Schematic or Arithmetic Viewpoints for regular polyhedra and in general all regular configurations.
Apart from the themes in item 12, a goodly portion of which first appeared in my thesis of 1953 and was further developed in the period in which I worked in functional analysis between 1950 and 1955, the other eleven themes were discovered and developed during my geometric period, starting in 1955
To appreciate my work as a mathematician, to "sense it", is to appreciate and to sense, as best one can a certain number of its ideas, together with the grand themes they introduce which form the framework and the soul of the work.
In the nature of things, some of these ideas are "grander than others"!, others "smaller". In other words, among these new and original themes, some have a larger scope, while others delve more deeply into the mysteries of mathematical verities. (**).
(**)To give some examples, the idea of greatest scope appears to me to be that of the topos, because it suggests the possibility of a synthesis of algebraic geometry, topology and arithmetic. The most important by virtue of the reach of those developments which have followed from it is, at the present moment, the schema. (With respect to this subject see the footnotes from to the previous section(#7)) It is this theme which supplies the framework, par excellence, of 8 of the others in the above list. (that is to say, all the others except 1,5 and 10), which at the same time furnishing the central notion fundamental to a total reformation, from top to bottom, of algebraic geometry and of the language of that subject.
At the other extreme, the first and last of these 12 themes are of much less significance. However, vis-a-vis the last one, having introduced a new way of looking at the very ancient topic of the regular polyhedra and regular configurations in general, I am not sure that a mathematician who gives his whole life to studying them will have wasted his time. As for the first of these themes, topological tensor products, it has played the role of a handy tool, rather than as the springboard for future developments deriving from it. Even so I've heard , particularly in recent years, sporadic echoes of research resolving ( 20 or 30 years later!) some of the issues that were left open by my discoveries.
Among the 12 themes, the deepestis that of the motifs, which are closely tied to those of an-Abelian Algebraic Geometry, and that of Galois-Teichmüller Yoga .
In terms of the effectiveness of the tools I've created, laboriously polished and brought to perfection, now heavily used in certain "specialized research areas" in the last 2 decades, I would single out schemas and étale and l-adic cohomologies.
For the well-informed mathematician I would claim that, up to the present moment, it can scarcely be doubted that these schematic tools. such as l-adic cohomology, etc., figure among the greatest achievements of this century, and will continue to nourish and revitalize our science in all following generations.
Among these grand ideas one finds 3 (and hardly the least among them) which, having appeared only after my departure from the world of mathematics, are still in a fairly embryonic state: they don't even exist "officially" , since they haven't appeared in any publication, (which one might consider the equivalent of a birth certificate)(*) .
(*)The only "semi-official" text in which these three themes are sketched, more or less, is the Outline for a Programme, edited in January 1984 on request from a unit of CNRS. This text (which is also discussed in section 3 of the
Introduction, "Compass and Luggage"), should be, in principle, included in volume 4 of Mathematical Reflections.
The twelve principal themes of my opus aren't isolated from each other. To my eyes they form a unity, both in spirit and in their implications, in that one finds in them a single persistent tone, present in both "officially published" and "unpublished" writings. Indeed, even in the act of writing these lines I seem to recapture that same tone- like a call! - persisting through 3 years of "unrewarded" work, in dedicated isolation, at a time when it mattered little to me that there were other mathematicians in the world besides myself, so taken was I by the fascination of what I was doing...
This unity does not derive alone as the trademark of a single worker. The themes are interconnected by innumerable ties, both subtle and obvious, as one sees in the interconnection of differing themes, each recognizable in its individuality, which unfold and develop in a grand musical counterpoint- in the harmony that assembles them together, carries them forward and assigns meaning to all of them, a movement and wholeness in which all are participants. Each of these partial themes seems to have been born out of an all-engulfing harmony and to be reborn from one instant to the next, while at the same time this harmony does not appear as a mere "sum" or "resultant" of all the themes that make it up, that in some sense are pre-existent within it. And, to speak truly, I cannot avoid the feeling ( cranky as it must appear) , that in some sense it is actually this "harmony", not yet present but which already "exists' somewhere in the dark womb of things awaiting birth in their time - that it is this and this alone which has inspired, each in its turn, these themes which acquire meaning only through it. And it is that harmony which called out to me in a low and impatient voice , in those solitary and inspired years of my emergence from adolescence ....
It remains true that these 12 master-themes of my work appear, as through a kind of secret predestination, to abide concurrently within the same symphony - or, to use a different image, each incarnates a different "perspective" on the same immense vision.
This vision did not begin to emerge from the shades, or take recognizable shape, until around the years 1957, 1958 - years of enormous personal growth. (*)
*1957 was the year in which I began to develop the theme "RIemann-Roch" (Grothendieck version) - which almost overnight made me into a big "movie star". It was also the year of my mother's death and thereby the inception of a great break in my life story. They figure among the most intensely creative years of my entire life, not only in mathematics. I'd worked almost exclusively in mathematics for 12 years. In that year there was the sense that I'd perhaps done what there was to do in mathematics and that it was time to try something else. This came out of an interior need for revitalization, perhaps for the first time in my life. At that time I imagined that I might want to be a writer, and for a period of several months I stopped doing mathematics altogether. Finally however I decided to return, just long enough to give a definitive form to the mathematical works I'd already done, something I imagined would take only a few months, perhaps a year at most ...
The time wasn't ripe, apparently, for a complete break. What is certain is that in taking up my work in mathematics again, it took possession of me, and didn't let go of me for another 12 years!
The following year (1958) is probably the most fertile of all my years as a mathematician. This was the year which saw the birth of the two central themes of the new geometry through the launching of the theory of schemes ( the subject of my paper at the International Congress of Mathematicians at Edinborough in the summer of that year) and the appearamce of the concept of a "site", a provisional technical form of the crucial notion of the topos. With a perspective of thirty years I can say now that this was the year in which the very conception of a new geometry was born in the wake of these two master-tools: schemes ( metamorphosed from the anterior notion of the "algebraic variety") and the topos ( a metamorphoses, even deeper, of the idea of space).
It may appear strange, but this vision is so close to me and appeared so "self-evident", that it never occured to me until about a year ago to give a name to it. (*)
*It first occured to me to name this vision in the meditation of December 4th, 1984, (in subnote #136-1) to the footnote "Yin the Servant"(2) - or Generosity" ( Récoltes et Semailles, pg. 637)
(Although it is certainly one of my passions to be constantly giving names to things that I've discovered as the best way to keep them in mind ...) It is true that I can't identify a particular moment at which this vision appeared, or which I can reconstruct through recollection. A new vision of things is something so immense that one probably can't pin it down to a specific moment, rather it takes possession of one over many years, if not over several generations of those persons who examine and contemplate it. It is as if new eyes have to be painfully fashioned from behind the eyes which, bit by bit, they are desitned to replace. And this vision is also too immense for one to speak of "grasping " it, in the same way that one "grasps" an idea that happens to arise along the way. That's no doubt why one shouldn't be surprise that the idea of giving a name to something so enormous, so close yet so diffuse, only occured to me in recollection, and then only after it had reached its full maturity.
In point of fact, for the next two years my relationship to mathematics was restricted ( apart from teaching it) to just getting it done- to giving scope to a powerful impulse that ceaselesly drew me forward, into an "unknown" that I found endlessly fascinating. The idea didn't occur to me to pause, even for the space of an instant, to turn back and get an overview of the path already followed, let alone place it in the context of an evolving work. (Either for the purpose of placing it in my life, as something that continued to attach me to profound and long neglected matters; or to situate it in that collective adventure known as "Mathematics")
What must appear even more strange, in order to get me to stop for a moment and re-establish acquaintance with these half-forgotten efforts, (or to think of giving a name to the vision which is its heart and soul), I had to face a confrontation with a "Burial" of gigantic proportions: with the burial, by silence and derision, of that vision and of the worker who conceived it ...
Promenade8
Promenade 8
The Vision-or 12 Themes for a Harmonization
Perhaps one might say that a "Great Idea" is simply the kind of viewpoint which not only turns out to be original and productive, but one which introduces into a science an extraordinary and new theme. Every science, once it is treated not as an instrument for gaining dominion and power, but as part of the adventure of knowledge of our species through the ages, may be nothing but that harmony, more or less rich, more or less grand depending on the times, which unfolds over generations and centuries through the delicate counterpoint of each of its themes as they appear one by one, as if summoned forth from the void to join up and intermingle with each other.
Among the numerous original viewpoints which I've uncovered in Mathematics I find twelve which, upon reflection, I would call "Great Ideas".(*)
(*) For the sake of the mathematical reader, here is the list of these 12 master ideas, or "master-themes" of my work, in chronological order:
- Topological Tensor Products and Nuclear Spaces
- "Continuous" and "Discrete" dualities ( Derived Categories, the "6 operations")
- The Riemann-Roch-Grothendieck Yoga ( K-Theory and its relationship to Intersection Theory)
- Schemes
- Topos Theory
- Etale Cohomology and l-adic Cohomology
- Motives, Motivic Galois Groups (*-Grothendieck categories)
- Crystals, Crystalline Cohomology, yoga of the DeRham coefficients, the Hodge coefficients
- "Topological Algebra":(infinity)-stacks;derivations;cohomological formalism of topoi, insipiring a new conception of homotopy.
- Mediated topology
- The yoga of un-Abelian Algebraic Geometry. Galois-Teichmüller Theory
- Schematic or Arithmetic Viewpoints for regular polyhedra and in general all regular configurations.
Apart from the themes in item 1, a goodly portion of which first appeared in my thesis of 1953 and was further developed in the period in which I worked in functional analysis between 1950 and 1955, the other eleven themes were discovered and developed during my geometric period, starting in 1955
To appreciate my work as a mathematician, to "sense it", is to appreciate and to sense, as best one can a certain number of its ideas, together with the grand themes they introduce which form the framework and the soul of the work.
In the nature of things, some of these ideas are "grander than others"!, others "smaller". In other words, among these new and original themes, some have a larger scope, while others delve more deeply into the mysteries of mathematical verities. (**).
(**)To give some examples, the idea of greatest scope appears to me to be that of the topos, because it suggests the possibility of a synthesis of algebraic geometry, topology and arithmetic. The most important by virtue of the reach of those developments which have followed from it is, at the present moment, the schema. (With respect to this subject see the footnotes from to the previous section(#7)) It is this theme which supplies the framework, par excellence, of 8 of the others in the above list. (that is to say, all the others except 1,5 and 10), which at the same time furnishing the central notion fundamental to a total reformation, from top to bottom, of algebraic geometry and of the language of that subject.
At the other extreme, the first and last of these 12 themes are of much less significance. However, vis-a-vis the last one, having introduced a new way of looking at the very ancient topic of the regular polyhedra and regular configurations in general, I am not sure that a mathematician who gives his whole life to studying them will have wasted his time. As for the first of these themes, topological tensor products, it has played the role of a handy tool, rather than as the springboard for future developments deriving from it. Even so I've heard , particularly in recent years, sporadic echoes of research resolving ( 20 or 30 years later!) some of the issues that were left open by my discoveries.
Among the 12 themes, the deepestis that of the motifs, which are closely tied to those of an-Abelian Algebraic Geometry, and that of Galois-Teichmüller Yoga .
In terms of the effectiveness of the tools I've created, laboriously polished and brought to perfection, now heavily used in certain "specialized research areas" in the last 2 decades, I would single out schemas and étale and l-adic cohomologies.
For the well-informed mathematician I would claim that, up to the present moment, it can scarcely be doubted that these schematic tools. such as l-adic cohomology, etc., figure among the greatest achievements of this century, and will continue to nourish and revitalize our science in all following generations.
Among these grand ideas one finds 3 (and hardly the least among them) which, having appeared only after my departure from the world of mathematics, are still in a fairly embryonic state: they don't even exist "officially" , since they haven't appeared in any publication, (which one might consider the equivalent of a birth certificate)(*) .
(*)The only "semi-official" text in which these three themes are sketched, more or less, is the Outline for a Programme, edited in January 1984 on request from a unit of CNRS. This text (which is also discussed in section 3 of the
Introduction, "Compass and Luggage"), should be, in principle, included in volume 4 of Mathematical Reflections.
The twelve principal themes of my opus aren't isolated from each other. To my eyes they form a unity, both in spirit and in their implications, in that one finds in them a single persistent tone, present in both "officially published" and "unpublished" writings. Indeed, even in the act of writing these lines I seem to recapture that same tone- like a call! - persisting through 3 years of "unrewarded" work, in dedicated isolation, at a time when it mattered little to me that there were other mathematicians in the world besides myself, so taken was I by the fascination of what I was doing...
This unity does not derive alone as the trademark of a single worker. The themes are interconnected by innumerable ties, both subtle and obvious, as one sees in the interconnection of differing themes, each recognizable in its individuality, which unfold and develop in a grand musical counterpoint- in the harmony that assembles them together, carries them forward and assigns meaning to all of them, a movement and wholeness in which all are participants. Each of these partial themes seems to have been born out of an all-engulfing harmony and to be reborn from one instant to the next, while at the same time this harmony does not appear as a mere "sum" or "resultant" of all the themes that make it up, that in some sense are pre-existent within it. And, to speak truly, I cannot avoid the feeling ( cranky as it must appear) , that in some sense it is actually this "harmony", not yet present but which already "exists' somewhere in the dark womb of things awaiting birth in their time - that it is this and this alone which has inspired, each in its turn, these themes which acquire meaning only through it. And it is that harmony which called out to me in a low and impatient voice , in those solitary and inspired years of my emergence from adolescence ....
It remains true that these 12 master-themes of my work appear, as through a kind of secret predestination, to abide concurrently within the same symphony - or, to use a different image, each incarnates a different "perspective" on the same immense vision.
This vision did not begin to emerge from the shades, or take recognizable shape, until around the years 1957, 1958 - years of enormous personal growth. (*)
*1957 was the year in which I began to develop the theme "RIemann-Roch" (Grothendieck version) - which almost overnight made me into a big "movie star". It was also the year of my mother's death and thereby the inception of a great break in my life story. They figure among the most intensely creative years of my entire life, not only in mathematics. I'd worked almost exclusively in mathematics for 12 years. In that year there was the sense that I'd perhaps done what there was to do in mathematics and that it was time to try something else. This came out of an interior need for revitalization, perhaps for the first time in my life. At that time I imagined that I might want to be a writer, and for a period of several months I stopped doing mathematics altogether. Finally however I decided to return, just long enough to give a definitive form to the mathematical works I'd already done, something I imagined would take only a few months, perhaps a year at most ...
The time wasn't ripe, apparently, for a complete break. What is certain is that in taking up my work in mathematics again, it took possession of me, and didn't let go of me for another 12 years!
The following year (1958) is probably the most fertile of all my years as a mathematician. This was the year which saw the birth of the two central themes of the new geometry through the launching of the theory of schemes ( the subject of my paper at the International Congress of Mathematicians at Edinborough in the summer of that year) and the appearamce of the concept of a "site", a provisional technical form of the crucial notion of the topos. With a perspective of thirty years I can say now that this was the year in which the very conception of a new geometry was born in the wake of these two master-tools: schemes ( metamorphosed from the anterior notion of the "algebraic variety") and the topos ( a metamorphoses, even deeper, of the idea of space).
It may appear strange, but this vision is so close to me and appeared so "self-evident", that it never occured to me until about a year ago to give a name to it. (*)
*It first occured to me to name this vision in the meditation of December 4th, 1984, (in subnote #136-1) to the footnote "Yin the Servant"(2) - or Generosity" ( Récoltes et Semailles, pg. 637)
(Although it is certainly one of my passions to be constantly giving names to things that I've discovered as the best way to keep them in mind ...) It is true that I can't identify a particular moment at which this vision appeared, or which I can reconstruct through recollection. A new vision of things is something so immense that one probably can't pin it down to a specific moment, rather it takes possession of one over many years, if not over several generations of those persons who examine and contemplate it. It is as if new eyes have to be painfully fashioned from behind the eyes which, bit by bit, they are desitned to replace. And this vision is also too immense for one to speak of "grasping " it, in the same way that one "grasps" an idea that happens to arise along the way. That's no doubt why one shouldn't be surprise that the idea of giving a name to something so enormous, so close yet so diffuse, only occured to me in recollection, and then only after it had reached its full maturity.
In point of fact, for the next two years my relationship to mathematics was restricted ( apart from teaching it) to just getting it done- to giving scope to a powerful impulse that ceaselesly drew me forward, into an "unknown" that I found endlessly fascinating. The idea didn't occur to me to pause, even for the space of an instant, to turn back and get an overview of the path already followed, let alone place it in the context of an evolving work. (Either for the purpose of placing it in my life, as something that continued to attach me to profound and long neglected matters; or to situate it in that collective adventure known as "Mathematics")
What must appear even more strange, in order to get me to stop for a moment and re-establish acquaintance with these half-forgotten efforts, (or to think of giving a name to the vision which is its heart and soul), I had to face a confrontation with a "Burial" of gigantic proportions: with the burial, by silence and derision, of that vision and of the worker who conceived it ...
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