>>: So for the last talk of the afternoon, we'll have Bryan Birch. First I wanted to
say a little bit about my own interaction with Oliver Atkin, which certainly wasn't
as extensive as Bryan's or Winnie's or Dan's.
But I first met Oliver in 1974, probably exactly the same time Winnie did when I
was a graduate student at Harvard, when he visited there. But I really that much
interaction with him until about 15 years later in the summer of think of 1989 I
came into my possession by a means I really don't remember a xerox of many
pages of computer output that Oliver had produced along with some scribbled
and notations along with some of his ideas written in rather terse but suggestive
style.
But knowing, you know, how much insight he had, I tried to pay attention to it
and, in fact, I spent probably a week poring over it until I understood his ideas,
and then I had some ideas. And I found his e-mail, wrote him an e-mail, this was
probably July or August. And I didn't hear from him for about a week. And then I
got an e-mail saying well, you've reached me during the summer and I have a
rule that I'm going to take a sabbatical from mathematics during the summer.
However what you've written interested me so much that I'll break my own rules.
[laughter].
So I get in retrospect I should feel very flattered. So he and I then spent the next,
you know, month or two, you know, exchange e-mails with some of my ideas.
And I had actually come up with pretty much the same ideas -- it seems that
Noam Elkies had also been in possession of this manuscript but wasn't in
communication with Atkin. And Noam basically came to the same conclusions I
did, and just as I was finishing writing my things up I received Noam's manuscript
who had actually done that in a little more. So at that point I just put the
manuscript back in the drawer.
But my interaction with Oliver was actually quite, quite interesting and pleasant.
And then four years later, when I started where I work now, Oliver spent the
summer with us in our summer program and, well, I can't talk about what we
spoke about, what we worked on there, but it was very interesting.
And in any case, here's Bryan Birch to speak about -- more about Oliver.
>> Bryan Birch: Yeah. Well, first I should thank you the [inaudible][applause] for
inviting me. And then a matter of white board technique is that -- from the back,
is that about the right size? Okay. Yes. And I want -- this will be much more
biographical talk than the mathematical lecture. So I won't really be using the
blackboard very much seriously.
And as a general guide, if I write something on the middle of the board, it's an aid
to people who can't hear what I'm saying otherwise. And when I write it at one
end or the other, it's probably something I want to keep. So the middle of the
board will probably look a horrible mess and will get rubbed out fairly frequently.
Okay. Now, which bits of Oliver's work am I going to talk about? There's rather
a lot of it. So I'll have to keep to the earlier parts I think. I think his life for
purposes of this talk it can be divided into about six sections. My writing is fairly
bad. 1925-2008.
Well, the first 20 years, 1925 to 1945 called education. The next few years, '45
to '61, early research. '61 to '70, Atlas. '70 to '72 I think I'd better call prices.
People may know why I say that. '72 to '85, renaissance. After the crisis.
And '85 to 2008, there exists elliptic curve cryptography which means in this
context that in particular Rene and Victor have said quite a lot about Oliver's work
in that period, and certainly they know more about it than I do. So I'll probably
stop there.
Now, in more detail I've -- I'm more exactly on net as having said quite a lot about
that period. And you suspect quite a lot of people here have read it because I
think of everything I've ever written is -- it appears to be the thing which has been
read most. And here Francois and Dan have said a certain amount, so I may not
say all that much there. So I'll concentrate on this part, which has the advantage
and I'm probably the only one in the room who knows all that much about Oliver's
early years. In 1961 he wasn't -- that's not all that early. Okay.
His education, his education until he was about 20 years old was really quite
extraordinary. A very, very hot house upbringing for the young mathematician.
He went to boarding school age nine. Then when he was 11, he got a
scholarship to Winchester and he was in Winchester age 12. That would be
1937 to 1942. I think.
Now, I should explain that Winchester is the oldest school in Britain and probably
the most academic. Really by most standards in all this time really quite odd the
scholars were -- lived apart from the other students. They lived together in very
beautiful medieval building called college. It was very beautiful of the it was
probably Tudor. It was extremely cold in winter. And they all lived together in the
strong sense there was no privacy.
They worked in the same room during the day and they -- it was like a dormitory
upstairs at night.
Oliver was pretty bright. Possibly even bright by the standards of Winchester.
However, he wasn't nearly as bright as one or two years ahead. There was a
little group of mathematicians including Freeman Dyson and three others who
became fellows of the Royal Society. That's to say accreditations. That was a
strong year.
Anyway, Oliver probably enjoyed this very rather extraordinary environment. As
you know, there's a war on. So university admissions were sort of odd and I
think in 1942 to '44 he went to Cambridge age 17 to 19 and Cambridge at the
time was a little odd because all the young lecturers were being conscripted into
the army. Or if they were unfit in some way, they had been conscripted into
some essential wartime work. So that the people teaching tended to be extreme
distinguished and decidedly elderly. [Inaudible], little word. Very fine but not
young.
And at that time there would be no opportunity to revise the [inaudible] so
probably what was taught was very classic analysis, applied mathematics only
epsilon algebra. Okay? So Oliver probably would not have learned much
algebra in Cambridge.
On the other hand, the classical analysis would probably including elliptic and
modular functions. Not that much other than he probably would know what they
were as a student.
Okay. What happened next in well, he graduated in 1944 and 1944 he was
posted to Bexley Park. While in Britain Bexley Park is considered to be where
computing began. And I think from really -- anyone would agree it was the first
major computing project which really made a difference to people. Which was
important.
When he got to Bexley Park he was posted to a small group called the
Newmanry. This was a small group I'll say a small research group of about 28
strong mathematicians. And while most of Bexley Park was engaged directly in
in the business of decoding German or possibly Japanese, but usually German
signals as being intercepted, the most important ones being the naval ones, the
duty of the small research group was to maintain the capability to decode of
those -- of the operational people. So that in particular if there was a German
naval code uncracked, crack it.
Now, of course in this essential task they weren't alone because the operational
people were pretty bright, too and even if one's job was actually decoding all the
messages, you could get pretty good at decoding other codes you hadn't even
before. Okay. Oh. And I should have said about 28 strong mathematicians, and
they were plus about 200 WRNS. The point is a WRN a member of the Women's
Royal Naval Service, girls who volunteered for the armed services were
screened. The bright, reliable ones tended to be posted to the WRNS and the
brightest and most reliable tend to be sent to Bexley, and in Bexley there were
several thousand WRNS did data processing. Essentially the input and output
system.
Okay. Who were these mathematicians? Well, I'll write down the names of
some of them. You won't know them. But let's see, there's Henry Whitehead. I
should have started off with Max Newman, who was in charge. Shaun Wylie who
was assistant director. Others included oh, Peter Hilton, Sandy Green and
others. The ones -- I'm not going to write down 28. The ones I've written down
were people I got to know myself afterwards.
A remarkable thing that no one working at Bexley has ever talked to me about
Bexley. Security really was secure.
Now, the particular people I've named. He was my professor in Manchester, he
was a topologist, pretty stern kind of guy. Henry Whitehead was a topologist,
founded topology at Oxford, essentially. Peter Hilton and Shaun Wylie, they
were algebraic topologies called algebraists. Wrote a nice book together
afterwards. Sandy Green excellent algebraist. Actually my senior colleague.
From Oliver's point of view, it probably cured his lack of algebra because these -these guys would have talked about communicative algebraic and vector spaces,
that sort of thing purely -- you know, it's part of their thinking process. So they've
got to go around that.
There's also the point that actually you don't have to go down and any store of
number theories to be good at decoding apparently. And of course, I should
have said that those Turing was very much the most important member of the
group. He wasn't strictly a member of the group because he was too important. I
mean, you know, you wouldn't have had Newman as Turing's boss. That would
be idiotic.
On the other hand, for this sort of job he was fairly essential. Okay.
So Oliver was there working there for a year. It's very difficult to think of a more
exciting mathematical environment. It's also true that the mathematicians Oliver
knew fairly well, Freeman Dyson, Turing, others, were actually a class or two
above those who they were lucky enough to work with afterwards. Okay.
So by the time he was 20, Oliver certainly knew what mathematics was about.
And to some extent, one could argue the thing that came out that was just a little
bit tame. And indeed, the next few years were by [inaudible] fairly routine.
Let's see. 1945. End of war. But Oliver still had two years national service to
go. So he went to the national physical lab and thought about aircraft wings. I
can't imagine that he enjoyed. [laughter].
Then in 1947 to 1952, he was a research student in Cambridge. And that must
have been pretty good because by 1947 the war had sort of warn off and all the
people like Peter Hilton and Shaun Wylie who ought to have been teaching him
when he was taking an undergraduate degree from '42 to '44 and who really did
teach him and who is in Bexley, all of them are back in Cambridge. And lots of
other bright people.
And he did under Littlewood on partitions. And then in 1952, he got his doctorate
and 1952 to -- he went to Durham as a lecturer. And now I'll talk about his thesis
at the end of the paragraph as it were. 1952 he went up to Durham as a lecturer
and, well, he's written a few decent papers. Durham it's a lovely little town.
Small town. Completely dominated by a cathedral, [inaudible] in the of the river
in the middle of the town. Very beautiful.
The important thing about the cathedral for Oliver, it had an organ. He was a fine
musician and he played the organ. And then in, let's see, 1959, he got married.
And, well, many people would have just stayed there for life. But not if you've
been in -- not if you'd had that sort of start at age 40. He was going to get
restless.
Now, so he only lasted in Durham until 1961. Now, what did he do his research
on? Well, it was time for mathematics and time for some notation. He did
research on partitions. So let's just remind ourselves PN if partition function. It's
got a generating function. I'll call it F of Q is sigma 1 to infinity, PNQ to the N
which of course is going to be the product from 1 to infinity, one minus X to the N
inverse, which is a very satisfactory way to define the partition function. And this
is really inverted commas a bunch of form. Because if we write Q to the 124th
product, 1 minus Q to the N, where Q is -- Winnie wrote from very convenient
notation for me. Q is eta 2 pi I zed. If each of zed is that, then everyone knows
[inaudible] identity, this is sum from minus infinity to infinity minus 1 to the N, Q to
the 6N -- let's get it right, 6N plus 1 over 2, 6N plus 1 squared over 24, that is
right, and this -- and then comparing that with that, F of Q I think is da, da, da Q
to the 124th times eta inverse, okay.
And then this is a -- eta itself is a modular form because it's a theta function. It's
a modular form of weight a half. With a multiplier. I'm not going to write down
the functional equation. It's quite complicated as a modular form. Everyone
knows and loves it because delta zed is eta of zed to the 24th. F itself is also the
modular form apart from the 24th root of Q. It was a very nasty modular form
because it's weight minus a half, which is dangerous.
However, being a glutton for punishment or obstinate, Oliver did research on the
coefficients of this rather perverse modular form eta of Q. Okay? The partition
function. A terrible function.
Yes. Now, I come to his -- ah, yes. While I'm here there's a conjecture by
Ramanujan, I hope I get it right, if 24N minus 1 is congruent to zero modulo 2 to
the A, 3 to the B, 5 to the A, 7 to the B, 11 to the C, then P of N is congruent to 0
modulo 5 to the A, 7 to the beta, 11 to the C. What actually [inaudible]
conjecture, it has a B where the beta is, but that was wrong, and beta is 1 or 2
when B is 1 or 2. Without that it only grows half as fast.
Okay? That's Ramanujan's conjecture. And this was proved for the 5 to the A, 7
to the B by Watson in, I'm not sure of the date, I'll say 1938, which is probably
right, and for 11 to the 1 and 11 to the 2 by Morris Newman I think. It may have
been Jermaine, but I'm pretty certain it was Morris Newman.
And Oliver's problem, his thesis problem, was complete the job. And of course
he did. But first Oliver's first paper was rather nice. First paper joint with
Swinnerton-Dyer who is a conjecture by Dyson. You may be thinking of Oliver's
-- it was a bit early for conjecture by Dyson. In fact, this conjecture was
conjectured by Dyson when he was still at school and published in Eureka, which
is the Cambridge students magazine. But it wasn't trivial.
Here a particular case I think says that a partition function for things of the form
5N plus 4 has always come to 0 modulo 5 and partition function for things of the
form 7N plus 5 is also congruent to 0 modulo 7. Okay? But there's particular
cases of that. And that as a conjecture said that if pi is a partition, define the
rank of pi equals mod largest bit minus the number of bits -- sorry, partition 7 plus
6 plus 2 plus 1, the rank is going to be that 7 minus number of parts, which is 4,
so you get 3. And then if N is congruent to mod 5, sold. Partitions modulo N into
classes according to rank modulo 5 and then classes all the same size.
So this gives you a simple combinatorial reason on a number of partitions of 4 or
9 or 14 the divisible by 5. The theorem itself, the proof of this is as proved by
Oliver and Peter Swinnerton-Dyer, they had to reinvent leverage and mark theta
functions. It was [inaudible] delighted Dyson. But completely nontrivial.
And then the same is true mutatis mutandis for mod 7 OK2. And that one didn't
work. Okay. So that's Dyson's conjecture and Oliver's first paper.
About 40 years later a guy called I think Garthin produced a different invariant of
partitions not the rank -- well, actually, it was used different variant of partitions
which Dyson called the crank and proved a similar theorem for 11 using the
crank instead of the rank. Okay.
Now, where were we? Oliver went on, and his -- I think his third paper was
called proof of conjecture from Ramanujan of -- and in a sense that was more
routine. Whereas prove this thing they had to use infernal ingenuity to prove the
Ramanujan conjecture, Oliver essentially followed the path followed by Watson
and Morris Newman, which was to manipulate modulo equations involving
various special modular functions. To prove an identity about -- to prove a
congruence for the coefficients of a particular rather difficult modular form.
And really the extra ingredient Oliver needed was simply knowing more about
modular functions and forms and anyone else and loving it. Okay?
So he'd written a thesis and he was doing good work on what I call a rather
narrow area which he loved. He's got a good job in a nice place. He'd got a new
wife. Why wasn't he happy? Well, it's obvious. I mean, he'd been in Bexley, he
knew what really mathematics was about. He knew he could do more. Also
looking at modular functions he felt though there's nothing is simple to prove
about the partition function, no obvious divisibility properties, he felt there must
be congruence properties and so on available. And I'll put something up which
I'll call Oliver's idea. Because the way I'm putting it up is much too sloppy to be
called a conjecture and it's probably about what he was thinking at the time.
So let F and zed exactly sigma 1 to infinity, AN Q to the N, the modular form or
function, not necessarily of positive weight. Probably not of positive weight. And
but vanishing at I infinity and invariant by [inaudible] group and N equals Q to the
K probably. Then there should be a congruence like to replace the heka
[phonetic] relation -- I tell you my notation is not actually the same as Winnie's,
but APN plus P to the K minus 1AN over P equals AP, AN -- yes, satisfied by
new forms. And let's just write the same thing down and just replace the equality
by congruent modulo level.
Now, as I've written it down, it's too sloppy to be right because new forms aren't
necessarily rational, so I should say -- I should be congruency is like and you
should probably say that the coefficients you've got are a linear combination of
things we satisfy an identity like that. Alternatively you could ask for a relation
with longer terms. So you -- so you have to -- so you took it to get a correct
sensible statement you've got to modify that. But this is -- it's his idea. And I
don't think he ever did formulate it absolutely right.
Now, the trouble was that he was -- he had this idea, and he was interested in
modular forms, horrible modular forms like eta of zed inverse which minus a half
or he got more modest in his thoughts and settled for the J function, which is a
nice function of level 1. And I should say that a good example for the sort of
thing in the theorem is going to be J of zed over N plus Y of zed plus N minus 1
over N or this with coefficient Nth roots of unity. And then that qualifies as one of
those. And this can be read -- this can be read off as convex relation between
the coefficient of the J function.
And he did a little bit of hand computation which suggested this might well be so.
But as everyone can see, the coefficients of the J function are far too large for a
computation to be of any use. And in 1961, Atlas Fellowship advertised. The
truth is that the Atlas Fellowship would be to work on one of the word's largest
computers at the time and essentially do what he likes along -- so long as he
worked hard.
Well, he thought about this question, which was -- it's a good idea, but completely
uncheckable without having computing resources. I've no doubt he remembered
Bexley and thought that mathematics is more exciting the Durham and so of
course he applied and of course he was appointed to the -- I think it jumped at
him.
So 1961, he moved on to Atlas. Well, at Atlas, excuse me why I check this out.
At Atlas he was in his element. He could work as hard as he liked, which was
pretty hard. On the whole he got on pretty well with other people. There were
those who didn't really enjoy it when he walked down the corridor singing
[inaudible] but [laughter] that was okay. And well, he worked. Mainly computing
modular forms, lots of modular forms, checking out this sort of thing, computing
other things as well. It's well known that for a while the check that the engineers
did on the Atlas computer at the beginning of the stay was to compute the first
page full of coefficients of eta of zed raised to the 25th hour and make sure this
was always right. It's a good test for a computer if it's got things like vowels
which can be a little bit rickety.
And well I arrived in Oxford I think it was 1964 and at that time I needed to learn
about modular forms because when well Shamir told me that the [inaudible]
which were parameterized by modular functions and, you know, know about
them and see if the conjecture was right. But at the time, the literature on
modular functions that was perfectly horrible. There are lots and lots and lots of
modular forms all running about connected by lovely relations and the literature
was completely unsystematic. And there are a lot of beautiful theorems known
about them. But the trouble is it is extremely difficult to find out from the literature
whether a useful fact that was known to be true was something which someone
had actually approved or it was a sort of folklore that no one ever got around to
checking. So and that was all of us. I asked him and he put me right.
What I can't quite understand was that the Atkin-Lana [phonetic] paper wasn't
actually published until 1970, and the things which Oliver told me about four, five
years earlier appeared to be straight out of Atkin-Lana, and I remember them.
But I think it was partly publication like and partly Atkin and Lana getting around
to the effort of writing it up. Anyway, as Winnie has already said, that was a -- it
wasn't a difficult paper, it wasn't a terribly new paper, but it completely
transformed the theory and made it easy for everyone to use. It was such an
important paper, in fact.
Then, well, I better keep things short because if I finish earlier everyone will be
very happy, and if I finish late, everyone will certainly be unhappy.
What else did he do at Atlas? Well, he -- yes, there was his -- one of the best
known things was is lovely essay on feasible computation, which I hope is
compulsory reading for everyone who does computational number theory. If
anyone hasn't read it, they'd better go and read it. It's easily available on the
Web.
He was generally helpful. There were a couple of things I mentioned in my 1995
talk. He did just extremely helpful computations. In particular there was a
conjecture to Tate which is -- was called the Birch-Tate conjecture because I
supplied him with the result of some computations. It should have been called
the Atkin-Tate conjecture after the guy who did most of the computation. I had
done some of them, but -- yeah. Those are the non-congruence groups, theta.
The biggest thing he did at Atlas really was what he planned to do, which was
computations relating to this idea. And the trouble there is that a very general
idea. There are and awful lot of cases one can check by look at the computer.
They all come out right. Proving a single case is actually rather difficult.
So Oliver -- let's see, 1968 there, was a paper essentially on the coefficients of J
of zed on those lines which checked the idea for powers of primes up to 37 which
fairly decided it was basically right. But going up to 37 there's only about one
case where the coefficient stopped being rational. So that I think the idea was
what he proved up to 37 except in one case was I think just as I've written it. But
for P equals 29, he had to get it right.
Except P equals 29 needed the adjustment, 2 to the space -- the space sort of
wasn't there being two dimensional instead of one dimensional.
And then in a paper with 1967 with O'Brien, the ditto for eta of zed inverse IE
congruences for P of N modulo I think 5 and certainly 13 and probably 7. Even
though there's not a divisibility theorem for 5 except for the RES U4 but the
congruence is much lower powers of 5 otherwise. So that I think summarizes
what he accomplished [inaudible].
Now, unhappily, all good things have an end. 1970 he left Atlas. More or less at
the same time, let's say 1968 to 1972 there was a complete revolution in theory
of modular forms which was more or less summarized at the 1972 Antwerp
conference after which the theory of modular forms became a matter of
representations as described -- as everyone thinks of nowadays. But of course
before it wasn't like that.
And so there's I'll say advent of representation theory and also advent of P ethic
modular forms. Now, in a sense, this was for Oliver a success, success,
because [inaudible] recognized the modular function as the prototype of a not
obvious p-adic modular form. Well, I suppose it's the first time of a whole covey
of p-adic modular forms because when we take these congruence conditions you
get a different setup for each prime. So that the P function gives rise to a family
of not particularly closely related p-adic modular forms.
And that in a sense made sense of that. On the other hand, from -- this is a bit of
a disaster because I don't think Oliver's background in algebra was ever that
good and having his beloved subject turned into representation theory and p-adic
hooha. Well, the [inaudible] was all right, but I don't think he ever liked the
representation theory.
So -- and the other disaster from that point of view is that representation theory is
very dependent on geometry. And to get some geometry out of the modular form
you've got to have some geometric objects and associated geometric object with
the modular form if a positive weight of [inaudible] variety or what have you, and
you can take Tate modules and represent the [inaudible] group and was acting
on the Tate module and all that jazz.
But if you take a form of negative weight it's horrible. So on the one hand, that
was dead right and on the other hand hardly anyone has every talked about a
modular form of negative weight since.
Much more disastrously when Oliver resigned from Atlas, he had taken a job in
Arizona and almost immediately after he got there his wife died. So he was in a
foreign country with children age 7 and 10, by all accounts very nice, sensible
children, but even so, and he somehow had to manage -- well, it's a -- his job in
Arizona lasted one year, and then he went to Brown for one year. And actually
my imagination boggles.
But then 1972 he went to, let's see -- he went Chicago circle, University of Illinois
actually Chicago, UIC, the standards initials are UIC, which at lunch time Winnie
didn't describe as being a particularly nice place, but he went to Chicago circle
and found a little house in the village of Oak Park and prior to the folks at Oak
Park took this mad Englishman to their house and he stayed there for the rest of
his life very happily.
So 1972 he at last got settled. I had loss contact completely and so did most
people who knew him. But still from 1972 he was okay. And which brings me to
this period. Well, because of the revolution in the theory of modular functions, he
tames his interest. He remains knowing everything about classical modular
function -- forms and modular functions. But for a little while there wasn't any
evidence that he was working on them.
Because -- well, he was always when he was -- that was quite reluctant to write a
paper unless else wrote it for him. So it's the harder one tries, the more one
finds. It's actually quite difficult to find what he was doing during that period. For
instance, he was doing quite a lot. Let's see, it was 1972 to 5 I think he was -Duncan Buell was his student. Wrote a very nice thesis.
I think that must have been Oliver algorithm for composition of quadratic forms.
In [inaudible] book there's a he refers to Oliver as the author of such an
algorithm, and I think it must have been that period because that's the sort of
thing that Duncan was working on.
Then a little bit later work with Winnie Li. That was modular forms again.
Classical. 1976, there was a conference always known as Antwerp 2 at Bonn.
[laughter] and he was certainly there. I can remember because he certainly
made inquiries about which is the best organ in Bonn. But Bonn is a bit of bond,
the other side of the river which is where the Max Planck Institute was in those
days. And he found the best organ, and he played with it and from what I
remember he seemed pretty well.
But I didn't talk to him very much because talking to other people, and [inaudible]
Stevens was computing something -- well.
1980 he was working with McKay producing the -- there was a moonshine
conjecture came out in 1978 and got proved by both in 1982 I think and in
between there's various verification work done and Oliver joined in this McKay
and somebody else producing the moonshine module, whatever that may be. He
-- his interests to [inaudible] so Francois [inaudible] then they thought about that.
Then finally in 1985, whoops, 1985 -- excuse me. In 1985 during his thesis
[inaudible] thesis reads Oliver [laughter] rather deliberately to [inaudible] practical
algorithm with Oliver. He looked at this and thought, ah, this is where I can use
my lovely modular forms. So he rushed into it and [inaudible] but for decent
[inaudible]. And then [inaudible] came along and got this. Anyway, at that stage
he thought [inaudible] how I got an e-mail from him. From then on, I was back on
his mailing list and [inaudible] from Oliver either directly or by [inaudible] some
[inaudible]. And that's about the time he stopped because from then on
[inaudible] knows more about Oliver than I do.
[applause]