Se vei que manejaba totalmente el tema. Un tipo de mediana o corta estatura y muy severo en su trato.
Me autógrafo el cartel
Sir Michael Atiyah obituary
One of the greatest British mathematicians since Isaac Newton
The last time I met Michael Atiyah, who has died aged 89, was at Tate Modern in London; not the most likely place to run into probably Britain’s greatest mathematician since Isaac Newton, but entirely consistent with his wide-ranging enthusiasm for his subject. It was June 2012, and I joined him and the flamboyant French mathematician Cédric Villani in a panel discussion: Mathematics, a Beautiful Elsewhere. The title says it all.
We have sulphuric acid to thank for Atiyah’s decision to become a mathematician. Early in 1940, as Britain and France fought over his homeland of Lebanon, his parents sent him to Victoria college in Cairo. In a 1984 interview he said that while there he got very interested in chemistry, but eventually decided that making “sulphuric acid and all that sort of stuff” was not for him: “Lists of facts, just facts ...” From that time on, mathematics became his passion. “I never seriously considered doing anything else.” Atiyah’s work was to have a profound influence on today’s mathematics.
Atiyah was a geometer, in the sense of visual thinking allied to abstract symbolism, a new attitude that swept through mathematics in the middle of the 20th century. You thought about it like geometry but wrote about it like algebra, and very esoteric algebra at that. His research divides into four main periods, to some extent overlapping – in the 1950s, algebraic geometry; in the 60s and early 70s, K-theory; the 60s to 80s, index theory; and the late 70s to mid-80s, gauge theory, where his ideas became extremely influential in quantum physics.
Algebraic geometry originally developed from a deep link between geometry and algebra promoted in the 1600s by René Descartes. Start with Euclid’s plane and introduce coordinates – pairs of numbers describing the location of a point, much as latitude and longitude determine a point on the Earth’s surface. Geometric properties of curves can then be described by algebraic equations, so questions in geometry can be tackled using algebra, and vice versa.
In the late 1800s and early 1900s, a new kid appeared on the mathematical block: topology, in which geometrical shapes can be deformed as if made of elastic. Classical features such as lengths and angles lose their meaning, and are replaced by concepts such as being connected, knotted, or having a hole like a doughnut.
Topology turned out to be fundamental to many areas of mathematics. Techniques were devised to associate with a topological space various “invariants”, which reveal when spaces can or cannot be deformed into each other.
One of the most powerful invariants, homology, was established by Emmy Noether, the greatest female mathematician of the late 1800s and early 1900s. She reinterpreted, in terms of abstract algebra, rudimentary methods for counting features such as the number of holes in a surface.
In effect, Noether explained that as well as counting holes and associated structures, we can ask how they combine, and extract topological information from the answer.
Atiyah began his research career in algebraic geometry, but under the influence of his supervisor, William Hodge, at Cambridge, he quickly moved into an adjacent field, differential geometry, which studies concepts such as curvature — how a space deviates from the flat plane of Euclid. There he made big advances in the interactions between algebraic geometry, differential geometry and topology.
Euclid’s investigations of a circle includes its tangents: straight lines that touch it at one point, like a road supporting a bicycle wheel. Similarly, a sphere has a family of tangent planes, one for each point on its surface. A general family of this kind is called a vector bundle: “bundle” because the sphere ties all the planes together, and “vector” because higher-dimensional analogues of lines and planes are called vector spaces.
The topology of a vector bundle provides information about the underlying space. The tangents to a circle, for example, form a cylinder. As proof: rotate each tangent line through a right angle, out of the plane of the circle, and you get a cylinder. There is another vector bundle associated with a circle, in which the lines are twisted to form the famous Möbius band, a surface that differs topologically from a cylinder since it has only one side. Atiyah applied these ideas to “elliptic curves”, actually doughnut-shaped surfaces with interesting number-theoretic properties.
His next topic, K-theory, is a far-reaching extension of Noether’s homology invariant. A cylinder and a Möbius band are topologically distinct because their associated bundles have different twists. K-theory exploits vector bundles to capture higher dimensional analogues of such twists.
The topic underwent a period of rapid development in the 60s, stimulated by remarkable links to other major areas of mathematics, and it provided topologists with a powerful toolkit of invariants.
Atiyah, often jointly with other leading mathematicians, was a driving force behind these developments. Important themes were the cobordism theory of René Thom (how one circle splits into two as you move down a pair of trousers from the waist to the leg holes, only done for multidimensional spaces) and the periodicity theorem, first proved by Raoul Bott, showing that higher K-groups repeat in a cycle of length eight.
Index theory has its origins in the observation that topological features of a landscape, such as the numbers of mountain peaks, valleys and passes, are related to each other. To get rid of a peak by flattening it out you must also get rid of a pass, for example. The index organises such phenomena, and can be used, in suitable circumstances, to prove that a peak must exist in some region.
A landscape is a metaphor for the graph of a mathematical function, and a sweeping generalisation relates the number of solutions of a differential equation to a more esoteric topological index.
Differential equations relate rates of change of various quantities to each other, and are ubiquitous in mathematical physics; the Atiyah-Singer Index Theorem, proved jointly with the American mathematician Isadore Singer in 1963, reveals a highly significant link between a topological index and the solutions of a differential equation.
In an appropriate mathematical setting this can lead to a proof that a solution must exist, so the Atiyah-Singer index has widespread applications to physics. Forty years after their discovery, the pair were jointly awarded the Abel prize of the Norwegian Academy of Science and Letters, in 2004.
Gauge theory arose in physics, formalising certain symmetries of quantum fields and particles. The first example arose from James Clerk Maxwell’s equations for the electromagnetic field (1861), where certain mathematical transformations can be applied without changing the physics.
In 1954 Chen Ning Yang and Robert Mills extended this idea to the strong interaction, which holds together each quantum particle in the atomic nucleus. Symmetry turned out to be vital to quantum mechanics – for example, the recently discovered Higgs boson, which endows particles with mass, acts by breaking certain symmetries – and gauge symmetries have huge importance.
Atiyah contributed key ideas to their mathematics, using his index theory to study instantons (particles that wink into existence and immediately wink out again) and magnetic monopoles (particles like a north magnetic pole without any corresponding south pole).
In 1983 his PhD student Simon Donaldson used these ideas to prove a remarkable theorem: contrary to what almost all topologists expected, four-dimensional space has infinitely many distinct differentiable structures – utterly different in this respect from any other dimension. The broader context for all this work is superstring theory, a conjectured unification of quantum theory and Albert Einstein’s relativity.
Atiyah was born in London, one of four children of Edward, a Lebanese civil servant, and his wife, Jean (nee Levens), who was born in Yorkshire of Scottish descent. The family moved to Khartoum, Sudan, where Michael went to school before boarding at Victoria College in Cairo and then moving to Manchester Grammar school at 16 to prepare for Cambridge. He was always keen on mathematics. An inspiring teacher introduced him to projective geometry and William Rowan Hamilton’s algebra of quaternions, and he read about number theory and group theory – all of which clearly influenced his later mathematical interests.
In 1949, after two years of national service, he studied at Trinity College, Cambridge, remaining there for his PhD. He held positions at the Institute for Advanced Study in Princeton (including a professorship 1969-72), and at Cambridge and at Oxford, where he was Savilian professor of geometry 1963-69 and Royal Society research professor 1973-90. He became a fellow of the Royal Society in 1962, and was the society’s president from 1990 to 1995. In 1966 he won a Fields medal, the highest honour for any mathematician.
In 1990 he became master of Trinity College, Cambridge, and director of the Isaac Newton Institute for Mathematical Sciences, Cambridge. He was knighted in 1983 and made a member of the Order of Merit in 1992. After retiring from Trinity in 1997 he moved with his wife, Lily (nee Brown), whom he had married in 1955, to Edinburgh.
Atiyah was always a keen advocate of public engagement, giving popular talks on the beauty of mathematics and his lifelong passion for the subject. Small and compact, with a quiet, precise delivery, he could nevertheless hold an audience spellbound. That is how I remember him, on that day in Tate Modern, telling non-mathematicians why we do it, what it is for, and what it feels like.
He and Lily had three sons: John, David and Robin. John died in a climbing accident in 2002; Lily died last year. Michael is survived by David and Robin.
Interview with Interview with
Michael Michael Atiyah and Isadore Singer Atiyah and Isadore Singer
Interviewers: Martin Raussen and Christian Skau
from left to right: I. Singer, M. Atiyah, M. Raussen, C. Skau
see what happens”.
ATIYAH No doubt: Singer had a strong
expertise and background in analysis and
differential geometry. And he knew certainly more physics than I did; it turned out
to be very useful later on. My background
was in algebraic geometry and topology, so
it all came together. But of course there are
a lot of people who contributed in the background to the build-up of the Index
Theorem – going back to Abel, Riemann,
much more recently Serre, who got the
Abel prize last year, Hirzebruch,
Grothendieck and Bott. There was lots of
work from the algebraic geometry side and
from topology that prepared the ground.
And of course there are also a lot of people
who did fundamental work in analysis and
the study of differential equations:
Hörmander, Nirenberg… In my lecture I
will give a long list of names2
; even that
one will be partial. It is an example of
international collaboration; you do not
work in isolation, neither in terms of time
nor in terms of space – especially in these
days. Mathematicians are linked so much,
people travel around much more. We two
met at the Institute at Princeton. It was nice
to go to the Arbeitstagung in Bonn every
year, which Hirzebruch organised and
where many of these other people came. I
did not realize that at the time, but looking
back, I am very surprised how quickly
these ideas moved…
Collaboration seems to play a bigger role
in mathematics than earlier. There are a
lot of conferences, we see more papers
that are written by two, three or even more
authors – is that a necessary and commendable development or has it drawbacks as well?
ATIYAH It is not like in physics or chemistry where you have 15 authors because
they need an enormous big machine. It is
not absolutely necessary or fundamental.
But particularly if you are dealing with
areas which have rather mixed and interdisciplinary backgrounds, with people who
have different expertise, it is much easier
and faster. It is also much more interesting
for the participants. To be a mathematician
on your own in your office can be a little bit
dull, so interaction is stimulating, both psychologically and mathematically. It has to
be admitted that there are times when you
go solitary in your office, but not all the
time! It can also be a social activity with
lots of interaction. You need a good mix of
both, you can’t be talking all the time. But
talking some of the time is very stimulating. Summing up, I think that it is a good
development – I do not see any drawbacks.
SINGER Certainly computers have made
collaboration much easier. Many mathematicians collaborate by computer instantly;
it’s as if they were talking to each other.
I am unable to do that. A sobering counterexample to this whole trend is
Perelman’s results on the Poincaré conjecture: He worked alone for ten to twelve
years, I think, before putting his preprints
on the net.
ATIYAH Fortunately, there are many different kinds of mathematicians, they work
on different subjects, they have different
approaches and different personalities –
and that is a good thing. We do not want
all mathematicians to be isomorphic, we
want variety: different mountains need different kinds of techniques to climb.
SINGER I support that. Flexibility is
absolutely essential in our society of mathematicians.
Perelman’s work on the Poincaré conjecture seems to be another instance where
analysis and geometry apparently get
linked very much together. It seems that
geometry is profiting a lot from analytic
perspectives. Is this linkage between different disciplines a general trend – is it
true, that important results rely on this
interrelation between different disciplines?
And a much more specific question: What do you know about the status
of the proof of the Poincaré conjecture?
SINGER To date, everything is working
out as Perelman says. So I learn from
Lott’s seminar at the University of
Michigan and Tian’s seminar at Princeton.
Although no one vouches for the final
details, it appears that Perelman’s proof
will be validated.
As to your first question: When any two
subjects use each other’s techniques in a
new way, frequently, something special
happens. In geometry, analysis is very
important; for existence theorems, the
more the better. It is not surprising that
some new [at least to me] analysis implies
something interesting about the Poincaré
conjecture.
ATIYAH I prefer to go even further – I
really do not believe in the division of
mathematics into specialities; already if
you go back into the past, to Newton and
Gauss… Although there have been times,
particularly post-Hilbert, with the axiomatic approach to mathematics in the first half
of the twentieth century, when people
began to specialize, to divide up. The
Bourbaki trend had its use for a particular
time. But this is not part of the general attitude to mathematics: Abel would not have
distinguished between algebra and analysis. And I think the same goes for geometry and analysis for people like Newton.
It is artificial to divide mathematics into
separate chunks, and then to say that you
bring them together as though this is a surprise. On the contrary, they are all part of
the puzzle of mathematics. Sometimes you
would develop some things for their own
sake for a while e.g. if you develop group
theory by itself. But that is just a sort of
temporary convenient division of labour.
Fundamentally, mathematics should be
used as a unity. I think the more examples
we have of people showing that you can
usefully apply analysis to geometry, the
better. And not just analysis, I think that
some physics came into it as well: Many of
the ideas in geometry use physical insight
as well – take the example of Riemann!
This is all part of the broad mathematical
tradition, which sometimes is in danger of
being overlooked by modern, younger people who say “we have separate divisions”.
We do not want to have any of that kind,
really.
SINGER The Index Theorem was in fact
instrumental in breaking barriers between
fields. When it first appeared, many oldtimers in special fields were upset that new
techniques were entering their fields and
achieving things they could not do in the
field by old methods. A younger generation immediately felt freed from the barriers that we both view as artificial.
ATIYAH Let me tell you a little story
about Henry Whitehead, the topologist. I
remember that he told me that he enjoyed
very much being a topologist: He had so
many friends within topology, and it was
such a great community. “It would be a
tragedy if one day I would have a brilliant
idea within functional analysis and would
have to leave all my topology friends and
to go out and work with a different group
of people.” He regarded it to be his duty to
do so, but he would be very reluctant.
Somehow, we have been very fortunate.
Things have moved in such a way that we
got involved with functional analysts without losing our old friends; we could bring
them all with us. Alain Connes was in
functional analysis, and now we interact
closely. So we have been fortunate to
maintain our old links and move into new
ones – it has been great fun.
We would like to have your comments on
the interplay between physics and mathematics. There is Galilei’s famous dictum
from the beginning of the scientific revoMathematics and physics
INTERVIEW
EMS September 2004 25
lution, which says that the Laws of Nature
are written in the language of mathematics. Why is it that the objects of mathematical creation, satisfying the criteria of
beauty and simplicity, are precisely the
ones that time and time again are found to
be essential for a correct description of the
external world? Examples abound, let me
just mention group theory and, yes, your
Index Theorem!
SINGER There are several approaches in
answer to your questions; I will discuss
two. First, some parts of mathematics were
created in order to describe the world
around us. Calculus began by explaining
the motion of planets and other moving
objects. Calculus, differential equations,
and integral equations are a natural part of
physics because they were developed for
physics. Other parts of mathematics are
also natural for physics. I remember lecturing in Feynman’s seminar, trying to
explain anomalies. His postdocs kept
wanting to pick coordinates in order to
compute; he stopped them saying: “The
Laws of Physics are independent of a coordinate system. Listen to what Singer has to
say, because he is describing the situation
without coordinates.” Coordinate-free
means geometry. It is natural that geometry appears in physics, whose laws are
independent of a coordinate system.
Symmetries are useful in physics for
much the same reason they’re useful in
mathematics. Beauty aside, symmetries
simplify equations, in physics and in mathematics. So physics and math have in common geometry and group theory, creating a
close connection between parts of both
subjects.
Secondly, there is a deeper reason if your
question is interpreted as in the title of
Eugene Wigner’s essay “The
Unreasonable Effectiveness of
Mathematics in the Natural Sciences”3
.
Mathematics studies coherent systems
which I will not try to define. But it studies coherent systems, the connections
between such systems and the structure of
such systems. We should not be too surprised that mathematics has coherent systems applicable to physics. It remains to be
seen whether there is an already developed
coherent system in mathematics that will
describe the structure of string theory. [At
present, we do not even know what the
symmetry group of string field theory is.]
Witten has said that 21st century mathematics has to develop new mathematics,
perhaps in conjunction with physics intuition, to describe the structure of string theory.
ATIYAH I agree with Singer’s description
of mathematics having evolved out of the
physical world; it therefore is not a big surprise that it has a feedback into it.
More fundamentally: to understand the
outside world as a human being is an
attempt to reduce complexity to simplicity.
What is a theory? A lot of things are happening in the outside world, and the aim of
scientific inquiry is to reduce this to as simple a number of principles as possible.
That is the way the human mind works, the
way the human mind wants to see the
answer.
If we were computers, which could tabulate vast amounts of all sorts of information, we would never develop theory – we
would say, just press the button to get the
answer. We want to reduce this complexity to a form that the human mind can
understand, to a few simple principles.
That’s the nature of scientific inquiry, and
mathematics is a part of that. Mathematics
is an evolution from the human brain,
which is responding to outside influences,
creating the machinery with which it then
attacks the outside world. It is our way of
trying to reduce complexity into simplicity,
beauty and elegance. It is really very fundamental, simplicity is in the nature of scientific inquiry – we do not look for complicated things.
I tend to think that science and mathematics are ways the human mind looks and
experiences – you cannot divorce the
human mind from it. Mathematics is part
of the human mind. The question whether
there is a reality independent of the human
mind, has no meaning – at least, we cannot
answer it.
Is it too strong to say that the mathematical problems solved and the techniques
that arose from physics have been the
lifeblood of mathematics in the past; or at
least for the last 25 years?
ATIYAH I think you could turn that into
an even stronger statement. Almost all
mathematics originally arose from external
reality, even numbers and counting. At
some point, mathematics then turned to ask
internal questions, e.g. the theory of prime
numbers, which is not directly related to
experience but evolved out of it.
There are parts of mathematics where the
human mind asks internal questions just
out of curiosity. Originally it may be physical, but eventually it becomes something
independent. There are other parts that
relate much closer to the outside world
with much more interaction backwards and
forward. In that part of it, physics has for a
long time been the lifeblood of mathematics and inspiration for mathematical work.
There are times when this goes out of fashion or when parts of mathematics evolve
purely internally. Lots of abstract mathematics does not directly relate to the outside world.
It is one of the strengths of mathematics
that it has these two and not a single
lifeblood: one external and one internal,
one arising as response to external events,
the other to internal reflection on what we
are doing.
SINGER Your statement is too strong. I
agree with Michael that mathematics is
blessed with both an external and internal
source of inspiration. In the past several
decades, high energy theoretical physics
has had a marked influence on mathematics. Many mathematicians have been
shocked at this unexpected development:
new ideas from outside mathematics so
effective in mathematics. We are delighted
with these new inputs, but the “shock”
exaggerates their overall effect on mathematics.
Can we move to newer developments with
impact from the Atiyah-Singer Index
Theorem? I.e., String Theory and
Edward Witten on the one hand and on
the other hand Non-commutative
Geometry represented by Alain Connes.
Could you describe the approaches to
mathematical physics epitomized by these
two protagonists?
ATIYAH I tried once in a talk to describe
the different approaches to progress in
physics like different religions. You have
prophets, you have followers – each
prophet and his followers think that they
have the sole possession of the truth. If you
take the strict point of view that there are
several different religions, and that the
intersection of all these theories is empty,
then they are all talking nonsense. Or you
can take the view of the mystic, who thinks
that they are all talking of different aspects
of reality, and so all of them are correct. I
tend to take the second point of view. The
main “orthodox” view among physicists is
certainly represented by a very large group
of people working with string theory like
Edward Witten. There are a small number
of people who have different philosophies,
one of them is Alain Connes, and the other
is Roger Penrose. Each of them has a very
specific point of view; each of them has
very interesting ideas. Within the last few
years, there has been non-trivial interaction
between all of these.
They may all represent different aspects
of reality and eventually, when we under
Newer developments
INTERVIEW
26 EMS September 2004
stand it all, we may say “Ah, yes, they are
all part of the truth”. I think that that will
happen. It is difficult to say which will be
dominant, when we finally understand the
picture – we don’t know. But I tend to be
open-minded. The problem with a lot of
physicists is that they have a tendency to
“follow the leader”: as soon as a new idea
comes up, ten people write ten or more
papers on it and the effect is that everything
can move very fast in a technical direction.
But big progress may come from a different direction; you do need people who are
exploring different avenues. And it is very
good that we have people like Connes and
Penrose with their own independent line
from different origins. I am in favour of
diversity. I prefer not to close the door or
to say “they are just talking nonsense”.
SINGER String Theory is in a very special
situation at the present time. Physicists
have found new solutions on their landscape - so many that you cannot expect to
make predictions from String Theory. Its
original promise has not been fulfilled.
Nevertheless, I am an enthusiastic supporter of Super String Theory, not just because
of what it has done in mathematics, but also
because as a coherent whole, it is a marvellous subject. Every few years new developments in the theory give additional
insight. When that happens, you realize
how little one understood about String
Theory previously. The theory of D-branes
is a recent example. Often there is mathematics closely associated with these new
insights. Through D-branes, K-theory
entered String Theory naturally and
reshaped it. We just have to wait and see
what will happen. I am quite confident that
physics will come up with some new ideas
in String Theory that will give us greater
insight into the structure of the subject, and
along with that will come new uses of
mathematics.
Alain Connes’ program is very natural –
if you want to combine geometry with
quantum mechanics, then you really want
to quantize geometry, and that is what noncommutative geometry means. Non-commutative Geometry has been used effectively in various parts of String Theory
explaining what happens at certain singularities, for example. I think it may be an
interesting way of trying to describe black
holes and to explain the Big Bang. I would
encourage young physicists to understand
non-commutative geometry more deeply
than they presently do. Physicists use only
parts of non-commutative geometry; the
theory has much more to offer. I do not
know whether it is going to lead anywhere
or not. But one of my projects is to try and
redo some known results using non-commutative geometry more fully.
If you should venture a guess, which
mathematical areas do you think are
going to witness the most important developments in the coming years?
ATIYAH One quick answer is that the
most exciting developments are the ones
which you cannot predict. If you can predict them, they are not so exciting. So, by
definition, your question has no answer.
Ideas from physics, e.g. Quantum
Theory, have had an enormous impact so
far, in geometry, some parts of algebra, and
in topology. The impact on number theory
has still been quite small, but there are
some examples. I would like to make a
rash prediction that it will have a big
impact on number theory as the ideas flow
across mathematics – on one extreme number theory, on the other physics, and in the
middle geometry: the wind is blowing, and
it will eventually reach to the farthest
extremities of number theory and give us a
new point of view. Many problems that are
worked upon today with old-fashioned
ideas will be done with new ideas. I would
like to see this happen: it could be the
Riemann hypothesis, it could be the
Langlands program or a lot of other related
things. I had an argument with Andrew
Wiles where I claimed that physics will
have an impact on his kind of number theory; he thinks this is nonsense but we had a
good argument.
I would also like to make another prediction, namely that fundamental progress on
the physics/mathematics front, String
Theory questions etc., will emerge from a
much more thorough understanding of
classical four-dimensional geometry, of
Einstein’s Equations etc. The hard part of
physics in some sense is the non-linearity
of Einstein’s Equations. Everything that
has been done at the moment is circumventing this problem in lots of ways. They
haven’t really got to grips with the hardest
part. Big progress will come when people
by some new techniques or new ideas really settle that. Whether you call that geometry, differential equations or physics
depends on what is going to happen, but it
could be one of the big breakthroughs.
These are of course just my speculations.
SINGER I will be speculative in a slightly
different way, though I do agree with the
number theory comments that Sir Michael
mentioned, particularly theta functions
entering from physics in new ways. I think
other fields of physics will affect mathematics - like statistical mechanics and condensed matter physics. For example, I predict a new subject of statistical topology.
Rather than count the number of holes,
Betti-numbers, etc., one will be more interested in the distribution of such objects on
noncompact manifolds as one goes out to
infinity. We already have precursors in the
number of zeros and poles for holomorphic
functions. The theory that we have for
holomorphic functions will be generalized,
and insights will come from condensed
matter physics as to what, statistically, the
topology might look like as one approaches infinity.
Mathematics has become so specialized, it
seems, that one may fear that the subject
will break up into separate areas. Is there
Continuity of mathematics
INTERVIEW
EMS September 2004 27
Isadore Singer and Sir Michael Atiyah receive the
Abelprize from King Harald. (Photo: Scanpix)
a core holding things together?
ATIYAH I like to think there is a core
holding things together, and that the core is
rather what I look at myself; but we tend to
be rather egocentric. The traditional parts
of mathematics, which evolved - geometry,
calculus and algebra - all centre on certain
notions. As mathematics develops, there
are new ideas, which appear to be far from
the centre going off in different directions,
which I perhaps do not know much about.
Sometimes they become rather important
for the whole nature of the mathematical
enterprise. It is a bit dangerous to restrict
the definition to just whatever you happen
to understand yourself or think about. For
example, there are parts of mathematics
that are very combinatorial. Sometimes
they are very closely related to the continuous setting, and that is very good: we have
interesting links between combinatorics
and algebraic geometry and so on. They
may also be related to e.g. statistics. I think
that mathematics is very difficult to constrain; there are also all sorts of new applications in different directions.
It is nice to think of mathematics having
a unity; however, you do not want it to be
a straitjacket. The centre of gravity may
change with time. It is not necessarily a
fixed rigid object in that sense, I think it
should develop and grow. I like to think of
mathematics having a core, but I do not
want it to be rigidly defined so that it
excludes things which might be interesting.
You do not want to exclude somebody who
has made a discovery saying: “You are outside, you are not doing mathematics, you
are playing around”. You never know!
That particular discovery might be the
mathematics of the next century; you have
got to be careful. Very often, when new
ideas come in, they are regarded as being a
bit odd, not really central, because they
look too abstract.
SINGER Countries differ in their attitudes
about the degree of specialization in mathematics and how to treat the problem of too
much specialization. In the United States I
observe a trend towards early specialization driven by economic considerations.
You must show early promise to get good
letters of recommendations to get good first
jobs. You can’t afford to branch out until
you have established yourself and have a
secure position. The realities of life force a
narrowness in perspective that is not inherent to mathematics. We can counter too
much specialization with new resources
that would give young people more freedom than they presently have, freedom to
explore mathematics more broadly, or to
explore connections with other subjects,
like biology these days where there is lots
to be discovered.
When I was young the job market was
good. It was important to be at a major university but you could still prosper at a
smaller one. I am distressed by the coercive effect of today’s job market. Young
mathematicians should have the freedom
of choice we had when we were young.
The next question concerns the continuity
of mathematics. Rephrasing slightly a
question that you, Prof. Atiyah are the origin of, let us make the following
gedanken experiment: If, say, Newton or
Gauss or Abel were to reappear in our
midst, do you think they would understand the problems being tackled by the
present generation of mathematicians –
after they had been given a short refresher course? Or is present day mathematics
too far removed from traditional mathematics?
ATIYAH The point that I was trying to
make there was that really important
progress in mathematics is somewhat independent of technical jargon. Important
ideas can be explained to a really good
mathematician, like Newton or Gauss or
Abel, in conceptual terms. They are in fact
coordinate-free, more than that, technology-free and in a sense jargon-free. You
don’t have to talk of ideals, modules or
whatever – you can talk in the common
language of scientists and mathematicians.
The really important progress mathematics
has made within 200 years could easily be
understood by people like Gauss and
Newton and Abel. Only a small refresher
course where they were told a few terms –
and then they would immediately understand.
Actually, my pet aversion is that many
mathematicians use too many technical
terms when they write and talk. They were
trained in a way that if you do not say it 100
percent correctly, like lawyers, you will be
taken to court. Every statement has to be
fully precise and correct. When talking to
other people or scientists, I like to use
words that are common to the scientific
community, not necessarily just to mathematicians. And that is very often possible.
If you explain ideas without a vast amount
of technical jargon and formalism, I am
sure it would not take Newton, Gauss and
Abel long – they were bright guys, actually!
SINGER One of my teachers at Chicago
was André Weil, and I remember his saying: “If Riemann were here, I would put
him in the library for a week, and when he
came out he would tell us what to do next.”
Next topic: Communication of mathematics: Hilbert, in his famous speech at
the International Congress in 1900, in
order to make a point about mathematical communication, cited a French
mathematician who said: “A mathematical theory is not to be considered complete until you have made it so clear that
you can explain it to the first man whom
you meet on the street”. In order to pass
on to new generations of mathematicians the collective knowledge of the previous generation, how important is it
that the results have simple and elegant
proofs?
ATIYAH The passing of mathematics on
to subsequent generations is essential for
the future, and this is only possible if
every generation of mathematicians
understands what they are doing and distils it out in such a form that it is easily
understood by the next generation. Many
complicated things get simple when you
have the right point of view. The first
proof of something may be very complicated, but when you understand it well,
you readdress it, and eventually you can
present it in a way that makes it look
much more understandable – and that’s
the way you pass it on to the next generation! Without that, we could never make
progress - we would have all this messy
stuff. Mathematics does depend on a sufficiently good grasp, on understanding of
the fundamentals so that we can pass it on
in as simple a way as possible to our successors. That has been done remarkably
successfully for centuries. Otherwise,
how could we possibly be where we are?
In the 19th century, people said: “There is
so much mathematics, how could anyone
make any progress?” Well, we have - we
do it by various devices, we generalize,
we put all things together, we unify by
new ideas, we simplify lots of the constructions – we are very successful in
mathematics and have been so for several
hundred years. There is no evidence that
this has stopped: in every new generation,
there are mathematicians who make enormous progress. How do they learn it all?
It must be because we have been successful communicating it.
SINGER I find it disconcerting speaking
to some of my young colleagues, because
they have absorbed, reorganized, and simplified a great deal of known material into
a new language, much of which I don’t
understand. Often I’ll finally say, “Oh; is
that all you meant?” Their new conceptuCommunication of mathematics
INTERVIEW
28 EMS September 2004
al framework allows them to encompass
succinctly considerably more than I can
express with mine. Though impressed
with the progress, I must confess impatience because it takes me so long to
understand what is really being said.
Has the time passed when deep and
important theorems in mathematics can
be given short proofs? In the past, there
are many such examples, e.g., Abel’s
one-page proof of the addition theorem
of algebraic differentials or Goursat’s
proof of Cauchy’s integral theorem.
ATIYAH I do not think that at all! Of
course, that depends on what foundations
you are allowed to start from. If we have
to start from the axioms of mathematics,
then every proof will be very long. The
common framework at any given time is
constantly advancing; we are already at a
high platform. If we are allowed to start
within that framework, then at every stage
there are short proofs.
One example from my own life is this
famous problem about vector fields on
spheres solved by Frank Adams where the
proof took many hundreds of pages. One
day I discovered how to write a proof on
a postcard. I sent it over to Frank Adams
and we wrote a little paper which then
would fit on a bigger postcard. But of
course that used some K-theory; not that
complicated in itself. You are always
building on a higher platform; you have
always got more tools at your disposal
that are part of the lingua franca which
you can use. In the old days you had a
smaller base: If you make a simple proof
nowadays, then you are allowed to
assume that people know what group theory is, you are allowed to talk about
Hilbert space. Hilbert space took a long
time to develop, so we have got a much
bigger vocabulary, and with that we can
write more poetry.
SINGER Often enough one can distil the
ideas in a complicated proof and make
that part of a new language. The new
proof becomes simpler and more illuminating. For clarity and logic, parts of the
original proof have been set aside and discussed separately.
ATIYAH Take your example of Abel’s
Paris memoir: His contemporaries did not
find it at all easy. It laid the foundation of
the theory. Only later on, in the light of
that theory, we can all say: “Ah, what a
beautifully simple proof!” At the time, all
the ideas had to be developed, and they
were hidden, and most people could not
read that paper. It was very, very far from
appearing easy for his contemporaries.
I heard you, Prof. Atiyah, mention that one
reason for your choice of mathematics for
your career was that it is not necessary to
remember a lot of facts by heart.
Nevertheless, a lot of threads have to be
woven together when new ideas are developed. Could you tell us how you work best,
how do new ideas arrive?
ATIYAH My fundamental approach to
doing research is always to ask questions.
You ask “Why is this true?” when there is
something mysterious or if a proof seems
very complicated. I used to say – as a kind
of joke – that the best ideas come to you during a bad lecture. If somebody gives a terrible lecture, it may be a beautiful result but
with terrible proofs, you spend your time
trying to find better ones, you do not listen
to the lecture. It is all about asking questions – you simply have to have an inquisitive mind! Out of ten questions, nine will
lead nowhere, and one leads to something
productive. You constantly have to be
inquisitive and be prepared to go in any
direction. If you go in new directions, then
you have to learn new material.
Usually, if you ask a question or decide to
solve a problem, it has a background. If you
understand where a problem comes from
then it makes it easy for you to understand
the tools that have to be used on it. You
immediately interpret them in terms of your
own context. When I was a student, I
learned things by going to lectures and reading books – after that I read very few books.
I would talk with people; I would learn the
essence of analysis by talking to Hörmander
or other people. I would be asking questions because I was interested in a particular
problem. So you learn new things because
you connect them and relate them to old
ones, and in that way you can start to spread
around.
If you come with a problem, and you need
to move to a new area for its solution, then
you have an introduction – you have already
a point of view. Interacting with other people is of course essential: if you move into a
new field, you have to learn the language,
you talk with experts; they will distil the
essentials out of their experience. I did not
learn all the things from the bottom
upwards; I went to the top and got the
insight into how you think about analysis or
whatever.
SINGER I seem to have some built-in sense
of how things should be in mathematics. At
a lecture, or reading a paper, or during a discussion, I frequently think, “that’s not the
way it is supposed to be.” But when I try
Individual work style
out my ideas, I’m wrong 99% of the time. I
learn from that and from studying the ideas,
techniques, and procedures of successful
methods. My stubbornness wastes lots of
time and energy. But on the rare occasion
when my internal sense of mathematics is
right, I’ve done something different.
Both of you have passed ordinary retirement age several years ago. But you are
still very active mathematicians, and you
have even chosen retirement or visiting
positions remote from your original work
places. What are the driving forces for
keeping up your work? Is it wrong that
mathematics is a “young man’s game” as
Hardy put it?
ATIYAH It is no doubt true that mathematics is a young man’s game in the sense that
you peak in your twenties or thirties in terms
of intellectual concentration and in originality. But later you compensate that by experience and other factors. It is also true that
if you haven’t done anything significant by
the time you are forty, you will not do so
suddenly. But it is wrong that you have to
decline, you can carry on, and if you manage to diversify in different fields this gives
you a broad coverage. The kind of mathematician who has difficulty maintaining the
momentum all his life is a person who
decides to work in a very narrow field with
great depths, who e.g. spends all his life trying to solve the Poincaré conjecture –
whether you succeed or not, after 10-15
years in this field you exhaust your mind;
and then, it may be too late to diversify. If
you are the sort of person that chooses to
make restrictions to yourself, to specialize in
a field, you will find it harder and harder –
because the only things that are left are harder and harder technical problems in your
own area, and then the younger people are
better than you.
You need a broad base, from which you
can evolve. When this area dries out, then
you go to that area – or when the field as a
whole, internationally, changes gear, you
can change too. The length of the time you
can go on being active within mathematics
very much depends on the width of your
coverage. You might have contributions to
make in terms of perspective, breadth, interactions. A broad coverage is the secret of a
happy and successful long life in mathematical terms. I cannot think of any counter
example.
SINGER I became a graduate student at the
University of Chicago after three years in
the US army during World War II. I was
older and far behind in mathematics. So I
was shocked when my fellow graduate students said, “If you haven’t proved the
INTERVIEW
EMS September 2004 29
Riemann Hypothesis by age thirty, you
might as well commit suicide.” How infantile! Age means little to me. What keeps me
going is the excitement of what I’m doing
and its possibilities. I constantly check [and
collaborate!] with younger colleagues to be
sure that I’m not deluding myself – that
what we are doing is interesting. So I’m
happily active in mathematics. Another reason is, in a way, a joke. String Theory needs
us! String Theory needs new ideas. Where
will they come from, if not from Sir Michael
and me?
ATIYAH Well, we have some students…
SINGER Anyway, I am very excited about
the interface of geometry and physics, and
delighted to be able to work at that frontier.
You, Prof. Atiyah, have been very much
involved in the establishment of the
European Mathematical Society around
1990. Are you satisfied with its development since then?
ATIYAH Let me just comment a little on
my involvement. It started an awful long
time ago, probably about 30 years ago.
When I started trying to get people interested in forming a European Mathematical
Society in the same spirit as the European
Physical Society, I thought it would be easy.
I got mathematicians from different countries together and it was like a mini-UN: the
French and the Germans wouldn’t agree; we
spent years arguing about differences, and –
unlike in the real UN – where eventually at
the end of the day you are dealing with real
problems of the world and you have to come
to an agreement sometime; in mathematics,
it was not absolutely essential. We went on
for probably 15 years, before we founded
the EMS.
On the one hand, mathematicians have
much more in common than politicians, we
are international in our mathematical life, it
is easy to talk to colleagues from other
History of the EMS
countries; on the other hand, mathematicians are much more argumentative. When
it comes to the fine details of a constitution,
then they are terrible; they are worse than
lawyers. But eventually – in principle – the
good will was there for collaboration.
Fortunately, the timing was right. In the
meantime, Europe had solved some of its
other problems: the Berlin Wall had come
down – so suddenly there was a new Europe
to be involved in the EMS. This very fact
made it possible to get a lot more people
interested in it. It gave an opportunity for a
broader base of the EMS with more opportunities and also relations to the European
Commission and so on.
Having been involved with the set-up, I
withdrew and left it to others to carry on. I
have not followed in detail what has been
happening except that it seems to be active.
I get my Newsletter, and I see what is going
on.
Roughly at the same time as the collapse
of the Berlin Wall, mathematicians in general – both in Europe and in the USA –
began to be more aware of their need to be
socially involved and that mathematics had
an important role to play in society. Instead
of being shut up in their universities doing
just their mathematics, they felt that there
was some pressure to get out and get
involved in education, etc. The EMS took
on this role at a European level, and the
EMS congresses – I was involved in the one
in Barcelona – definitely made an attempt to
interact with the public. I think that these
are additional opportunities over and above
the old-fashioned role of learned societies.
There are a lot of opportunities both in terms
of the geography of Europe and in terms of
the broader reach.
Europe is getting ever larger: when we
started we had discussions about where
were the borders of Europe. We met people
from Georgia, who told us very clearly, that
the boundary of Europe is this river on the
other side of Georgia; they were very keen
to make sure that Georgia is part of Europe.
Now, the politicians have to decide where
the borders of Europe are.
It is good that the EMS exists; but you
should think rather broadly about how it is
evolving as Europe evolves, as the world
evolves, as mathematics evolves. What
should its function be? How should it relate
to national societies? How should it relate
to the AMS? How should it relate to the
governmental bodies? It is an opportunity!
It has a role to play!
Could you tell us in a few words about your
main interests besides mathematics?
SINGER I love to play tennis, and I try to
do so 2-3 times a week. That refreshes me
and I think that it has helped me work hard
in mathematics all these years.
ATIYAH Well, I do not have his energy! I
like to walk in the hills, the Scottish hills – I
have retired partly to Scotland. In
Cambridge, where I was before, the highest
hill was about this (gesture) big. Of course
you have got even bigger ones in Norway. I
spent a lot of my time outdoors and I like to
plant trees, I like nature. I believe that if you
do mathematics, you need a good relaxation
which is not intellectual – being outside in
the open air, climbing a mountain, working
in your garden. But you actually do mathematics meanwhile. While you go for a long
walk in the hills or you work in your garden
– the ideas can still carry on. My wife complains, because when I walk she knows I am
thinking of mathematics.
SINGER I can assure you, tennis does not
allow that!
Thank you very much on behalf of the
Norwegian, the Danish, and the European
Mathematical Societies!
The interviewers were Martin Raussen,
Aalborg University, Denmark, and
Christian Skau, Norwegian University of
Science and Technology, Trondheim,
Norway.
1 More details were given in the laureates’
lectures.
2 Among those: Newton, Gauss, Cauchy,
Laplace, Abel, Jacobi, Riemann,
Weierstrass, Lie, Picard, Poincaré,
Castelnuovo, Enriques, Severi, Hilbert,
Lefschetz, Hodge, Todd, Leray, Cartan,
Serre, Kodaira, Spencer, Dirac,
Pontrjagin, Chern, Weil, Borel,
Hirzebruch, Bott, Eilenberg,
Grothendieck, Hörmander, Nirenberg.
3 Comm. Pure App. Math. 13(1), 1960
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