Having aleady OCRed Borel's article on Weil (see Theo's reply), I cannot resist posting a larger excerpt here, since it provides much further context that I suspect will help readers to better appreciate Weil's "search for elegance, beauty and hidden harmonies". Perhaps it will help motivate some readers to join in such fruitful endeavors. I too strongly endorse the literatured cited by Theo. It can prove highly inspirational to budding mathematical minds.
Edit: I just noticed that the AMS has a nicer typeset version of Borel's article from 2009 BAMS.
His output offers an extraordinary combination of foundational work, to secure a solid
basis in some area, of often decisive contributions at the cutting edge, solving old or new
problems, and of forays into unknown territory, in the form of problems or conjectures,
guided by a seemingly infallible sense for the directions into which one should forge ahead.
Of course, I feel quite uncomfortable in making such a statement without backing it up
in any way, so allow me to turn to the mathematicians to give an idea of these facets of his
output in at least one area, algebraic geometry. The theorem he had proved in 1940 (see
above) relied on some facts of algebraic geometry for some of which there was no solid
reference. Moreover, the development of algebraic geometry, from "classical" (i.e. projective
or affine complex varieties) to "abstract" (varieties over arbitrary fields), was also crying out
for reliable foundations. It took him several years to supply them in a massive (and rather
arid) treatise "Foundations of algebraic geometry" (1946), the only comprehensive basis for
algebraic geometry for a number of years. Although dealing with a very general "abstract
situation", he developed it in part in analogy with the theory of differentiable manifolds in
differential geometry, and also with some constructions in algebraic topology. It was
followed, among other items, by a monograph proving in full his 1940 result, by foundations
for abelian varieties, fibre bundles in algebraic geometry, algebraic groups, the advocacy of
the use of analytic fibre bundles in several complex variables, and in 1949, in a short Note,
by a series of conjectures (soon called the Weil conjectures) which were to have an enormous
impact on algebraic geometry. In particular, he postulated the existence of a cohomology
theory in this set up, with properties allowing one to transcribe known arguments in algebraic
topology, such as the Lefschetz fixed point theorem, a bold idea, unique to him, way ahead of
its time. It was implemented some ten years later by A. Grothendieck (etale cohomology),
and it took twenty-five years before Deligne proved the last, and by far hardest, of these
conjectures, with far reaching consequences, not yet exhausted.
So far, I have said little of what has arguably been Weil's most abiding interest in
mathematics: "Zeta functions". The first one was used by B. Riemann in 1857 to study the
distribution of prime numbers among positive integers. The "Riemann hypothesis" about the
zeroes of this function is still unproved and generally viewed as the Holy Grail of
mathematics. The introduction of this function to study the discrete (the integers) in a
continuous framework (real or complex numbers) was quite revolutionary and proved to be
immensely fruitful. Zeta functions, with corresponding Riemann hypotheses, have proliferated
in analysis, algebraic geometry and number theory, and have always been on Weil's mind.
(His 1940 theorem dealt with one kind and his 1949 conjectures with generalizations of it.)
He was convinced that the problem of the Riemann hypothesis, even in the original case, had
to be attacked broadly. How broadly can be only explained in mathematical terms of course,
but he drew an analogy with the Rosetta Stone, which seems to me so typical of his thought
processes and of the aesthetic component in his approach to mathematics that I cannot resist
trying to give an idea of it, as imprecise as it has to be. It is developed in a short article: De
la metaphysique aux mathematiques, (From metaphysics to mathematics), Science 1960, 52-56;
Collected Papers II, 406-412.
"Metaphysics", he explains, is meant here in the sense of the 18'th century
mathematicians, when they spoke of, say, "the metaphysics of the theory of equations":
"... a collection of vague analogies, difficult to grasp and difficult to formulate,
which nevertheless appeared to them to play an important role at certain
moments in the research and discovery in mathematics".
and then he elaborates.
"Nothing is more fecund, all the mathematicians know it, than those obscure
analogies, the blurred reflections from one theory to another ... nothing gives
more pleasure to the researcher. One day the illusion drifts away, the
premonition changes to a certitude: the twin theories reveal their common
source before disappearing; as the Gita teaches it, knowledge and indifference
are reached at the same time. The metaphysics has become mathematics, ready
to form the subject matter of a treatise, the cold beauty of which cannot move
"Fortunately for researchers, as the fogs clear away on some point, they
reappear on another. A major part of the Tokyo Colloquium  was
devoted to the analogies between number theory and the theory of algebraic
functions. There we are still fully in metaphysics..."
"Algebraic functions" alludes here to a theory built up by Riemann by analytical,
transcendental means. To link it to number theory, guided by "obscure analogies", is a
problem which had fascinated Weil early on (as already hinted by the title of his Thesis), and
he felt that progress was still scant by 1960. Meanwhile, a third topic had appeared:
"algebraic curves over finite fields" (the subject matter of his 1940 theorem), which was
easier to relate to the other two and thus served as an intermediary. These items and many
generalizations or related results formed an enormous amount of mathematics naturally
divided into three parts, each with its own framework, (in brief, transcendental, arithmetic and
algebraico-geometric) and techniques. As Weil puts it, we are faced with a text in three parts
(he calls them columns), each written in its own language, called by him Riemannian,
arithmetic and Italian respectively, in analogy with the Rosetta Stone. However, there is an
huge difference: the latter contains the same text in the three languages (or rather, assuming
this, Champollion was able to decipher Egyptian hieroglyphic writing), while we have here
only in each column fragments of what is hoped to be similar texts, once completed.
The task of the mathematicians, then, is to add translations of a given fragment into
the other columns, to transform those obscure analogies into mathematics, and eventually
build a dictionary which would allow one to pass from one column to the others. If it were
sufficiently complete, then the Riemann hypothesis would be proved, Weil
concludes, wondering how long mathematics will have to wait for a Champollion.
As an illustration of his outlook, let me mention a paper (, p. 249-64, in his
Collected Papers III), where he formulates a statement in "Riemannian" language, the truth of
which would imply that of the Riemann hypothesis (for many zeta functions), points out that
it has an analogue in "Italian" which, in view of his earlier work, is a proven theorem, and
comments that this provides for him, perhaps, the strongest evidence in favor of the original
Riemann hypothesis, one of many examples of his unshakable belief in the unity and harmony
Weil was indeed fluent in the three languages and many of his works can be
interpreted as contributions to the dictionary, but not all, though. In particular, as befits a
man with his cultural interests, he had a strong commitment to the history of mathematics,
which culminated in a history of number theory from 1800 B.C. to 1800 A.D. (from
Hammurapi to Legendre). Much earlier it had been at the origin of the Historical Notes in
Bourbaki, to which he was a main contributor until he retired.
As a mathematician, his work shows him to be at the same time an architect, a builder
and a poet: an architect for fostering a global view of mathematics and striving to display its
fundamental unity, a builder by his specific, often decisive, contributions to a great variety of
topics and a poet by his search for elegance, beauty and hidden harmonies.
School of Mathematics
Institute for Advanced Study
Princeton, NJ 08540