KR: Well, the usual way that people, um, discuss
it is to talk about perfect squares first. So, if you have studied the Pythagorean
Theorem there's a formula that the square of the hypotenuse is the sum of the two squares of the sides of a right triangle. So you might
ask if the, err, the sides of a right triangle are whole
numbers might it happen that the hypotenuse is also a whole number.
They're called Pythagorean triples and the Greeks figured out all of the Pythagorean Triples. Okay,
so what happened next is I think it must have been very natural
to look at the equation A cubed plus B cubed equals C cubed and A to the fourth plus B fourth equals C to the fourth. And so on. So, this is now in the realm of
algebra and you can plug in values for 'A' and 'B'
you'll see that, err, these equations, infinite number of
equations, seem not to have solutions. Well, so the
story is that Fermat was reading the text of Diophantus, who was the Greek mathematician best known
for this type of problem. And in the margin of his copy of Diophantus, he wrote some marginal note
saying that I can prove it for higher exponents
bigger than two there are no positive integer solutions and then famously he said but my solution is too long to write in this margin. And in his copy of Diophantus which
was inherited by Fermat's son Samuel de Fermat there are many, many marginal notes and
Samuel published a revised version, annotated version of Diophatus's Book in which all his father's writings
are reproduced. And the reason that Fermat's
Last Theorem was called Fermat's Last Theorem, is that all the previously on examined
marginal notes proved to be correct for they were, ermm prove to be correct or incorrect mostly
correct by mathematicians after the
seventeenth-century and there was this only one assertion that remained it was the last
one was hit list. And you know again, I'm not an expert in the history of mathematics but I think most
people are convinced that when Fermat was
a mature mathematician he realized he did have no proof of this thing that he wrote in the marginal
note and in fact later in his life he devoted a lot of
attention to special cases of that type of equation. It's not just the cubes but it's the cubes
the fourth powers and the fifth powers and so on. And Fermat himself later in his life took care the fourth powers. Well, the cubes
certainly must've bothered him because after all he knew
about squares and he knew about fourth powers and cubes, you know, right in the middle. BH: The theory is that
there is no solution. There's no whole number solution. KR: That's right. BH: Would it have been or wouldn't it have been more exciting or more attractive to a mathematician to prove him wrong and to find these three
numbers that you could plug in that would do it? Would that have been more triumphant? KR: Well, it would have been- would have been very exciting and would
have made lots of news. And then the proof, so to speak, would
have been simply on a single counter-example that would
have fit, you know, very neatly in a newspaper
column, you known, there would have been some story, you know. 'Mathematician in Australia finds the
first counter-example to Fermat's Last Theorem'. Certainly was big news either
way. Ermm, the main thing about Fermat's Last
Theorem is that over the centuries it gave rise to a tremendous amount of new, and fruitful mathematics. That people
really develop new techniques to try to solve that equation that proved to be, you know, infinitely versatile. BH: So a bit like going to
the moon and let[ting] them develop other lots of other technologies along the way. This was for going to the moon. KR: That's a-that's a fine
analogy. I mean, you, when you formulate a problem in
mathematics no one knows whether this problem will really be
helpful for mathematics as a general enterprise. And, you know, for whatever reason Fermat's equation has led to tremendous amount of- of new mathematics KR: So, when I was a graduate student. We're
talking, you know, roughly forty years ago or forty-five years ago. Fermat's Last Theorem
was kinda some curiosity. Everybody knew about
the problem but no one really had any approach to it and there was no kinda fundamental reason why the problem
was going to have a positive solution. And when I was a student there was a
tremendous interest in what are called elliptic curves and it was realized around nineteen sixty-eight, nineteen sixty-nine, nineteen seventy by André Weil and Goro Shimura in
Princeton who built on work of Yutaka Taniyama who was his buddy in Tokyo in the nineteen-fifties.
Taniyama had passed away and there was this
gradual understanding that elliptic curves really were related modular forms and again that was some
completely are apparently inaccessible outstanding
conjecture analysts thought to be incredibly hard
conjecture. And then in the early nineteen-eighties a
mathematician came around, and I say that literally because you know he was
sitting here in this office his name is Gerhard Frey. At the time he
was at the University of Saarland in Saarbrücken on and he came- BH: Is that a famous one? I don't know that one. KR: Well, um, I didn't know it either and came around
and he had the scheme for relating Fermat's Last Theorem and the modularity conjecture. Somehow he realized for the first time that, um, if- if he had a solution to Fermat's equation this would give him an elliptic curve
which apparently could never be modular. I was a little
skeptical because I knew that some of the things
that he said that he needed in his argument, um, were
very hard. Well, for me a really important thing was in nineteen eighty-five Jean-Pierre Serre, who was one of my mathematical heroes in Paris
sat down and kind of tried to distill everything that had been said
about the Prey's idea and he came out with a very clear letter and this letter basically said if
you can prove this tiny little result which he called epsilon because the number in calculus, epsilon is a small number. He
said if you can prove this tiny little epsilon then you can really show that on the
modularity conjecture, which he called Weil at the time, implies Fermat's Last Theorem. So, epsilon became, for me, a holy grail because not only was it something fairly well
encapsulated and very clear, and this was the kind of thing that
I'd thought about before without success but nonetheless while
thinking about yet another problem, it occurred to me that I really
had some new information that I could possibly bring to bear on this epsilon. When I first did was say to myself: "What's
the simplest possible case where I can think about this? What's the simplest
possible prediction?" I tried to settle that. I said "Well, you
know, if you have to start somewhere. can I-" Well, I know about this simplest
case so I have this thing in my mind and I
thought about it, off and on, during the academic year- nineteen eighty-five and eighty-six. And during this year I was teaching calculus and I was busy with things, there are, you know, colleagues come in and they want you to be
on committees. And finally, um, my final grades were submitted in May nineteen eighty-six and I started thinking about this problem again, first when I was on the east
coast for a while, going to Europe on and I found myself
in Paris. And finally I found myself at an empty desk and on the Max Planck
Institute in Bonn where I can just sit and think and I
thought about this yet again - BH: Is that what you do? Do you just sit at a desk with your head and your hands, and think, think, think, I mean - what if I was a fly on the
wall when you were trying to crack this problem. What would I have seen? What was it looked like? Well, it's probably not have probably had
a pad of paper in front of me and a pencil or a pen we didn't really have computers
quite yet although they were starting to come, and when you visit some institute you
don't have all your stuff with you don't have lots of books you don't have all your
paraphernalia. So I was really just kind of sitting there like that, and I started writing, you know.
And I- If you look in my office you'll find
lots of old pads where I sit and write, and, you know, a page might start what do we
know what are we trying to do summarize the situation so far. "Let's-
let's try this", and to my astonishment, I was finally able to- to cracked that problem - the simplest case of that epsilon. BH: It's like inspiration
comes at preordained times.
Like you sit down and say, okay for the next two hours before I make my friend Bob a hot dog, I'm gonna have an inspiration and trying to solve the problem, and that's when it happens. It doesn't happen on time, so you're lying on bed-
Oh it can, I mean, if you
if you think about something very intensively then you can just carry
around with you and you know lots of my colleagues say
that while they're swimming had some insight or you know lots of
people say in the middle of the night they jumped up and they said my god you
know why don't I try a minus sign instead of instead of a plus sign and this- this can indeed happen,
you know. BH: Not for you though. You are an at-the-desk kind of guy? I-I think in this particular situation I
was at the desk I knew for the first time that I could do something that I hadn't done before that no one had done before. This is a
special case is this so-called epsilon, And I kinda wrote and rewrote it just to make sure I was really right. And finally, using a very first
Macintosh that I had access to I typed out a letter to Barry Mazur at Harvard, and I sent him this letter explaining what I had done. And then I spent weeks and weeks trying to do the general case. I couldn't see what to do and then fortunately, there was an international congress of
mathematicians. They occur once every four years on the campus of UC Berkeley. So I was right here.
I came back in August for this International Congress and Barry Mazur was there. And on I spoke to him, and I said Barry I sent you this letter
explaining how I did a special case, and I really wonder, you know, what I
have to do now in the general case. And he looked at me completely quizzically.
He said: "Well, you've already done the general case." and basically, he- when he read my letter to seemed obvious to him that there was this extra little thing you carry alone as a supplemental
object. And this extra thing which is do the bookkeeping. And, so we sat down at this Caffe Strada
which is at the corner of the Berkeley campus, and we had a Cappuccino and he kinda said well you just carry this and run with you and I was astonished you know it's like this amazing moment were all the sudden I realized that in principle at least there was
no obstacle to proving this epsilon conjecture. I mean, it like, you
know, my god, you know, it's some special thing, you know, an angel comes down
and the light shines from the heavens or something. So this was really quite a thing and on since or my colleagues were walking around the campus in this
big international congress I kind of told, you know, a number of them
that I have done this thing. You know of course
in beginning I was cautious I said well I think I've done this thing, and- but
people kept running up to me. They said was it really
true you've done epsilon? I lived in a world and think I still do
were my colleagues are very generous with
attributions and praise and there isn't too much of this kinda race to be the
first to publish and people basically- I had little fear that
someone would try to scoop me on the other hand what was a little
worrisome was that I hadn't written down all the
details, you know, so I kinda written down very closely the special case. But I hadn't kind of worried about writing
thing more generally so what happened was that during the
year following that, nineteen 86-87 there was a special program on algebraic
geometry and arithmetic geometry at the Mathematical Sciences Research
Institute in Berkeley, the MSRI. So I was up there with my callings
and I gave lectures every week I gave
another lecture I was kinda giving like a min course on my proof and when I wasn't lecturing I was furiously
typing into the computer to try to expand and
revise my manuscript in kinda more and more
details got put in. There were certainly people in the
audience who were appropriately skeptical. They asked me hard questions and I
didn't always know the answer to our questions you know like "How do we really know that this
thing is in such-and-such way", and I would say well
you know this was proved in the 1960s by Grothendieck but it isn't quite written down in the place you
expect then I have find the argument. Well epsilon is small. It was clearly a misnomer. And let me just get up and pull off a reprint. Here's my article. It was published
in 1990 which really makes are a lot of time between 1986 and 1990. BH: How many pages? It's like a book. Well it's a more than a pamphlet. If you
open up this paper on the very first or second page there's Fermat's equation and an elliptic curve but then they disappear. If you look at
the actual article it's more about the
technicalities of the subject. It was a big deal in mathematics. It got write ups in the press,
including Science magazine if I'm not mistaken. You know, whenever there's a serious
announcement, of a major theorem that gets a lot of
attention. And people stop what they're doing to try to
see what the basis of proof is. BH: who called this Ribet Theorem? you know obviously it to make that name
that is above all other whenever addition both well the yeah that's right so you certainly there
there's a general aversion to naming things after yourself
very often people give lectures and I'll say such-and-such
the room and princes the right name the person who on proves it and if it's a fair
amount they prove themselves in typically right there the first
initial their last name or something like that for all say they
don't on they have no attribution listeners
understand all this is the fear that the persons is providing in this lecture. - You've done
this thing which is important if you have a bridge
between two things which no other which also Granja. - Well that's
right so the um what what I when I showed was that
this modularity conjecture which was thought to be was
was felt to be true but um wasn't expected to be provable I show that that thing actually implies
there was less there I think the main takeaway from my
theorem was that since the very conjecture short on the OMA was likely to be true
that now you should think that for miles last year it was likely to be true for example it probably wouldn't have
been a really good idea to spend a lot of computing
power looking for a counterexample farrell's less there well as anyone else
who's in the room well I was the whole thing so any Wells
you know coursing came around a few times I guess but
basically he was holed up in his attic and I didn't know that and I think most
people didn't know that so he was on I mean it's his I'm it's a statement that he was
certainly intrigued by Fermat's last theorem as a child and then as soon as he heard
about this Ridge he decided to try to prove the
modularity theorem because of its relation to the Fermat's last theorem. so was it is a sort of the situation that Andrew Wiles wanted to prove Fermat's last theorem by any means possible he given him the means that's what he says I mean if this is
surprising to me because modularity theorem by itself is a
tremendous price and you know it seems kind of strange
that you needed the motivation to Fermats to go and
try to prove it if you thought that there was some way side well anyway that's that's that's
that's what he says yet we were all together meaning kinda
lots of people in the subject in a conference in Cambridge England in June 1993 and usually go to a
conference and you see the list speakers and you know you speaker has an hour and the thing and then set up so that
Andrew Wiles had three hours I'm so here's the organizers he said I
have something I want to discuss it will take me three hours and I like those
slots I said okay from and then in the first lecture you could see he
was proving He was trying to prove the modularity
of some class elliptic curves but the class elliptic curves at least in
the first lecture had no intersection with the class that you
needed for Fermat's last theorem within the second lecture got a lot
closer and it became more and more parents have
people at the conference third lecture he was going to be able to
do enough to get fair was last year the crowd was
growing so love the mathematicians you know brought their colleagues who
happen to be in Cambridge in you know some other family members and I think there was a little doubt that he
would announce approval for was left there I'm so I can i sat in the front row center and
brought my camera and snap the picture of him I wasn't another moment when you know
those cuts time stop and when I first realized what was going
to happen .. and then you know when it actually did
happen there was this some frends of activity I mean the director love of the Isaac Newton Institute where these lectures were held on had ordered a case a Napa Valley brute
champagne for the end the lecture is after the
lecture we came out into the you know common area only is all the
champagne flutes were produced in his bottle started popping you know this is kinda really an amazing
thing between that time when you- originally published a paper and this
lecture it was a cold or write you a nasty lol nah to work not a word you know it i no idea what he
was doing on you he whole who absolutely he was so he was a a friend and a
colleague I first met him when he was a graduate student in Cambridge and in fact when I first came to
berkeley the first year that I was here he was one of the first people that I
invited came to spend a week and gave a lecture to on do you that delight he kept a secret
that make a surprise I to this day you battle by I'm using a
little collegiate you're telling me when you thought you
open your running around telling everyone NEC said well it's very unusual behavior and the you know you can say in
retrospect that it was really um justified because he was afraid that
people would kind of jump all over it because %uh the important so the modularity conjecture and fair I was
last there on button he did not on tell me what he was doing he did from confide in to this calling to princeton
as he was giving is graduate course where he was working
out the ideas that on were instrumental in this paper
I think it was only a few days later that I got a copy of this manuscript and the manuscript was very thick nose like 300 pages from single space you know typeset by test mathematics typesetting program and on we were supposed to be the referee is on looking at different parts of the paper
and I was assigned to a par that be turn out to be problem-free arm and I'm the cats look at a park but turn out to have problems a problem was
there shattering for you a do you like that doesn't make
you think this is good I like the reader a lot about it really
deflating up to I have champagne mean finding the problem well from want more of the aspects of the
whole thing is that I had gone out pretty for a limb from telling everybody that Andrew Wiles
improve this thing on in the way that started was that when the New York Times first heard
about his announcement may ask to speak to him and returns me and said and you know you speak to Gina Kolata so
I went and on went upstairs and spoke to her for a very long time
explaining the whole thing and then I'm somehow the media kept
calling me because it was understood from that
somehow I was is designated spokesperson so I nice spent the entire summer I'm lecturing about his theorem and I'm fielding phone calls from I'm the
press hello for was announcement cast oh I think it was like two weeks what
happens in mathematics is if if you have a proof that seems to really be on seems to have a lot or from internal logic to it and there's something that's wrong
with the proof your immediate reaction is to say well you know this is just some technical
problem that I need to fix and I think that are Andrew Wiles when
he found out about the gap arm expected
that you would be able to repair insured or so it really took a while before there
was some realization on his part that this was a
serious problem couldn't be fixed on immediately just by
their tools improper if you love the person
who was taking all the calls from the press whatever reason did that mean you will
you are contacting and while get updates on the game
because my i don't know what to tell them what we call him but those you want to say it
was a he gone right not especially Inc he kind
of went dark he was working on on mathematics and I wasn't wasn't getting a lot of
information out of him at that time you know it was all poison about right i
mean I'm wasn't clear whether or not this
will get repaired or not and then you know wonderfully it turned
out that there was this so repair to the proof from release in completely new insight
actually by Andrew Wiles and Richard Taylor who work together over the summer and
this will be the summer 1994 and they had a or preprint out in in the fall 1994 really explain their
new method in which is now call Taylor Wiles method and that really I'm circumvented a lot or the
more complicated arguments in the original preprint and the two articles were
published together on the following summer which is a
summer 1995 in the annals of mathematics it was a relief but it was it was not
our kinda joyous relief you know and Richard
Taylor came to me one day again MSRI where I happen to be in fall far 1994 any sand we have some good news for you and then on the next day and make up in this
paper so that was kinda very good and you know I don't remember there
being a huge celebration on the moment the releases that reprint been just
kinda gradually build people understand that the proof is complete and then there was a very large
instructional conference at Boston University in summer 1995 different people got up and explain
the different elements approve well it was a fabulous
pieces mathematics arm as impressed arm with by what what I did and presses I am by my
my own were coming I'm even much more impressed by what
they did on they had some really knew inside and this inside has I'm percolated through our whole subject for the last twenty
years arm one way to say what they did is they
showed that somehow modularity is contagious in the sense that if you have something
that hope has a little pieces that the modular you can parlay that into modularity over
forty first evolved Wiles was obviously intensely driven to do this he
was really motivated and he said you know
I'm gonna do this one way or another and me brought in all sorts a different techniques that hadn't been
related on to the the problem before on what some what's amazing for example
there's a whole subject an algebraic geometry call deformation theory that was going on when I was a graduate
student that I didn't learn I thought that was really far removed
from anything involving elliptic curves and on in the years before wells announcement
Harry Mazer had developed this new technique of
deformation theory in gallery presentations and it turned out to be
exactly what Wiles needed the language in the
techniques the median for his discussion on and so you know something that seems to be far
away technically turn out to be crucial well I think I'm you know just
informally everyone refer to it as the Fermat's last theorem but in fact when you when you look at
the article just like my article was something internal to on series modular forms and get our
presentations and Wiles article in my article share the trait that for was a question covers
exactly once in the article in the beginning and then you get down to the real stuff
on very happy with the recognition that I've gotten and I very proud of what I did and be I when students come and my classes they
may know about my role for Fermat's last theorem. A lot of them have seen the video that was broadcast on over and BBC and a I think you know if you do
something very very very very famous and you have an encore problem where people saying you know what great
thing you're going to do next and I try to be on more or less a normal
guy and not you know walk around with angels on
your shoulder britt to rediscover what FAMAS proof
might it be is that they have widely-helds that he
had done it already disclaim lot was he telling the truth ? I think by the
time we get to 20th century it's quite clear that this is an incredibly complex problem.
A random question that dawned on me during the video: If Wiles had proven the Taniyama-Shimura conjecture and then Ribet subsequently showed it implied Fermat's last theorem (i.e the opposite of what happened in real life) would we say that Ribet proved FLT?
At the end, where he says modularity is contagious, what is modularity?
I'm not a mathematician, and don't know exactly what to search for to find a basic intro. I have a master's in engineering and can "read math" but would like to only have one or two things to read.
Ken Ribet looks like Al Michaels.
My math prof!
Just finished Simon Singhs "Fermat's last theorem" what a coincidence. So that's a really good timing, Brady :-)