Promenade 10
The New Geometry: or the Marriage of Number and Magnitude.
But here I am, digressing again! I set out to talk about the "master-themes", with the intention of unifying them under one "mother vision", like so many rivers returning to the Ocean whose children they are..
This great unifying vision might be described as a new geometry. It appears to be similar to the one that Kronecker dreamed of a century ago(*)
(*) I only know about "Kronecker's Dream" through hearsay, in fact it was when somebody ( I believe it was John Tate) told me that I was about to carry it out. In the education which I received from my elders, the historical references were very rare indeed. I was nourished, not by reading the works of others, ancient or modern, but above all through communication, through conversations or exchanges of letters, with other mathematicians, beginning with my teachers. The principal, perhaps the only external inspiration for the sudden and vigorous emergence of the theory of schemes in 1958, was the article by Serre commonly known by its label FAC (Faisceaux algébriques cohérents) that came out a few years earlier. Apart from this, my primary source of inspiration in the development of the theory flowed entirely from itself, and restored itself from one year to the next by the requirements of simplicity and internal coherence, and from my effort at taking into account in this new context, of all that was "commonly known" in algebraic geometry ( which I assimilated bit by bit as it was transformed under my hands), and from all that this "knowledge" suggested to me.
But the reality is ( which a bold dream may sometimes reveal, or encourage us to discover) surpasses in every respect in richness and resonance even the boldest and most profound dream. Of a certainty, for more than one of these revelations of the new geometry, ( if not for all of them), nobody, the day before it appoeared, could have imagined it - neither the worker nor anyone else.
One might say that "Number" is what is appropriate for grasping the structure of "discontinuous" or "discrete" aggregates. These systems, often finite, are formed from "elements", or "objects" conceived of as isolated with respect to one another. "Magnitude" on the other hand is the quality, above all, susceptible to "continuous variation", and is most appropriate for grasping continous structures and phenomena: motion, space, varieties in all their forms, force fields, etc. Thereby , Arithmetic appears to be ( overall) the science of discrete structures while Analysis is the science of continuous structures.
As for Geometry, one can say that in the two thousand years in which it has existed as a science in the modern sense of the word, it has "straddled" these two kinds of structure, "discrete" and "continuous". (*)
(*)In point of fact, it has traditionally been the "continuous" aspect of things which has been the central focus of Geometry, while those properties associated with "discreteness", notably computational and combinatorial properties, have been passed over in silence or treated as an after-thought. It was therefore all the more astonishing to me when I made the discovery, about a dozen years ago, of the combinatorial theory of the Icosahedron, even though this theory is barely scratched (and probably not even understood) in the classic treatise of Felix Klein on the Icosahedron. I see in this another significant indicator of this indifference ( of over 2000 years) of geometers vis-a-vis those discrete structures which present themselves naturally in Geometry: observe that the concept of the group ( notably of symmetries) appeared only in the last century ( introduced by Evariste Galois), in a context that was considered to have nothing to do with Geometry. Even in our own time it is true that there are lots of algebraists who still haven't understood that Galois Theory is primarily, in essence, a
geometrical vision, which was able to renew our understanding of so-called "arithmetical" phenomenon.
For some time in fact one can say that the two geometries considered to be distinct species, the discrete and the continuous, weren't really "divorced". They were rather two different ways of investigating the
same class of geometric objects: one of them accentuated the "discrete" properties ( notably computational and combinatorial) while the other concerned itself with the "continuous" properties ( such as location in an ambient space, or the measurement of "magnitude" in terms of the distances between points, etc.)
It was at the end of the last century that a divorce became immanent, with the arrival and development of what came to be called" Abstract (Algebraic) Geometry". Roughly speaking, this consisted of introducing, for every prime number p, an algebraic geometry "of characteristic p", founded on the model (continous) of the Geometry ( algebraic) inherited from previous centuries, however in a context which appeared to be resolutely "discontinuous", or "discrete". This new class of geometric objects have taken on a growing significance since the beginning of the century, in particular owing to their close connections with arithmetic, which is the science par excellence of discrete structures. This appears to be one of the notions motivating the work of André Weil (**) , perhaps the driving force ( which is usually implicit or tacit in his published work, as it ought to be): the notion that "the" Geometry (algebraic), and in particular the "discrete" geometries associated with various prime numbers, ought to supply the key for a grand revitalization of Arithmetic.
(**)André Weil, a French mathematician who emigrated to the United States, is one of the founding members of the "Bourbaki Group", which is discussed in some length in the first part of Récoltes et Semailles (as is Weil himself from time to time).
It was with this perspective in mind that he announced, in 1949, his famous "Weil conjectures". These utterly astounding conjectures allowed one to envisage, for these new " discrete varieties" ( or "spaces"), the possibility for certain kinds of constructions and arguments(*) which up to that moment did not appear to be conceivable outside of the framework of the only "spaces" considered worthy of attention by analysts - that is to say the so-called "topological" spaces (in which the notion of continuous variation is applicable).
One can say that the new geometry is, above all else, a synthesis between these two worlds, which, though next-door neighbors and in close solidarity, were deemed separate: the arithmeticalworld, wherein one finds the (so-called) spaces without continuity, and the world of continuous magnitudes, "spaces" in the conventional meaning of the word. In this new vision these two worlds, formerly separate, comprise but a single unity.
(*) (For the mathematical reader) The "constructions and arguments" we are refering to are associated with the Cohomology of differentiable and complex varieties, in particular those which imply the Lefschetz fixed point theorems and Hodge Theory.
The embryonic vision of this
Arithmetical Geometry" ( as I propose to designate the new geometry) is to be found in the Weil conjectures. In the development of some of my principal ideas(**) these conjectures were my primary source of inspiration, all through the years between 1958 and 1969.
(**)I refer to four "intermediate" themes (nos. 5 to 8) that is to say , the
topos,
étale and l-adic cohomology,
motives and (to a lesser extent)
crystals. These themes were all developed between 1958 and 1966
Even before me, in fact,
Oscar Zariski on the one hand and
Jean-Pierre Serre on the other had developed, for certain "wild" spaces in "abstract" Algebraic Geometry, some "topological" methods, inspired by those which had formerly been applied to the "well behaved spaces" of normal practice.(***)
(***) (For the mathematical reader) The primary contribution of Zariski in this sense seems to me to be the introduction of the "Zariski topology" ( which later became an essential tool for Serre in FAC), his "principle of connectedness", and what he named the "theory of holomorphic functions" - which in his hands became the theory of formal schemes, and the theorems comparing the formal to the algebraic ( with, as a secondary source of inspiration, the fundamental article by Serre known as GAGA). As for the contribution by Serre to which I've alluded in the text, it is, above all, his introduction into abstract Algebraic Geometry of the methodology of
sheaves, in FAC ( Faisceaux algébriques cohérents )the other fundamental paper already mentioned.
In the light of these 'reminiscences", when asked to name the immediate "ancestors" of the new geometric vision, the names that come to me right away are are Oscar Zariski, André Weil, Jean Leray and Jean-Pierre Serre.
Serre had a special role apart from all the others because of the fact that it was largely through him that I not only learned of his ideas, but also those of Zariski, Weil and Leray which were to play an important role in the emergence and development of the ideas of the new geometry.
Their ideas, without a doubt, had played an important part from my very first steps towards the building of the new geometry: furthermore, it's true, as points of departure and as tools ( which I had to reshape virtually from scratch in order to adapt them to a larger context) , and a sources of inspiration which would continue to nourish my projects and dreams over the course of months and years. In any case, it's self-evident that , even in their recast state, these tools were insufficient for what was needed in making even the first steps in the direction of Weil's marvellous conjectures.
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