of x squared plus y squared equals Z squared and then it plugs various numbers in and gets particular solutions have a 3,600 years ago another home unit sort of dates from roughly the same time is another Babylonian clay tablet this one you can actually read if you like if you if you know that the sort of v-shaped symbol means one and a v-shaped symbol on its side means ten you can actually read this it means there's one part and then 24 parts of the next degree down which differs by a factor of 60 and so on and you've work it out what this is telling you is the length of the side of this thing is the particular number 1.4142 etcetera which is very very close to root 2 so that tells us people interested in this one one root two triangle 3,600 years ago by the way these dates are not at all exact I mean might have been 3601 years ago but it could easily be several hundred years out nobody knows exactly when Diophantus lived he could have lived in the second third or fourth century AD supposing he lived in the second then it's about half the time ago 1800 years ago that he wrote his book which proves that that Babylonian formula gives all the solutions to the to the square problem I'll put on very quickly the 392 years ago Pierre de Fermat was born and he's greatly the greatest number theorist after diophantus and the very great number theorist and all done three hundred and fifty-six years ago roughly we don't exactly know well basha is a edition of diophantus a new translation of Diophantus appeared a few years previously and fomo writes that famous marginal note now you know you've heard the history Fermat's Last Theorem very well expounded by the dog bloom and a few other things by the people here so I'm going to omit the FIR no problem itself here the last theorem but you know it's only one of many theorems that he enunciated both in a in as marginal notes in his Diophantus which were later published by his son and also in letters to other people and these are the theorems lasted quite a respectable time 17:49 Euler proved Fermat's theorem that tells you when a prime number is the sum of two squares the answer is if it leaves the remainder one when you divide it by four it is it needs to remain two minus one it isn't Elfi in thermo says that every integer was the sum of either three triangular numbers I'm not going to stop and tell you what that means or four square numbers or five pentagonal numbers and so on well the 4-square case was proved first by the Grouch in 1772 the pea triangles is one of the things published by Gauss in 1801 he actually found it a year to her Co she who's also been mentioned proved to five Pentagon's theorem and all the others and the other theorems are not going to mention them in detail the last few of them I will believe apart from the last one was proved by a Kobe in the 4018 40s so you can see they lasted 200 years okay the last one lastly 350 as well big deal I'm jumping to 10 years ago now roughly and these dates are just about as inexact as my previous ones by the way it may be 12 fall things now Princeton but that was what got him his job at Princeton and the Fields Medal proved the Modell conjecture I won't tell you what that is exactly but it does follow that for any particular end there are only finitely many really different solutions of the firm our problem so you can't have more than a finite number the problem aren't any in fact we knew now that there are but more models conjecture entails of there were only there couldn't be infinitely many for any kidnapper eight years ago Andrew told me he started working he seriously started working a lot harder seven years ago after Ken Gilbert proved the theorem you've heard him talking about now to six months ago when Wiles tells two colleagues in Princeton you know I'm an emissary to msri from Princeton really mother thinks I've got to do is point out there a few Princeton names as well as bloody names well people are now told me that one day in January whilst earned up and late at night and that I've got something to tell you and he then made him sit down and tell him and then he worked with Nicholas Katz the other person he told at the same time he actually announced the lecture course only Katz knew that it was going to prove Fermat's Last Theorem at the end with ease so the audience rapidly shrank to one personal Katz and they worked through and the walls of difficulty perhaps I should mention that there was a portion that Andrew didn't feel terribly secure about he wasn't entirely familiar himself with the concepts that's why he got cats to help out and there was a difficulty and 12 weeks ago in the middle of May he found what he calls the 10-minute argument that solved that difficult and I don't whether he calls in 10 minutes argument cuz he found it in 10 minutes but he told me it took him only 10 minutes to explain it to cats where he'd been used for several months on this very point before then now this is the first time I come in six weeks ago home I heard about the proof as a party in Princeton and then I asked all my friends in Cambridge to tell me what went on at was his talks well you know you heard five weeks ago to this day he announced the proof his lectures third lecture was to start at 10:00 and as 11 that's 11 a.m. British time that's 6 a.m. in Princeton time and indeed I heard it shortly before 6:00 he must have finished early I heard it at 5:50 a.m. in Princeton I couldn't sleep tell us five weeks ago four weeks ago Wiles came back to Princeton and was a lovely party we had instead of tea instead of tea at 3 p.m. in the math department we had champagne the main difference also the president of university for some reason turned up and I was talking to him and I said where is Andrews about five minutes after teatime was officially due to start and then suddenly everybody started clapping still couldn't see him but then he walked in it was very nice touching him because he walked in and he was leaving one little girl by the hand and holding his other little girl in his arms and they were little bit frightened by the cameras that were flashing and everything but a very very nice moment he said he was well now passed a few weeks ago the foam at this firm fest is plan 20 hours ago I arrived in San Francisco well rather more than one hour ago since we're running away this firm affair started about 10 minutes ago I hope I started talking and now and stop it we're going to take an intermission now please come back we've got a panel discussion that will be starting and if you have questions find one of the cards out in the lobby and bring it to the side of the stage there and we'll include those questions in the panel discussion thank you we're going to have kind of a panel discussion now get at some of the implications and take some questions you've met all the panelists so far except for lead-in Bart Lee is a journalist and science writer and book reviewer he's been a reporter for the New York Times Wall Street Journal LA Times San Francisco Examiner as well and he's also an attorney and you've met the other speakers so lately maybe maybe we should start with you since you haven't made a presentation outside of mathematics what is the significance of this why should people care who are not mathematicians about this kind of work how would you explain it to the people well if your question is what's the good of this as a lot of people have asked me in recent weeks my only answer would be gee that's like asking what's the good of the Sistine Chapel what's the good of Beethoven's ninth these are great achievements of the human mind which which we can all revel in and enjoy for their sheer beauty think of what we have heard here tonight this is a problem whose origins go back into antiquity many thousands of years ago it's a problem that lots of people have thought hard about the specific problem with three hundred and fifty years and we now know it's true furthermore for Moz Last Theorem itself has the property that it's accessible unlike much of what mathematicians do and talk about and think about the Weiman hypothesis which is still outstanding would be very hard I understand I don't understand it and it would be very hard I understand to explain to non mathematicians what it is but firm AHS last theorem can be explained and we've heard it explained tonight at least what the statement is and so this is an opportunity for non mathematicians to understand to get a glimpse into the great beauty the sheer beauty of mathematics and what mathematicians do and how they think about it and this great intellectual structure that has been being developed since formally since the time of the Greeks and that continues to be developed today is there is it a cure for cancer obviously not but there are much more things in the in the firmament of human endeavor than cures for cancer and this and this is one of the great ones the mathematicians agreed basically with the that being the right way to tell the world why you're excited there's this great thing you know you don't understand something until you've proved it and I feel this great desire to understand something and surely that's you know a thing that we can appreciate in itself just we want to understand we want to know what's going on and now a few people will be in the position of understanding why this particular well this question was asked by a lot of people and I this is one of many versions of it but considering that it took three months to find the error and some other proofs and many false proofs have been advanced lenore why why are there confidence that this proof is really going to hold up how do you know that FAMAS theorem has been proved it's really interesting because in fact you hardly hear any dad at all and I think one of the things and ken can probably talk to this even more is that this is being proved within a context of a program of the 20th century I mean there's a whole lot of structural mathematics series it's almost true for structural reasons the ideas are not coming out of the out of left field at all it's coming through a very direct pattern and program of mathematics for but on the other hand there is a lot of vested interest in it in the sense that because so many people are involved in so many parts we don't have as many critical eyes there is a I think that is something to to be cautious of but I think it's just very slight I think there is a sense that this is really solid ken yeah it's the level of confidence and it just grows from day to day because people are really thinking about the major issues involved and what they see is approved where the major ideas are very clear and if there's something wrong well there may be some problem with really justifying fully one of the steps but then we have the feeling that we can come in and we can really sure that up because the thing structurally just makes sense more broadly there is a serious problem in recent years which has been discontinued in which scientists announce their results alleged results about one thing or another literally at a press conference and not in a refereed journal and so forth the most egregious recent example is the cold fusion announcement of I guess four years ago now which turned out to be complete nonsense the press should be faulted in this regard because many many newspapers and reporters and so forth simply cannot tell the difference and have no means of telling the difference between what's right and what's wrong somebody makes an announcement you don't want to be scooped by the opposition by the other papers and so on and so forth and and it should be underlined I don't mean to in any way to equate wiles proof of Fermat's Last Theorem with ColdFusion but it should be underlined that every line of this proof has not yet been checked that's perfectly true and of course there could be a mistaken there's a a very definite difference between this and some other things you know the basic thing is you ask could you conceivably prove it this way with good arguments of this type getter and every now and then there's a calculation involved could you calculate that thing and if you did and got a set answer would it be right so is this what ken was saying before it is an extent and the mathematicians in a position to know can see that a program like this might succeed and you know then under was a very careful cautious conservative person that goes into the equation everything does after all we're humans I mean even Andrew Wiles is human in mind and you know we can make mistakes and he's you know he's made some but you can see that he could be done like this you know with a recent flawed attempt to prove the theorem dirtball things a great expert said if he could have been done like that I would have done it he knew straight away this time the experts right from the start are believing it why well because they see it's part of an intellectual program that could succeed and at Wiles has good credentials you know etc I mean there are a lot of things but yes there could be some things that need patching maybe there's something that can't be patched maybe this is a really serious hole in this proof it's conceivable but we have good reason here to believe this is a pretty soundproof we got this question a few times - did Verma really have approval and what what is the general thinking is there something that perhaps has been missed all these years or was a robot just making an informal comment what was the what's the view of the professionals earlier actually I think most people feel that from I do not have improved but on the other hand John really believes the other way of you were trying to argue that the others just been opening my mouth a lot so I was hoping to have a chance to rest for a time but but since I'm on the underdog side here yeah I I'm a Qi sort of perhaps 5050 the general consensus among those in the know seems to be that firma probably deceived himself but there are some difficulties with this point of view first of all he wasn't writing for anybody else it was a note to himself so there's no question of his reputation as it were being on the line at the moment that he wrote that marginal note it's a memorandum to himself it's possible that he just made a sign error and you know so he only possibly wrote that note only a minute after he found the proof there was a sign error and I think that's unlikely because he comes back several times and he did produce a proof for N equals four later on you have to answer this question if you believe he didn't have a proof what did he think he had what was the proof that deceived firma that firm are deceived himself with it certainly wasn't these later proofs that do exist all over the place that used non unique factorization all sort of use unique factorization when it doesn't exist it's that an extensive number fields those ideas weren't available to Burma it's a very interesting question and I don't know the answer we shouldn't take the fact that I don't feel that we should take the fact that mathematicians haven't found one until now as sort of evidence that there wasn't one that was available - can you look like you're like to amplify that now that the fact that we know Andrew Wiles has approved and that this wouldn't have been accessible to Fermo doesn't mean that fair amount improve it if he had a proof it was certainly a different one obviously surprised by the way so to speak unless it really something still an open question here wherever anybody looking for some work what was firm osburgh it was not the proof of Andrew Wiles what's the simpler proof even if it was a fallacious / what was it that would be a very interesting that would be there's a life's work for someone in the audience remember that some of the other theorems of therma lasted for 200 years and some of them have a simple proof oh you mean some of 350 years you know less than a factor of 2 bigger it might seem quite a lot to you people but for squaring a circle to 2,000 years yes yes this question was aimed at Ken rivet but maybe everybody can have what what is the if if from Oz theorem itself is not interesting what is the significance of the taniyama conjecture and the kind of you know we people say that Wiles work fits into the larger flow of mathematics talk about that larger flow if Verma was not an implication would this still be exciting work that Wiles had done well it would certainly be exciting to professional mathematicians but I think honestly we wouldn't be here tonight there are many people who are fascinated by pheromones Last Theorem and I think among mathematicians some are and some aren't taani honest conjecture is fascinating to me because it represents a connection between two different kinds of mathematics and two different kinds of objects namely you have the elliptic curves given by simple algebraic equations and then you have what Carl Rubin called recipes for finding numbers of solutions to these equations modulo 5 and other numbers and these recipes they fit into the theory of modular forums someone talked to me during the intermission about elliptic functions this belongs to another branch of mathematics called analysis and the fact that there is a connection between these two different branches is astonishing and if we understood it better I think we'd know really a lot more than we know today here's a question in a different direction most of the speakers tonight have been men and how what is the picture for women in mathematics and what is the the audience seems to be a little more heterogeneous than the speakers what what how is mathematics as a field maybe at the risk of throwing the question in the wrong person or where you want to get us started at least well I probably am the right person there are many many issues here I mean is mathematics a good field for women that's a that's a particular issue and I would say absolutely it's great I mean I hope that people here are starting to see the excitement of it where again mathematics is just wonderful what about the situation for women in mathematics I mentioned Sophia Germain and her her experiences have things changed since the French Revolution American dependence and there has been an interesting point in history of women in mathematics women could not become a graduate student at Princeton University mathematics until 1968 there's very very recent but the situation has changed a lot in this century at the turn of the century and in the United States before World War two the situation for me the United States was quite good and that sort of lasted through the war effort and women were very involved in mathematical technical fields there was a historically afterwards some kind of effort to put women back at home I think that affected women in many fields including mathematics when the numbers of math women getting PhDs plummeted down to 6% now again we're up to about 20 25 percent so there is a very significant number of young women going into mathematics today past 20 years have been many programs to encourage women in mathematics I've been involved quite a few is the association of women mathematics which is a professional organization right here in the Bay Area there the math-science Network there are summer programs there's the Mills College summer Institute for undergraduate women at MSR I we've been this year it's part of our emissary effort to be much more involved with social issues we have been really paying attention to participation women and minorities in mathematics people often say you know mathematicians do great work when they're young but when they get to be 3040 they're over the hill does anybody agree or disagree mom mention Waller well how old is Andrew Wiles for 44 absolutely that's important for the fields so he's a counterexample to the burn out well I think there are a lot of counter examples yes it is true that in mathematics as in music as in chess there are child prodigies and those I think are the only fields in which there are real child prodigies you could not imagine I mean Mozart wrote great great music when he was eight you could not imagine somebody an eight year old writing a great novel it's not that it's not that the that that the eight year old couldn't have mastered the techniques of composition and so forth but would not have the life experience to write a great novel it's less to now than it used to be partly because to do really good work at the forefront of mathematics now you just have to know such a lot and it takes a long time to learn it so the that age the age at which mathematicians are productive is creeping upwards so could there ever be another Ramanujan a person who comes from a untutored background but has such enormous talent that they can really participate or do you have to go through a standard academic career to really become a mathematician today I think this still do I mean I think you know there are lots of different people in the world and mathematicians are very different to each other that there's not the standard model for mathematicians at home think well here's the question is is this a golden age of mathematics that were in today yes the only thing left to do is to thank our speakers and panelists or counting sheep when you're trying to sleep being fair when there's something to share being neat when you're folding a sheet that's my farm is when a ball bounces off of a wall when you cook from a recipe book when you know how much money you owe that's my mattock so much gold can you hold in an elephant's ear when it's noon on the moon then what time is it here if you could count for a year would you get to infinity or somewhere in that vicinity when you choose how much postage to use when you know what's the chance it would snow when you bet and you end up in debt or try as you may you just can't get away from math over Andrew Wiles gently smiles does his thing and voila QED we agree and we all shout hoorah as he confirms what Fermo jotted down in that margin which could have used some enlargen tap your feet keep in time to a beat of a song while you're singing along harmonize with the rest of the guys yes try as you may you just can't get away from my man

Channel: Graduate Mathematics

Views: 449,181

Rating: 4.7840419 out of 5

Keywords: Fermat's Last Theorem, andrew wiles

Id: 6ymTZEeTjI8

Length: 96min 40sec (5800 seconds)

Published: Sat Sep 24 2016