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 at the public library that talked about this problem, and like many other mathematicians, that's one of the things that got me excited about mathematics. I discovered it when I was 10, obviously when I was 10 years old. I didn't do much with this problem. I kept trying in my teens and then gave up when I was a student. This problem has a long history. It was proven by many mathematicians in the 19th century, but most mathematicians seem to have done it. abandoned in the 20th century until about 8 years ago there was a breakthrough someone related to the

last

deore of fma this is the problem I have been working on another well known problem in number theory and since that time it was made that link I've worked on it, so it's been seven years now.

I didn't start with any idea how to solve this problem so it was a problem and it took me five years to make the first real breakthrough and then I finished it probably around 3 or 4 weeks ago there are many more problems in mathematics there are many unsolved problems uh 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 the next project, the main meaning of this problem is symbolic. One doesn't expect any real application of this kind of result, but you never know.

It is mainly symbolic because many mathematicians have tried it and it has led to the creation of so much mathematics itself that everyone. I have thought that the dream of mathematics is to solve such a problem. Welcome to tonight's special on the Last Theorem of Form and I see some seats and some people coming in, so feel free to take a seat while I make a couple of introductory comments. I'm Will Hurst, the editor of the San Francisco Examiner, and I'll be moderating tonight's session and introducing some of the speakers. On June 23,

1993

, something extremely exciting in the mathematical world happened to some of us.

They were innocently reading the New York Times that day, others found out by email or maybe a day later and that was the announcement that a mathematician named Andrew WS had proven something that mathematicians had been trying to prove for 350 years and that had finally resolved. Oz's

last

theorem

is affirmative. Now tonight's presentation is really aimed at a general audience, so we're hoping that we'll have some students in the audience, professionals from other allied fields, maybe also some professional mathematicians, and members of the general community. audience who are interested in finding out what all the fuss is about this

theorem

what's exciting how Andrew WS did it what it means and we're going to try to answer some of those questions and we have some very talented and brilliant speakers who I've condensed a lot of complicated mathematics into some simple talks about 10 minutes each.

I don't think anyone should worry if they miss a detail here or there, even professional mathematicians go to talks and miss a detail here and there, but uh. try to understand the general concepts, feel the flow and you will have some questions at the end and don't worry there won't be a test. Thanks, I should go to the Mathematical Sciences Research Institute, which is our main host tonight, msri, as it's known, is a research institution a little bit like the Princeton Institute for Advanced Studies located in the Berkeley Hills. They have a Zen monastery type building overlooking the bay area and a lot of interesting people stop by.

Mathematical researchers who give talks listen to other people's talks and talk to each other. Bill Sddon, director of MSRI, set a goal for the institution to do some outreach. This was before the announcement of the Fas test. Last. Theorem, so this show tonight is an effort to try to make msri the agent of public awareness about mathematics and we hope you enjoy the show. I guess I should also answer the question of why I'm here, why is Will Hurst here. and I can only offer two answers, one is that about 25 years ago I was a college math student and I had a high school teacher who told me look at all these things you learn in elementary school arithmetic fractions geometry algebra all these things are the preparation for you to discover what mathematics really is and, although I did not go to the field, I think that at some point in my university education I realized that there was a whole world of mathematics beyond the elementary, a world of beauty and logic . and a kind of Adventure and I hope that the speakers tonight convey that to you in a way that I couldn't, but I guess my second reason for being here as an editor I believe in public outreach and providing information to the broader public about not just news but also

ideas

the speakers tonight are the real experts and they will tell us a little about the history, the prehistory of verma's last theorem and the part between fromma doing his conjecture and his

proof

and they will walk you through some of the Important points.

ideas

that are within this

proof

and will try to answer the question why is this important, what is it connected to, if anything, in ordinary life and, along the way, what is mathematical culture like, what do mathematicians do, for example? what they do, what is so exciting before. we begin I have been asked and this is the difficult part for me to remind the audience what exactly the OAS theorem says and perhaps the easiest way to try to illustrate this theorem and it has a very simple statement: it is the proof that It is difficult to remember something from the Pythagorean theorem from high school algebra and geometry and maybe you have come across or remember the triangle uh 345 this is a right triangle uh with three sides most triangles have three sides has this side next to the right angle and this side and then it has this hypotenuse and Pythagoras noticed and 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 them and added those two together, you got the length of this side squared and the triangle 345 illustrates this because 3 squared is 9 4 squared is 16 the sum of 9 and 16 is 25 and that is 5^ squared the interesting problem the problem that makes this a number The problem of theory is that these are integers they are not just some kind of arbitrary dimensions and the Ancients understood that you had a 345 triangle and a 512 13 triangle and in fact you have an infinite number of triangles with integer sides that are in this relationship of one square plus one square is equal to another Square what FMA noticed was that you could not do this if instead of squares you tried with cubes or with fourth powers and a cube is something multiplied by itself multiplied by itself again x x x time x and it He conjectured that he actually thought that He had a proof 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 look for, you will not find a triple that is in this relationship, not for cubes, no. for the fourth powers, not for any power, except two people who have been trying for 350 years to give a mathematical proof that there are no other solutions except the ones we have talked about.

That's the easy part, the statement of the theorem. Now comes the fun part. And the interesting part, how do we know it's true? We only found out it was true on June 23, so it's relatively new knowledge to begin with. Our first speaker is Robert Osman and Bob Osman is the deputy director, one of two from MSRI. professor of mathematics at Stanford University, his research interests are in the direction of geometry and Professor Osman is going to talk about the Pythagorean theorem, FMA Bob's kind of prehistory, as you heard, what I will talk about was basically the equation A2 + B2 = c^2 the original case that led FMA to think what happens if you go to a puppy plus b cubed c cubed and fourth power and so on to infinity and subsequent speakers will start from the third power and will they tell you what happens? the rest of the way, but I'll stick to the original equation, but what I'll really do is start with a very practical problem that I think many of you will have faced at one time or another and I'll tell you how a solution can be found and the problem It's just this Let's say you have a choice and you want to decide which is the best deal: would you like a large pizza or a small plus a medium?

Now, as always, we will make life simpler and let's just assume that the price is the same, they charge you the same for the small and medium as for the large. What we really want to know is which one gives us more pizza. I found well, this is a very old problem. The Greeks also thought about it and about 2,000 years ago they recently came up with a method to solve it, which I will let you know so that the next time you face the problem you can do it. You bring your pizza knife and you just cut each one in half, right down the middle, like this. and then put it there, cut the medium place that is next to it and finally you cut the large one and place the three with the straight sides together so that they form a triangle and then the trick is this: you look at the opposite angle to the large pizza which is this one down here and check if it's bigger or smaller than a right angle now in this case it's smaller than a right angle and that means you better take the small and the medium on the other hand if you had a pizza and you did this and it was extra large there the angle is larger than a right angle and in that case you want the large instead of the small and the medium now there is only one case that is right in the middle where you carry out this process and here is what you get is a right angle here opposite the large side and that leads you to what we call a Pythagorean theorem and as no doubt everyone learned in school, what the Pythagorean theorem says is that the pizza in the hypotenuse is the sum of the pizzas on both sides.

Now it is possible that you have learned it with other words. They often don't use any words, they just use this formula A S plus b^2 = c^2 which in fact simply refers to the fact that a s if sides A and B are the sides the two small sides C is the hypotenuse then it 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 there and I'm so sorry, you all know that the Area of ​​a circle is p r squ, so if you are faced with a 6 inch pizza, an 8 inch pizza and a 10 inch pizza, then you see the 10 inch pizza would have a radius of five Pi * 5^ SAR would be You put half of the whole pizza on it and that is what you have on your hypotenuse.

You do the same on both sides and you have to verify if it is or not. Isn't it true that the hypotenuse adds? You write the equation and it ends up being 3^2 + 4 S is 5^ s and as mentioned above, this is a correct equation, it is an example of the Pythagorean theorem. I wanted to mention a special case that comes up and is particularly interesting, which is where you happen to have two small pizzas of the same size, so you get what's called an isosceles right triangle, maybe where these are equal, the hypotenuse is C, if you do. the Pythagorean theorem you will find that the ratio between the length of the hypotenuse and the side is just the square root of two now the square root of two is a very interesting number you can show that it is what is called an irrational number, it is not the ratio between two whole numbers and two whole numbers is not a fraction, so if you try to make a triangle with whole numbers you will never get a right angle, so that was an interesting fact that the Greeks also discovered.

I think it's that much. like I want to say about the Pythagorean theorem, but I wanted to say a little more about Pythagoras and other things that he did. Pythagoras founded a philosophy or a religion whatever you want to call it a cult, sometimes the Pythagorean Brotherhood of which he is a basic element. The tenant was that nature which often seems totally irrational and unpredictable before the whims of these gods who are full of human weaknesses; in fact when you look closer it has numerical and mathematical foundations and you can often find mathematical reasons to explain things that on the surface you wouldn't understand at all and one of the examples that was given most surprisingly was that of Music people who had noticed that If you take a string, tighten it and pluck it, you get a certain note and if you write it in different ways.

Sometimes the notes you get will sound very consonant or harmonious and sometimes they will sound discordant and this is a kind of aesthetic judgment or an emotional reaction and yet what they discovered there was a simple mathematical reason and to illustrate that I would like to show you a instrument that is quite simple known as a mono chord. It looks like this. You have one string but you can split it with a sort of movable fret so that the proportion of the two sides can be split. any way you want now I put it here where there is a marker and if it is marked correctly it divides so that the ratio of the length of the two sides is 4 to three, that is a simple numerical ratio if the Pythagorean theory is correct , sothat should give you two notes that sound consonant, so let's try to make one side the other, and in fact, that's an interval that we're familiar with called fourth and modern notation, but in any case it's a consonant sound between those two now if you move. a little bit so that the proportion is not simple numerical then it probably shouldn't sound so good let's try that's the kind of thing you're going to get in general when you don't have a simple interval now if we turn it up further so you get now it's divided 3 to two so again one would expect something simple and again you have a harmonious uh consonant sound, that's actually the interval that we call musical fifth with a ratio of 3:2 so what I wanted to do was just ask what happens when you combine the theory pythagorean theory of music with a pagan theorem about right triangles and of course you can guess that what you get is a musical triangle, that's not exactly the type I was referring to but in fact what do I mean by a musical triangle ?

I would be referring to a right triangle where the sides have lengths that are in simple proportions, so if you built such a triangle it would produce nice harmonious sounds and in fact I think there may be one in the wings. Here comes one. There are two musical triangles here. I would like to thank the Exploratorium store. for this and all the other great contributions now, first of all, let's look at this triangle, this one you notice has two equal sides, it is an example of what we call an isosceles right triangle and therefore there is a string simply wound with a pulley to make I am sure that the tension is the same on all three sides, so if we try these two sides you will get what is called Unison, it is the same note due to the same length, on the other hand, if you try these two sides Now I have one side on the hypotenuse.

What do we get? That is the interval you get when the ratio of the lengths is the square root of 2:1. Remember how I said that when you have two equal sides then the ratio of these lengths is irrational? Not only is it not a simple number. ratio of 3:4 or 5:3 is not the ratio of two integers, so it should be maximally discordant. That particular chord you might be interested in knowing since the Middle Ages was considered the most dissonant in existence. was known as the diabolis in Musa the devil in music and they told you to avoid that at all costs on the other hand we can look for a triangle where we have a SIMPLE proportion of sides and again in this this is the one I have mentioned several times that the sides they have a length of three, four and five, so the proportions are nice and simple and if we now play, for example, the three and the four, we get exactly what we had before a musical interval of the four or the four and five a third and two of them that you play give you a good musical sound, thank you so the question is is this the only one or are there many of those musical triangles and the answer is yes, there are many and uh in fact, a Greek mathematician named diophantus wrote in a book called arithmetica a system to find all those triangles, that is the book that farar was reading when he made his famous conjecture and that is what the next speaker will tell you, thanks to the square of the The hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides.

You wouldn't tolerate your entry hanging, so please show the same respect for your geometric slides. Old Einstein said it when he wasn't getting anywhere. Go ahead. credit Eureka was heard declaring the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides sure as a shot when problems get into your hair be like Newton who was she to declare to Eureka the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides now the two right Brothers before conquering the air like those others Orville Hollard look here the square of the hypotenuse of a right triangle It is equal to the sum of the squares of the two adjacent sides thanks to that wonderful performance was by Morris Babro, who is a writer, composer, lyricist of numerous musicals and reviews, as well as special entertainment for large corporations.

Our next speaker is Lenor Blum and she is the other one. Deputy Director of MSRI, she founded the mathematics and computer science department at Mills College and was its director or co-director for 13 years. Her talk will be about the history of efforts to prove the theorem and she will also talk briefly about uh. computer efforts to solve the Brute Force theorem Lenor well, that's an act to follow and it's also surprising to see all these people here for mathematics Andre places the initial birth of modern number theory around 1630, when a student of French law Pierre Duar received a copy of a Greek to Latin translation of the book of Dian after Arithmetic Diaphanus was widely thought to have written his study of numbers around 250 AD A typical problem taken from book two would be to divide a given square into two squares and Bob discussed that problem in the late 1630s.

FAS's correspondence makes it clear that he absorbed Diaphanus's ideas and that he had a series of ideas and creative results that went far beyond diaphanus. My job tonight is to explain the meaning and context of a particular statement that FMA wrote in the margin of his copy of diaphanus FMA was certainly not written for posterity but his annotations come to us from an edition of diaphanus published by his son his son major after the death of FMA by this curious route a casual statement by FMA has given rise to an enormous amount of mathematics including the recent enthusiasm of Proof V I would like to describe the context of this statement starting with with FMA as a person FMA was born in 1601 and studied law in 1631 he was appointed judge in the French city of Tulo he held this position throughout his life Verar had an excellent classical education and had a talent for languages ​​and in fact wrote poetry in several different languages, he was certainly part of the ferments scientists and mathematicians of the time and worked in many areas of science.

Mathematics and indeed Newton later commented that FMA's method of tangents led directly to differential calculus, to put this a bit in the historical framework T. The early 17th century was when the Pilgrims settled in America , most of FMA's scientific and mathematical ideas must be inferred from his correspondence in modern Parliament. He had serious writer's block, although the issue of publishing his ideas, especially in number theory, recurred in his correspondence. Basically, he never posted anything. I know about the ownership of this guy, too, thanks to my travels, very little before. Even leaving Southwest friends aside with one or two possible exceptions, he never had face-to-face meetings with other math competitions, which is pretty unremarkable.

I mean, contrast that today with so many international meetings and places where mathematicians come together like in mathematical sciences research. Institute in Berkeley or the Newton Institute in Cambridge, where Wildes recently announced his results. Forma died in 1665, at which time Samuel, the eldest of five children, began work on the project of publishing his father's mathematical work. Forma's most famous annotation, in fact, surely the most famous marginalia. Note throughout the story is the following quote, but this is just a photocopy of an addition to the diaphanous published by Samuel, this is the closest quote in Latin and since I suspect some of you may not have a classical education equal to FMA I will 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 the same degree. I have discovered something truly remarkable. Proof that this margin is too small to contain. Forma didn't have the benefit of modern algebraic notation, but we do have it. Let's try to express well the idea that a cube cannot be written as the sum of two cubes. What is a cube? is a number like eight 2*2*2 that is a product of a number times itself three times, for example a cubic number naturally measures the volume of a cube, it is not just a coincidence, we call it raising the third exponent to the third power.

Okay, so we have an algebraic statement of at least the first, so FMA's first statement was that it was impossible to find three integers a b and c such that a cubed a * a * a plus b cubal c cub and when we are talking about integers we mean positive numbers 1 2 3 4 not zero suppose we are trying to disprove or show this empirically well we have some props here and we could do some experiments. I have a collection of cubes, so I might be able to take a cube and see if it balances the sum of two cubes, we have a scale here and we have several cubes lined up, one inch by one inch by one inch.

I can't see, I guess so, so that's a volume. Well, we could imagine that weight is proportional to volume, that's how we would use a scale to check things. So we have, I guess, a 2 by two by two that would have a weight of 2 inches x 2 inches What I'm going to start with is one, we have a six, here, on the right, 6 cubed. *6*6 so how much should it be? We have so much audience that it's 216 well and then I'm going to take a 5 cubed which is 5 * 5 * 525 and that weighs in there, the sum is 341 and let's see, let's suppose.

We want to prove to mom, so I take a seven 7 * 7 * 7 is what is almost correct, let's go to three, so we had the sum, there was 341 and now 343, let's see what, oops, almost, but not of the everything, if I were an experimental mathematician in At least I haven't found a counterexample yet, so it's close but not exact, and I was saying that no matter how hard I tried, it would be impossible to find two cubes with integer sides that balance another cube with one integer side. and We affirm more generally that the impossibility is valid for all largest exponents.

This is the most general statement and notice to show that, because of my mistake, all you have to do is find a single example of integers a, b, c and n greater than two that satisfy this. equation, although many people have tried it, no one has found the solution and it is greater than two and now Andrew W says no one ever will. It's impossible to know what was going through FMA's mind. The only proof of FMA that we know of is the proof of his The n equals 4 theorem and through a clever argument using a technique called infinite descent, it is a kind of technique that shows that if you have an example you can get a lower one and one lower, showed that the equation a 4º plus b 4º = C 4 is impossible in whole numbers.

He frequently refers to cases Nal 3 and nals 4 in his later correspondence, but never returns to the general case. More than 100 years later, the Swiss mathematician Oiler, who spent much of his time in Russia, gave a proof of FMA for n equals 3 FMA apparently innocent REM Mark has given rise to an enormous amount of mathematics, although the theory of numbers was considered almost a recreational endeavor in the days of Fi and Oiler in the early 19th century. The profound work of the German mathematician Cedri Gal gave respectability to number theory. and importance that it retains to this day in the middle of the century the statement of fma is known as the last theorem of fma because it was the last of the statements of fma that remained unsolved not because it was the last that gaus ever stated, perhaps The greatest mathematician of all time considered the question to be of little importance in itself, but rather the tip of an iceberg of a much broader mathematical domain.

In 1860, the French Academy offered a prize for his work on the FAS theorem and this sparked a flurry of activity that has again continued unabated to the present day. Some of THM's first general work was done by Sophie Germaine Gerain. She was the first to progress for a large number of end classes at once, so up to this point she has only been shown for particular particular numbers. three and four and Sophie starts to look more generally, she takes a more general approach and to give an idea of ​​what it was like to be a woman interested in mathematics at that time, it is perhaps worth taking a few minutes to say a little about this. extraordinary women, so I'll digress for a minute or two.

Sophie Germain was born in Paris in 1776, the year of American independence, and she was a child during the French evolution to protect her from it, she was kept at home without mental stimulation. Young Sophie began to read. Mathematics books in the family library and this became a passion for her. Her family vehemently disapproved of such studious tendencies and, concerned about her mental well-being, took desperate measures: they denied light and heat in her bedroom to force her to sleep at night. evening. She was going to study, but after Sophie's parents fell asleep, she went to school.sheltered under duvets, took out a stash of hidden candles and worked all night after finding her asleep at her desk in the morning with frozen ink and calculations on her blackboard; her family finally relented.

When Sophie Germaine was 18, Paris was returning to normal and the EO polytechnic was founded, although women were not accepted. Sophie Germaine collected lecture notes from various professors and began communicating with mathematicians using the pseudonym M LeBlanc to be taken seriously. In 1804 after reading G. Leblan's disquisition she began a long correspondence with Gal. It was not until 1807 that Gal discovered her true identity. He commented that the question of gender mattered little to him and he let me read a few lines of his letter to her at that time, but how can I describe my admiration and punishment at seeing my esteemed mure correspondent LeBlanc metamorphose into this illustrious character who sets such a brilliant example?

What I would find difficult to believe apparently I had sent him a theorem a taste for abstract science in general and especially the mysteries of numbers is excessively rare it is not a topic that surprises everyone we know it but when a person of the sex that According to our customs and prejudices, he must find infinitely more difficulties than men to become familiar. However, with these investigations he manages to overcome these obstacles, then without a doubt he must have the noblest courage. A lot of progress has been made over the years to demonstrate M. I'll put a table here, let me mention it and so we can see the progress, we see the Sophie Germaine French Academy and other things, but let me mention the surge of enthusiasm in the spring of 1847.

Warning note on March 1. she announced to the Paris Academy that she could solve Maas theum for all n by introducing complex numbers into the problem. Lame enthusiastically told the academy that he could not claim full credit for this idea since the mathematician Leoville had casually suggested the idea some months earlier and immediately Fille Rose fell to the ground saying that he did not share L's enthusiasm and refused any credit for himself. same; in fact, he suggested that any competent mathematician approaching the problem for the first time would have thought of that idea and what, but then the eminent mathematician Koshi spoke up indicating that he believed Lame would succeed, noting that he himself had presented to the academy the fall previously an idea that he believed would lead to a resolution, but unfortunately said he had not found time to develop his idea further.

On March 22, both Lame and Koshi deposited secret packages with the academy. This was a convention that allowed one to claim priority of ideas without having to reveal them in the following weeks, each publishing their notices in the readings of the notices of the academy which, according to some of the books, were annoyingly vague on May 24, a letter from the German mathematician Kummer was read, yes, at the Academy proceedings, pointing out how his Pro had failed, but Kummer added by introducing a new type of complex number and an ideal complex number so you can get the dynamics of how this mathematician and mathematicians work. here, but in fact Cumer developed a completely new approach that laid the foundation for what is now called algebraic number theory, particularly he Pro the theorem is true if N is a so-called regular prime and established a much more efficient way of verifying the theorem for individual primes those were techniques that were used in this century, so moving forward to the 20th century, Vaniver along with graduate students and desktop calculators at the University of Texas and that is, um, in the '30s he used the idea of Kummer to verify the Mazas theorem for n minus. more than 600 and then came the arrival of computers and here at Berkeley Derek Lamer, a veteran Berkeley professor, was one of the first to use computers to work on mathematics and in this way he was able to confirm from uh to n equals 4000 and, more recently, using a computer network, Joe Bueller, who is in our audience somewhere here, verified that the FAS Last Theorem was true up to n equals 4 million, so we're getting very far and, In fact, if you think about it, try to come up with a counterexample of more than 4 million to even prove that it was a counterexample.

Doing the math would be quite difficult. One could imagine a computer test that could prove an error. It's not impossible to imagine, but the techniques that were used. Just try it for one and do the methods that have been used at the same time. They could never be used to verify the entire theorem at once, but now they do just five weeks in one fell swoop using mathematics and not computers. Everything Andrew Wilds has shown is true for everyone, thank you. There is a Delta for each Epsilon. It's a fact you can always count on. There is a Delta for each Epsilon.

And from time to time there is also an N, but I must give a condition. Epsilon must be positive, a solitary life, everyone else lives without theorem, a Delta for them, how sad, how cruel, how tragic, how pitiful, another adic Ives. I might mention that the issue deserves our attention if an Epsilon is a hero just because it is greater than zero. It must be very discouraging to lie to the left of the origin. This rank discrimination is not for us. We must strive for an enlightened calculus in which Epsilon, both less and more, has Deltas to call its own.

Our next speaker is Professor Carl Rubin, who is. a professor at Ohio State University and is visiting msri right now. Carl was also a PhD student under Andrew ws and has done important work on his own in the theory of elliptic curves and now we're getting into the kind of meat of how Wilds accomplishes this feat and one of the essential ideas, a development mathematician that has its own history and was not considered linked in any way to the uh Fma conjecture, but a very important part of number theory is the theory of elliptics. curves and you have to get a small part of this to really understand the large scale architecture of how this theorem was proven, so professor screw up elliptic curves, well let's start with an example of an elliptic curve, here we have the equation y^ 2 = x Cub - x and then we have the graph of this equation, so we have the horizontal x axis, the vertical and AIS and on these axes we plot all the X Y pairs that satisfy this equation y^2 = xb- x and you get this curve with those two pieces you are probably familiar with for graphing equations if you start with a very simple equation that has x and y but no higher powers then you will get a straight line if you have X squared and Y squared and maybe XY but no higher powers than that, you get what are called conic sections.

You can get bars or circles if you make it a little more complicated like we did here by adding an X to the cube. This is an elliptic curve. It's a little more complicated. It's a little more mysterious and a little more interesting. Faira studied this particular elliptic curve. You may have already heard about how he was reading in his copy of diaphanus where there are many results on right triangles that FMA had wondered about. there are right triangles whose sides are integers and whose area is a perfect square simple question about right triangles FMA was able to prove that the answer is no, there is no such triangle and he proved it by showing that this equation has no solutions with fractions X and Y, except the three points up there where Y is zero and all this and we know that he was able to prove this because he managed to put the proof in the margin of his diaphanous, so this is an elliptic curve, but there are many more if you take two different integers. non-zero integers, so two different positive or negative integers we call A and B.

Look at the equation y^2 = x * x - A * x - B. That is an elliptic curve if you take a as 1 and B like minus one and multiply that product and you get y^2 = , how can you? find all the solutions maybe you want to find all the solutions where testing to identify important properties of elliptic curves and this problem can help you with the first one because identifying the right type of properties can help you in this matter of finding solutions. There is a particular property of elliptic curves that plays a crucial role in the proof of Wild's pit theorem. and that's what we want to talk about now, this is the property of an elliptic curve called being modular and I would like to explain what that means when mathematicians have a question that may be too difficult to answer with a strategy, those are the ones that something like answer our question now that we are going to mark them there are seven in this image now you might wonder why I stopped at four for X and Y why I didn't continue well I leave it to you to check that if I continued which numbers are multiples of five would simply be repeated, the numbers themselves would change but this pattern of the seven numbers that are divisible by five would remain the same well, there is nothing special about five, you can do this for any number you would be interested in prime numbers like 2 3 5 7 and so on that have no divisors except one and they themselves take one of these prime numbers, call it P make the same type of table count the number of values ​​that are multiples of p and call that number n subp this table we made for five shows that n sub5 has the value seven well, you can calculate this for the other primes here is a table for some of them.

I have listed the primes up to 31 and then I have listed some larger prime numbers, if you start looking at this table you will notice some patterns. One thing you may notice is that often, but not always, the number in the top row is the same as the number in the bottom row, multiple too if you look. If you look at it for a while, you will see that after the first, after the number two, all the numbers in the bottom row are one less than a multiple of four. What do these patterns mean? Well, it turns out that these numbers have even more structure than you can guess that in 1814 the mathematician Carl Friedrich Gaus, considered one of the most eminent mathematicians of all time, found a formula, a recipe for calculating this number in sub P, the recipe says that these two are always a little bit apart, so keep them separate. and you realize that N Sub 2 is equal to two; otherwise you look at your Prime, if it is one less than a multiple of four, then the number n subp for that Prime is just the number itself and if it is one more than a multiple of four, there is a slightly more complicated formula, but if you look at this table here I promise you that I calculated the numbers in the table below using Gus's formula and that is much easier than writing all the values ​​of X up to a million and all the values ​​of Y up to a million and filling out the table what the G formula tells us is that this sequence of numbers N Sub 2 N sub3 N Sub five and so on has a very special structure and because of this special structure we say that this elliptic curve y^2 = XB - x is modular, well, you can do the same with any elliptic curve.

Remember that we have a whole family of elliptic curves. What that means is that if you start with your equation, you can construct a sequence like this in exactly the same way. way by making a table counting the number of divisibility that you find to obtain a sequence N Sub 2 N sub3 and so on and we define this elliptic curve to be modular we say that it has the modular property if the sequence that you obtain in this way has the special type of structure analogous to the structure that Gauss's formula gave for the elliptic curve y^2 = X CU minus X, so being modular is a special property of elliptic curves that is related to this sequence that is obtained from the curve elliptical for a sequence like this to have this modular property is actually very special, very few sequences if you were to write a sequence would have this property.

We know that the sequence coming from y^2 = - x has this property, but it was very surprising when in 1955 the Japanese mathematician Utaka Tanama suggested that perhaps 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 it should be that way. True, so the suggestion didn't spark many people's interest for a while. Taniyama died in 1958, but some time later, a colleague of Goro Shamura, who is now a professor at Princeton, researched this Samour and thought about what Taniyama had suggested he do.

This guess was more accurate and also made it more credible. He came up with some reasons why it might be expected to be true, so mathematicians call thisA conjecture, meaning it's a guess, is something you believe is true but don't know how to prove it. The conjecture over the years gained more and more support in the sense that many people believed it until recent years, certainly mathematicians believed it universally, but still no one had any idea how to prove it well. It is this conjecture that every elliptic curve is modular that plays a crucial role in proving the FMA lesson.

Our next speaker is Ken Ribbit and Ken is a professor at Berkeley across the bay. His specialties are number theory and arithmetic algebraic geometry, which is a difficult field for Ken Ribbit himself. Research has played a big role in the pieces that come together and are established from the OAS theorem, so he will talk about how all of these pieces fit together, including his piece, and tell us a little about what it was like to be there in June. 23 in Cambridge when the theorem was announced Ken Ribbit thank you very much I would like to take you from utaka taniyama in 1955 to June 23 of this year possibly a little further if one asked 20 years ago or even as recently as 1981 what was the conjecture of tama had to do with the last theorem of FMA the answer would have been nothing there is a man who is very important in history his name is Gard Fry he is a mathematician in Essen who really saw that there was a connection for the first time Fry was in Seattle the last week and was hoping to bring him here unfortunately he is back in Europe.

I don't even have a photo of him to show you what I do have to show you is a letter he wrote me in 1981 where he proposed coming to Berkeley for two months to talk about modular elliptic curves and modular curves in the last theorem of fma. He was very busy at that time. I thought maybe I would discourage him from coming, but in fact he was welcomed with open arms at Berkeley and we had many discussions about elliptic curves. and he wrote quite a few forums linking elliptic curves and FMA's last theorem and, frankly, none of us really saw what he was doing.

He had the connection between elliptic curves and the last theorem of FMA, but he didn't see how. relate the tama conjecture to this important statement of FMA now if we jump a little further we see that in 1985 the situation had changed drastically fry announced to the mathematical public in fact a very small group of mathematicians that indeed the tama conjecture had as a consequence the last fma theorem and this was the first time that someone had really managed to link the last fma theorem with the tools of modern mathematics, the success was unfortunately not complete because Fry gave his lecture at a small retreat at the volak work in the Jungle Black, there were about 20 people present, I was certainly among them and the vast majority of the 20 people realized at the end of the conference that there was a big mistake in what Fry was saying.

The error occurs in his manuscript, which was only 2 A2 pages, um, it's a little off. focus, but in fact that is how most mathematicians saw it, there was a very important idea in the manuscript, the link between the two, but it was not actually completed, many of the people who were present in the Black Forest, This was in January. 1985 quickly arrived in Paris and tried to deepen the link that Fry had first seen and there were a number of people who had ideas that made partial progress in the direction of actually establishing what Fry had hoped for. um, I was busy teaching calculus. in Paris and he hoped that people would get somewhere, but it really didn't seem clear how the result was going to turn out, but again in the summer of that year the situation had changed.

There was a mathematician in Paris named Jean Pier s who is certainly one of the greatest living mathematicians and won the Fields Medal, which is our version of the Nobel Prize. Writing from his mountain house in August during the holidays he realized that there was a precise method for linking Fair's Last Theorem and Tama's conjecture. Unfortunately, this introduced a new layer of complication because in his letter that he actually wrote to a young colleague in Paris, Jean Francois Mess, he explained that the link that Fryan initiated could not be directly established, but rather what would happen is that he would only two small conjectures could be proved and if these new s conjectures were established, then you would know that the tama conjecture implied fma's last theorem.

Now the news of such things spreads very quickly among professional mathematicians. This was the case because while Mler was receiving this letter, he was boarding a plane to California and came to a workshop we had at Humbal State University in Arcada where once again there were about 60 mathematicians gathered and this caused a huge line at the photocopier. coin-operated because everyone was very eager to receive news of this deeper bond. that Sarah had established now one thing I should point out is that this indicates, at least it seems illustrative to me of the way that mathematicians work during the year, we are busy teaching courses and we don't really have time to interact with each other. and it has been a pattern that we have week-long lectures, usually when the universities are on vacation, so that we can live in the dorms and enjoy dorm food and we usually have four or five lectures a day surrounded by walks around the forest and debates. between us and this is a very intense period for people who cannot see their colleagues during the academic year.

Email has changed that to some extent because you can now send news to everyone, but it's still very important to talk to people face to face. confront and see exactly what the other person has now done for about a year. Sarah's conjectures, which he called C1 and C2 in the letter, were not resolved and the person who could really prove them is me, yours truly. Pro the conjecture in 1986. And the way it happened is that at the time I was proving this conjecture there was an International Congress of Mathematicians, there's one every four years, the next one will be next summer in Zur and it happened to be in Berkeley , California, there I was sitting.

With Barry Maser, who is a professor at Harvard, we were having cappuccinos on the South Side and we were talking about these C1 and C2 conjectures and I told Barry that I had more or less proven them but I still didn't understand to what extent my techniques would work. and Barry was the person who said well, you're just being s because in fact what you're doing is testing these conjectures directly and as I licked the foam off my cup I realized that, in fact, he was He's totally right and like there were 2,000 mathematicians present at Berkeley I was able to confide in some of them that this was going to work and then the news spread and it was kind of embarrassing because there was a lot more interest in FMA than we realized at the time and people were really pressuring me to the details of the test that were still being washed out of 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 is sometimes called an Emissary because it extends and tries to convey mathematics outside of professional mathematicians and I gave a lecture once a week over a period of several months at the msri to specialists in my field and this was absolutely brilliant because as I tests were taken I received a lot of direct criticism from people who really knew all the ins and outs of the mathematical objects I was working with and at the end of that special year I had an airtight manuscript, at least I thought it was airtight, but I do. I want to point out that the process of publishing a proof is very long because there are referees who intervene and make changes that the author may or may not agree with, but a negotiation still takes place and a final manuscript is obtained to which It sends. the printer and then there's the concept of a delay in publication before it goes out on the local newsstand and actually there were four years between the time I realized I was going to try this and the time I was actually print, well that's the main sort of thing that happened before

1993

, but then the next chapter of the story begins, this is Andrew WS in a photograph I took of him quite a few years ago, maybe it's a photograph from 1980.

I met to Andrew for the first time in 1975. When he was a research student at Cambridge, his advisor John Coats is a very close friend of mine and was busy solving all the problems Coats posed to him. We certainly became friends and colleagues, especially as Andrew has spent most of his career at Cambridge. United States after his thesis he became a postdoc at Harvard and after his three years at Harvard he stayed as a permanent professor at Princeton in Princeton New Jersey I have some other photos to show you it was his way of making a joke but I think He was very present in his mind, he explained to British television on June 24 that he had been thinking about FMA's last theorem since he first encountered it as a statement as a child and when he learned that FMA and Tanama were connected in 1986.

He basically shut himself away, he actually worked in an attic for seven years trying to get a proof of the tama conjecture and this is really something unparalleled in the history of mathematics that I know of because generally people work closely with their colleagues, go to conferences and They talk and Wes thought this was so important that he actually had to work alone until he knew he could solve the next problem. I just want to tell you, of course, that Andrew announced that he had solved the sama conjecture, I guess I already know that and one thing I want to emphasize is that there is an enormous amount of modern mathematics involved in proving it.

You may have gotten the impression from preliminary accounts that he had somehow sat down and written 200 pages that were independent of all of them. developments you've heard about tonight, but in fact what he did was take everything he needed from the most powerful techniques in modern number theory, including many that hadn't really been devised when he started working on the problem. in 1986. I would just like to tell you who the personalities involved are so that you have an idea of ​​the global sweep. One hero is a professor at UCLA, but of course he's from Japan. He is from Saporo, where many of the ideas first linked modular shapes and elliptical curves. started Barry Maser that I mentioned before and then there's a topic called iawa theory in kichi iawa, who was a close colleague of shimura and the people whose work was used by Andrew WS, of course, Carl Rubin, who you just heard about and Ralph Greenberg, who is a professor at the University of Washington in Seattle and then there's something called Oiler Systems.

Oiler is the person who first saw FMA's last theorem for the exponent 3 and Oiler's name appears in many different places in number theory. There is a man named Victor Coy Vagin. who is from Moscow who found a new technique in the theory of elliptic curves and called it Oiler Systems because he thought it was very appropriate in the context in which he worked and it was really Wilds working for the first time with some ideas from Matas. Flock, who is another student of John Coats at Cambridge and who really adapted the greasing systems to the context that was needed.

My name also appears in the article because I demonstrate something that involves congruences between modular forms, these appear in the proof and then there is another aspect that involves modular forms that is actually based on the work of Robert Langin, a professor at the Institute for Advanced Study in Princeton , and Gerald Tunnel, a professor at Ruter University also in New Jersey. Well, what I'd like to do next for you is simply summarize the logic. the way the proof goes if you have at your disposal the tools that have emerged since 1985 if you want to prove fma's last theorem you do it by contradiction you assume it is false and that means that there really are integers a b and c with a the N plus b raised to N equal to a perfect nth power, of course, n is a number that is greater than 4 million.

You know when you start because of the computer calculations, I think you just use that is greater than five, so you start with this. counterexample and you do this important thing that Fry wrote, you make the elliptic curve y^2 = X and so on. This is just a variant of the elliptic curve that Carl Rubin told you about. Now it's a little more complicated and Gaus isn't. I'm not telling you that it satisfies the Tanana conjecture, but Andrew WS does, that's what we learned on June 23, that this elliptic curve satisfies the Tanana conjecture, on the other hand, if you believe what I told people in 1986, you will know that it is not like that, so it is impossible.

Situation, you have an elliptic curve that has contradictory properties and the only thing that makes this elliptic curve is the supposed solution to the last theorem of FMA. It's wrong? There was no solution and this is how the test ends. Now you can see this from another perspective. From the point of view, you can have the chain of ideas in terms of when they occurred. Tanama made his conjecture in 1955, of course,before that we could have put FMA at 1637, it is said, but there is a lot of math in between and it didn't fit in the margin of the slide, um, so we skip to 1955 and then we have Tanama at 198, excuse me, we have FR ideas at 1985 that weren't fully realized until 1986 and then there's a big jump of seven years before Andrew's conference in 1993, well.

In conclusion, I would like to say that this proof of Fair's Last Theorem is a tremendous triumph for modern mathematics, it is a tremendous triumph for number theory, of course, it is a wonderful personal triumph for Andrew Ws and I am 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, a mathematics professor at Princeton, and Ken Ribet told you a little bit about what it was like to be with Andrew and John tells me he's going to Tell us a little bit about what it was like not being there, that's because John Conway is one of Andrew Wild's closest friends and coworkers and they're actually working on a book together and John Conway was probably the first person that came out. from Cambridge to find out by email.

He is also the inventor of the Game of Life, which you may have heard of if you are a computer expert and after John's talk, there will be a 15 minute break, 10 minutes or so and then we will be back for a panel discussion , so here's John Conway with his personal story of Verma's Last Theorem. Well, you know, one of the first things I want to say is that I'm actually not Ed with this at all. I'll get to 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 Wilds the longest.

And I'm very, very delighted, of course, that it was successful, but I'm going to give you a very quick story. I'm going back to 3,600 years ago. Because you know it's really very interesting to me. I am very interested in mathematical history. This problem really has a history as old as this clay. The tablet actually contains essentially a formula, it's the formula from the book of Dantes that gives you integer solutions of x^2 + y^2 = z^2 and then you plug in various numbers and get particular solutions that were 3,600 years ago, something else which is dates from about the same time, um, it's another Babylonian clay tablet, this one you can read if you want, if you know, a kind of V-shaped symbol means one and a V-shaped symbol on its side means 10 In fact, I can read this, it means that there is one part and then 24 parts of the next degree down, which differs by a factor of 60 and so on, and if you calculate it, what this tells you is the length of the side of this particular thing.number 1.4142 ET which is very, very close to < tk2, which tells us that people are interested in this triangle 1 one < tk2 600 years ago.

By the way, these dates are not entirely exact, I mean, they could have been 361 years ago, huh, but it could easily be several hundred years, no one knows exactly when Dantas lived. He could have lived in the second, third or fourth century. Assuming you lived in the second, then it was about half the time 1800 years ago that he wrote the book of it, proving that that Babylonian formula gives all the solutions to the square problem. I'll put it very quickly that 392 years ago Pierre Defero was born and he is really the greatest number theorist after Diantus and a great number theorist of all time 356 years ago.

Generally speaking, we don't know exactly, well, the most basic edition of diantus a new translation of diantus appeared a few years earlier and FMA writes that famous marginal note. Now you know that you have heard the story of the last theorem of FMA explained very well by Lenol. Blum and some other stuff from people here, so I'm going to skip the ferma issue itself here. The last theorem, but you know, is just one of the many theorems that he announced both in marginal notes in his book that were later published. by his son and also in letters to other people and um these other theorems lasted a fairly respectable time 7 1949 Oiler approved the just mass theorem which tells you when a prime number is the sum of two squares the answer is if you leave the remainder when you divide it by four, if you leave the remainder minus one, it is not or three in FMA stated that every integer was a sum of three triangular numbers.

I'm not going to stop and tell you what that means or four squares. numbers or five pentagonal numbers and so on, the case of the four squares was first demonstrated by lrange in 1772, the three triangles were one of the things published by House in 1801, in fact found it a year or two before, Kosi, who has also been mentioned, demonstrated all five. pentagon theorem and all the others and the other theorems I'm not going to mention in detail, the last ones I think, apart from the last ones, were proven by yobi in the 1840s, so you see, they lasted 200 years, okay , the last one lasted 350 years, big deal, I'm jumping to about 10 years ago and these dates are just as inaccurate as the previous ones, by the way, maybe 12 fingers now at Princeton, but that's what got him his job at Prinston and Fields' medal proved Modell's conjecture.

I won't tell you what it is exactly, but it follows that for any particular purpose there are only a finite number of really different solutions to the fmma problem, so there can't be more than a finite number there. There probably aren't any, in fact we now know there aren't any, but the model's conjecture implies that there was only infinite money for a given eight years ago. Andrew told me that he started working, seriously he started working a lot. harder seven years ago, after Ken Ribbert demonstrated the theorum I had heard him talk about now, until six months ago, when Wes told two colleagues at Princeton that they know that I am an msri emissary from Princeton, really one of the things I have Tri a Point out some names from Princeton and Berkeley.

Well, Peter Sak told me that one day in January he showed up late at night and said, I have something to tell you, and then he made him sit down and tell him. him and then he worked with Nicholas Cats, the other person he told at the same time, uh, he actually announced a course of lectures that only the cats knew was going to prove F's last theorum at the end, so the audience quickly dwindled to one person, Cats. and they worked it out and there was a difficulty, maybe I should mention that there was a part that Andrew wasn't very sure about, he himself wasn't entirely familiar with the concept, so he got the cats to help him and there was a difficulty and 12 weeks ago in mid-May um he found what he calls the 10 minute argument um that solved that difficulty and I don't know if he calls it the 10 minute argument because he found it in 10 minutes but he told me It took only 10 minutes to explain it to Cat, where I had been working for several months at this very point before that.

This is the first time I come. Six weeks ago I heard about the test at a party in Princeton and then. I asked all my friends in Cambridge to tell me what happened at Wild's talks, well, you know, you heard it five weeks ago to this day, he announced proof that his lecture, his third lecture would start at 10 : and it would end at 11: it's 11:00 a.m. m. great, it's 6:00 a.m. m. on Pinston time and, in fact, I heard it shortly before 600, it must have ended early. I heard this at 5:53am. m. in Princeton, I couldn't sleep, so that was five weeks ago, four weeks ago Wes I came back to Princeton and we had a lovely party instead of tea at 300 p.m. in the mathematics department we drink champagne, it is the main difference.

Also, the president of the university, for some reason, came up to math and I was talking to him and I said, where's Andrew? It's about 5 minutes until tea time officially started and then Suddenly everyone started clapping and they still couldn't see him but then he walked in. It was a very nice and touching moment because he came in and he was holding a girl by the hand and he was holding his other girl in his arms and they were a little scared because of the cameras that were flashing and everything, but very, very nice moment, he said some words, well, now Pastor, a few weeks ago, the FMA, this FMA Fest was planned 20 hours ago, I arrived in San Francisco, well, more than an hour.

We have been working well for this first year now, I started about 10 minutes ago. I hope I've started talking and now I'm stopping we're going to take an intermission now please come back we have a panel discussion Start and if you have questions look for one of the cards in the lobby and bring it to the side of the stage there and we'll include those questions in the discussion panel. Thanks, we're going to have some kind of panel. Discussion now. Go through some of the implications and answer some questions. He has met all the panelists so far, except Lee.

Demart. Lee is a journalist, science writer and book reviewer. He has been a reporter for the New York Times. Wall Street Journal. LA Times. San Francisco Examiner as well and he's also a lawyer and you've met the other speakers so, um, leave it, maybe we should start with you, since you haven't given a presentation outside of mathematics, what's the meaning of this? Why should I do it? Non-mathematicians care about this kind of work, how would you explain it well to people? If your question is what is this for?, as many people have asked me in recent weeks.

My only response would be, wow, that's like asking. What is the use of the cinema chapel? What good is Beethoven's Ninth? These are great achievements of the human mind that we can all delight in and enjoy for their sheer beauty. Think about what we've heard here tonight. This is a problem. whose origins date back to ancient times, many thousands of years ago, it is a problem that many people have thought a lot about the specific problem with 350 years and now we know that it is true, in addition, Fasl's theorem itself has the property that it is accessible, unlike much of what mathematicians do, speak and think, the Reman hypothesis, which is still pending, would be very difficult, I understand it, I do not understand it, and it would be very difficult, I understand, to explain to non- Mathematicians what is it, but Fasl's theorem. can be explained and we have heard him explain tonight at least what the statement is, so this is an opportunity for non-members to understand, to glimpse the great Beauty, the sheer beauty of mathematics and what mathematicians do and as.

They think about it and this great intellectual structure that has been developing since the time of the Greeks and continues to develop today. Is there a cure for cancer? Obviously not, but there are many more things. in the firmament of human endeavor that cures for cancer and this and this is one of the big ones, do mathematicians basically agree that that's the right way to tell the world why you're excited? thing you know you don't understand something until you've proven it you know it and I feel this great desire to understand something and surely that's what you know something that we can appreciate in itself we just want to understand we want to know what's going on and now some people will be in the position to understand why this particular thing is not right.

This question was asked by many people and I this is one of many versions of it, but considering it took three months to find the solution. bug and some other tests and many fake tests have been advanced Lenor, why is there confidence that this test will actually hold up? How do you know that Veras' theorem has been proven? It's really interesting because you actually hardly hear any of that at all and I think one of the things and Ken can probably talk about this even more is that this is being demonstrated within the context of a 20th century show.

I mean, there's a lot of structural math there, it's almost true for structural reasons. The ideas don't come at all from something out of place, but they come through a very direct pattern and there is program mathematics, but on the other hand there is a lot of vested interest in it in the sense that because there are so many people involved In many places we don't have as many critical eyes and I think it's something to be careful about, but I think it's very slight. I think there's a sense that this is really solid. Ken, yes, I think the level of confidence in it is just growing day by day because some people are really thinking about the main

issues

involved and what they see is proof that the main ideas are very clear and if there is something wrong, there may be some problem to really justify it. completely one of the steps, but then we feel like we can go in and we can really shore that up because things structurally just make sense in broader terms.

There is a serious problem in recent years, which has been in science in general, which has been described as a press conference publication in which scientists announce their results, or supposed results about one thing or another, literally in a press conference and not in a reference newspaper, etc. The most egregious recent example is cold fusion. announcement from uh, I guess, four years ago, which turned out to be complete nonsense, uh, the press, um, should be criticized in this sense becausemany, many newspapers and reporters etc. simply cannot tell the difference and have no means to tell the difference. difference between what is right and what is wrong someone makes an announcement that you don't want the opposition to take over the other newspapers and so on and uh and it should be underlined uh it is not my intention in any way to equate the wild test of the form's last theorem with cold fusion, but it should be stressed that every line of this proof has not yet been proven to be perfectly true, and of course there could be an error in the sense that there is a very definite difference between this and some . other things you know, um, the basic thing is you ask: could you try it this way?

Would arguments of this type arise? And occasionally there is a calculation involved. Could you calculate that? What if you did it and got a certain response. that's right, I mean, this is what Ken was saying before, to some extent, mathematicians in a position to know can see that a program like this could be successful, and so Andrew Wild is a conservative, cautious, very careful person. that goes into the equation everything does everything we are human I mean, even Andre W is human, even I am human and you know we can make mistakes and he knows he has made some, but you can see that it could be done like this you know, with a recent failed attempt to prove the theorem, uh, Gert Fing is a great expert, he said that if it could have been done that way, he would have done it, he knew it immediately, this time the experts from the beginning believe it, why, well . because they see that it's part of an intellectual program that could be successful and Wild 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 is something that can't be patched, maybe there is a really serious hole in this test, but we have good reasons here to believe that this It's pretty solid evidence. Now we also get this question several times. Did FMA really have a test? And what is general? There is something? that maybe has been missed all these years or was uh uh true just making a casual comment what is the opinion of the professionals on this before actually I think most people feel that they had no proof, but on the other hand On the one hand On the other hand, John really believes otherwise.

You were trying to argue differently. I've just been opening my mouth a lot, so I was hoping to get a chance to rest for a while, but, um, uh, but since I'm on the underdog side here. um, yeah, I'm actually kind of a 5050, the general consensus among those in the know seems to be that Ferma probably fooled himself, um, but there are some difficulties with this view, uh, first of all, I wasn't writing for nobody. Otherwise, he was a note to himself, so there is no doubt that his reputation was at stake at the time he wrote that marginal note.

It's a memo to himself. It's possible that he just made a sign error and you know, so he just possibly wrote that note just a minute after he found the proof and there was a sign error. I think that is unlikely because he comes back several times and presented proof for n. same 4 later you have to answer this question if you think he didn't have proof, what did he think he had? what was the test that fooled FMA that ferma fooled himself with that it certainly wasn't these later tests that exist all about the place that uses non-unique factorization or sorry, it uses unique factorization when it doesn't exist, it's true that the fields of extended numbers those ideas were not available for FMA it's a very interesting question um and I don't know the answer um we I shouldn't take the fact that I don't think we should take the fact that mathematicians haven't found one so far as sort of of evidence that there were none available to you.

I would like to amplify that and that the fact that we know that Andrew WS has proof and that this would not have been accessible to the FMA does not mean that the FMA has not proven it. If he had proof, it was certainly something else. I wouldn't be surprised in any way. to talk on this really, there is still an open question here for anyone looking for some job what was the FMA test was not the Andrew Wilds test what is the simplest test even if it was a conscious test what would be a very Interesting test, that would be interesting.

There's a lifetime's work for someone in the audience. Remember that some of the other FMA theorems lasted 200 years and some of them have very simple proofs. Interesting. Oh, you mean some 350 year old ones. You know, it's less than the factor of two largest may seem like a lot to you guys, but hey, squaring a circle took 2000 years, yeah, yeah, this question was directed at Ken Ribbit, but maybe everyone can have a question about what is it, if, if, if, the fma theorem itself is. It's not very interesting what the meaning of Tanama's conjecture is and the kind of thing we know that we people say that Wild's work fits into the broader flow of mathematics, we talk about that broader flow, if FMA were not an implication, it would still be an exciting job.

If he had done it right, it would certainly be exciting for professional mathematicians, but I think we honestly wouldn't be here tonight. There are many people who are fascinated by the Ace Theorem and I think that among mathematicians some are and some are not. The Tanam conjecture is fascinating to me because it represents a connection between two different types of mathematics and two different types of objects, that is, you have the elliptic curves given by simple algebraic equations and then you have what Carl Ruin called recipes for finding numbers of solutions to these equations modulo 5. and other prime numbers and these recipes fit into the theory of modular forms someone talked to me during 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 amazing and If we understood it better, I think we would know a lot more than we know today.

Here's a question in a different direction. Most of tonight's speakers have been men. What is the outlook for women in mathematics and what is the situation? The audience seems to be a little more heterogeneous than the speakers. That? What is mathematics like as a field? May, at the risk of throwing the question at the wrong person. Lenor, you want us to at least get off to a good start. I'm probably the right person. There are many, many

issues

here, I mean, is mathematics a good field for women? It's a particular theme and I would say it's absolutely brilliant.

I mean, I hope people here are starting to see the excitement of working in math is just wonderful, what's up? The situation of women in mathematics I mentioned to Sophie Germaine, in her experiences, things have changed since the French Revolution, the American dependency and there has been an interesting and powerful history of women in mathematics. Women could not become graduate students in mathematics at Princeton University until 1968 is very, very recent, but the situation has changed a lot in this century, at the beginning of the century and in the United States, before World War II , the situation for women in the United States was quite good and that lasted until during the war effort and women were very involved in mathematical and technical fields, historically there was some kind of effort to return women home and I think that affected women in many fields, including mathematics, when the number of women in mathematics who obtained PhDs plummeted to 6. % now again we are around 20 25%, so there are a very significant number of young women who are dedicated to mathematics today.

Over the last 20 years, there have been many programs to encourage women in mathematics that I have been involved in for quite some time. few, um, there's the association for women in mathematics, which is a professional organization here in the Bay Area, there's the Mathematical Sciences Network, there's summer programs, there's the Mills College Summer Institute for undergraduate women at MSRI , we've been this year as part of our Emissary, uh, effort to get much more involved in social issues, we've been paying a lot of attention to the participation of women and minorities in mathematics. People often say that they know that mathematicians do great work when they are young, but when they reach their 30s or 40s.

We are over the hill. Does anyone agree or disagree? We are over the hill. Well, mention well how old Andrew WS 40 Epsilon is. That's important for fields now, so it's a counterexample to depletion. Oh, I think there is a. many counterexamples, yes, but it is true that in mathematics, as well as in music and chess, there are child prodigies and I think that those are the only fields in which there are real child prodigies, you couldn't imagine it. I mean, Mozart wrote great music when he was eight, you couldn't imagine an eight year old writing a great novel, not that the 8 year old couldn't have mastered the techniques of written position and so on, but he did. would do.

Not having the life experience to write a great novel is less true now than it used to be, partly because to do really good work on the cutting edge of mathematics now you only have to know a lot and it takes a long time to learn it. so the age at which mathematicians are productive is increasing, so could there ever be another ramanujan, a person who comes from an uneducated background but is so enormously talented that he can actually participate or has to pursue a career standard academic? To actually become a mathematician today, I think it's still good.

I mean, I think you know there are a lot of different people in the world and mathematicians are very different from each other. I don't think there is a standard model for mathematicians. Here's the question: are we in the golden age of mathematics today? Yes, the only thing left to do is thank our speakers and panelists for tonight counting sheep when you're trying to sleep, being fair when there's something to do. share being neat when you fold a sheet that's math when a ball bounces off a wall when you cook with a recipe book when you know how much money you owe that's math how much gold can you keep in an elephant's ear when it's noon on the moon, then , what time is Here?

If you could count for a year, would you get to infinity or somewhere close? When you choose how much postage to use. When you know what the probability is of snow. When you bet and finish. in debt, oh try as you might, you can't get away from the math and as Smiles gently does his thing and voila, QED, we agree and we all shout hurray as he confirms what Fair noted in that margin that could have used a touch wider. your feet keep the rhythm of a song while you sing at length you harmonize with the rest of the guys, yes, try as hard as you can, you just can't get away from math, all we have to do is thank our speakers and panelists