Alexander Grothendieck was born in Berlin March 28, 1928. His father, Sascha Shapiro Russian-born anarchist, took an active part in revolutionary movements in Russia rhyme and then in Germany, in the '20s, where he met Hanka Grothendieck, Alexander's mother. After the advent of Nazism in Germany was too dangerous for a revolutionary jew and the couple moved to France, leaving Alexander in foster care to a family in Hamburg. In 1936, during the English Civil War, Alexander's father was associated with the anarchists in the resistance against Franco. In 1939, Alexander joined his parents in France but was arrested and his father - also a result of the racial laws promulgated by the Vichy government in 1940 - sent Auschwitz where he died in 1942. And Alexander Hanka Grothendieck were also deported but survived the massacre. Alexander was able to attend high school at the College Cévenol in Chambon-sur-Lignon Secours Suisse staying in the house for refugee children, separately from the mother, but was forced to flee into the woods to each of the Gestapo raid. It was then a student at the University of Montpellier and the fall of 1948 he arrived in Paris with a letter of introduction to Elie Cartan. He was then accepted at the Ecole Normale Supérieure as auditeur pounds for the year 1948-49 witnessing the debut of the algebraic topology at the seminar of Henri Cartan (son of Elie). The first interest of Grothendieck, however, were directed to functional analysis and on the advice of Cartan moved to Nancy. Under the guidance of J. Dieudonné and L. Schwartz received his doctorate in 1953. Grothendieck, in the years of high school and college, took little satisfaction from the courses and curricula and institutional can not say that was a model student. His curiosity, combined with dissatisfaction, led him to develop independently, not yet twenty, a theory of measure and integration, which then learned, in Paris, was already being written by Lebesgue. "I was then in the solitude that is essential in the profession of mathematics - one that no teacher can really teach" so Grothendieck. The productive period of Grothendieck officially attested by an impressive body of writings, stands over the years 1950-70. If the subjects of research in the early 50's functional analysis, the major themes of algebraic geometry, its foundations, such as the redefinition of the concept of space, are the basis of research of the years 1957-70. In 1959, he became a professor at the Institut des Hautes Etudes Scientifiques nascent (IHES) in Bures, near Paris, a seminar in which soul and suggests offers to students and colleagues - with a generosity exemplary - his ideas of research, sharing his unqualified enthusiasm and creativity. In these early years also frequent and intense with Jean Pierre Serre, as evidenced by their correspondence, are a source of inspiration and a mutual exchange of ideas. In the decade 1959-69 the results are mainly disseminated, on the one hand, such as' Eléments de Géométrie Algébrique (EGA) - written in collaboration with Dieudonné - and with the help of participants in the Séminaire de Géométrie Alge ; brique (EMS) using the notes to the seminar, and the other in the Exposés Bourbaki seminar. In the original draft of the Grothendieck Séminaire was regarded as a preliminary form of 'Eléments intended to be incorporated in the latter, which are initially published in a variety dall'IHES ponderous tomes. In 1966 he received the Fields Medal (the highest honor for a mathematician). Grothendieck in 1970 at the age of 42 years, left the official stage. The reasons that led him to withdraw from the academic world are many, but certainly his radical anti-militarism is a stated reason. In fact, he realizes that the IHES receives funds from the Ministry of Defence - for more than three years without his knowledge - and as an answer leaves the Institute and the formal publication of EGA and SGA, assigning the new edition of the latter to Springer-Verlag. Having lived as a refugee with a UN passport, without citizenship - his official papers disappeared Nazi apocalypse - gives life to the peace movement and environmental Survivre. During the war years in Vietnam and the proliferation of nuclear weapons - as also in our current landscape of conflict always live - pacifism Grothendieck appears as an assumption of significant responsibility and not insignificant that the institutions involved, the Conversely, even today they continue to receive these funds. Following this choice Grothendieck spent a couple of years at the College de France, then in Orsay and, finally, in 1973, Back at the University of Montpellier, rejecting the Crafoord Prize in 1988, the year of his retirement. In recent years, retired to private life at Mormoiron in the country, having given up travel, is devoted to correspondence and writing Récoltes et Semailles, a long reflection and testimony about his past as a mathematician, in the words of Grothendieck, a long meditation on life, or "inner adventure that has been and that is my life." I received some parts of Récoltes et Semailles in 1991, along with a letter in which I also Grothendieck Aldo Andreotti indicated as "a good friend and a person truly precious: I am come to appreciate its special qualities much better now that it's not that in the 50s and 60s when he was still alive." I am not knowledge of Italian mathematicians who have collaborated with Grothendieck in those years, the Italian school has assimilated very slowly his methods in algebraic geometry, although some have Italian roots, in Severi and Barsotti, for example. The Présentation des Thèmes et Semailles of Récoltes is the invaluable source - together with the letter - some considerations for history and the canvas for a fresco of his mathematical thinking that now I am about to outline. The excellence of Grothendieck, his mathematical genius, is recognizable in its natural propensity to visibly reveal the crucial issues that no one had highlighted or recognized. Its fecundity has deep roots and is expressed through new language always emerges as a stream of new concepts and abstractions-set-formulations. Ben often stated so perfectly expressed by a lively imagination and relentless are found to be the foundation of a whole theory that Grothendieck himself has outlined, developed and accomplished, and in other cases just mentioned. This propensity the creation of mathematics, even before solving mathematical problems, makes a mathematician Grothendieck extremely unusual and extravagant, if we athematic dexterity as the human capacity to solve problems. The layman who approaches the work of Grothendieck mathematics will have to abandon common sense to look at math as a problem solver and really try to look at mathematics as an art and the artist as a mathematician. An art of particular importance, for which the invention is to borrow the demonstrations or the imagination is accorded with reason and his works are theories in a plot, a design that allows you to capture more unity in diversity. As Grothendieck himself writes, "this act is to go further, not of being closed in a circle is imperative that we fix, is primarily a loner who is this act of creation." To Grothendieck, mathematical theories are also opportunities for reflection in the broad sense and a meditative exercise, a form of contemplation that accompanies the adventure within. Mathematics is therefore a yoga that is different and different theories proliferate but that has very solid foundations unit. The differentiation of these themes old and new also interwoven with a history of ideas to which they are inspired. In the words of Grothendieck itself, there are traditionally three aspects of the things that are the subject of mathematical thinking: the number or appearance of arithmetic, the metric measure or the appearance (or analytical) and the geometric shape or appearance. "In most cases studied in mathematics, these three aspects are present simultaneously and in close interaction." In the following we will examine some of these issues in their perspective of algebraic geometry that Grothendieck has revealed. An eye that focuses on form and structure and thus the geometric aspect and arithmetic, in a unifying vision that gave birth to a new geometry to arithmetic geometry.
We can say that the number is likely to grasp the structure of aggregates
discontinuous or discrete: systems, often
finished, formed by elements or objects as it were isolated
in relation to each other, without any principle of continuous passage
from each other. The size of the contrary is
quality par excellence, capable of continuous variation;
through what is likely to grasp the structures and phenomena
continue: movements, spaces, varieties of all kinds, etc.
force fields. Thus, the arithmetic is (roughly)
as the science of discrete structures, and analysis, as the science of
continuous structures.
As to geometry, we can say that after more than two thousand years
that exists in the form of a science
in the modern sense, is riding on these two types of structures
, those discrete and continuous ones. Moreover,
for a long time, there was really a "divorce" between two
geometries that were different in nature
a fair and the other continues. Rather, there were two different points of view
investigating the same
geometric shapes: an emphasis on discrete properties
[...] the other properties constant [...].
is to the 800 who appeared in divorce, with the advent
and development of what is sometimes referred to as the geometry
(algebraic) abstract. Roughly speaking, this was as
aim to introduce, for any prime p, a geometry
(algebraic) of characteristic p, modeled along the lines
(continuous) of geometry (algebraic)
inherited from centuries earlier, but in context, however, that appeared as
irreducibly discontinuous, discreet. These new
geometric objects, have become increasingly important at the beginning of
900, and this, in particular, in view of their close relationship with the arithmetic
[...] seems to be one of the ideas
guiding the work of [...] AndréWeil
is in this spirit that he made in 1949, the famous Weil conjectures. Conjectures
absolutely stunning, indeed, they do
glimpse of these new varieties (or spaces) of a discrete
the possibility of certain types of constructions and arguments
that until then seemed imaginable only in the framework of the only areas considered
as worthy of the name
analysts
[...] We believe that the new geometry is first and foremost,
a synthesis between these two worlds, the world [...] [...]
arithmetic and the world of greatness continues [...]. In this new vision,
once separated the two worlds, they form only one.
discontinuous or discrete: systems, often
finished, formed by elements or objects as it were isolated
in relation to each other, without any principle of continuous passage
from each other. The size of the contrary is
quality par excellence, capable of continuous variation;
through what is likely to grasp the structures and phenomena
continue: movements, spaces, varieties of all kinds, etc.
force fields. Thus, the arithmetic is (roughly)
as the science of discrete structures, and analysis, as the science of
continuous structures.
As to geometry, we can say that after more than two thousand years
that exists in the form of a science
in the modern sense, is riding on these two types of structures
, those discrete and continuous ones. Moreover,
for a long time, there was really a "divorce" between two
geometries that were different in nature
a fair and the other continues. Rather, there were two different points of view
investigating the same
geometric shapes: an emphasis on discrete properties
[...] the other properties constant [...].
is to the 800 who appeared in divorce, with the advent
and development of what is sometimes referred to as the geometry
(algebraic) abstract. Roughly speaking, this was as
aim to introduce, for any prime p, a geometry
(algebraic) of characteristic p, modeled along the lines
(continuous) of geometry (algebraic)
inherited from centuries earlier, but in context, however, that appeared as
irreducibly discontinuous, discreet. These new
geometric objects, have become increasingly important at the beginning of
900, and this, in particular, in view of their close relationship with the arithmetic
[...] seems to be one of the ideas
guiding the work of [...] AndréWeil
is in this spirit that he made in 1949, the famous Weil conjectures. Conjectures
absolutely stunning, indeed, they do
glimpse of these new varieties (or spaces) of a discrete
the possibility of certain types of constructions and arguments
that until then seemed imaginable only in the framework of the only areas considered
as worthy of the name
analysts
[...] We believe that the new geometry is first and foremost,
a synthesis between these two worlds, the world [...] [...]
arithmetic and the world of greatness continues [...]. In this new vision,
once separated the two worlds, they form only one.
This unified view has run embodied in the concepts of schema and topos revealing hidden structures: the wealth of the world discrete geometric came to light in all its beauty and structure, allowing the demonstration of these conjectures of Weil by the same Grothendieck and Pierre Deligne, one of his students. The concept of schema is a large magnifying or generalization of algebraic varieties as it was designed by Italian and German schools of the early twentieth century. The idea of Grothendieck scheme and the basic outlines of a theory of schemes, using the concept of morphism between them or by appropriate transformations of schemas, back to the years 1957-58 and are briefly exemplified at the World Congress of Mathematicians in Edinburgh in 1958. The very concept of the beam - already introduced and studied by Leray Serre and - here is essential as it allows to reconstruct a global data from a range of local data and allows continuous type of reasoning within discreet. If the algebraic geometry is the study of polynomial equations and the locus defined by these bundles and the theory of the forms is the fast and natural language in which to express it faithfully, tastefully explicit language capable of the intimate structure of these geometric entities .
Montpellier, April 19, 1988
Ganelius Dear Professor, thank you for your letter of April 13
, which I received today, and the telegram. The Crafoord Prize
insignitomi with Pierre Deligne (who was my student
) this year by the Royal Swedish
accompanied by a large sum of money, I was very honored.
However, I regret to inform you that I do not want to accept this award
like no other, for the following reasons:
1) The salary of a professor and the board, starting from next October
, are more than adequate for my needs materials
and those of my employees, so I do not need
money. As for the honors conferred
to some of my work on the fundamentals, I am convinced that only time will
test the fertility of new ideas or visions. Fertility
measure with the result and not with recognition.
2) I note also that all high-quality research, to
such a prestigious award like that is directed Crafoord,
have a social status that gives them more and more material wealth
scientific prestige to what is necessary
with the power and privileges that this entails. Yet,
is not clear that the abundance of some is possible only at the cost
the needs of others?
3) The work that brought me to the kind attention of the Academy
finished five years ago, when I
the scientific community and essentially endorses
the spirit and values. I came from that environment
in 1970 and, although scientific research has continued to thrill
inwardly I retired from the increasingly
milieu of science. Meanwhile, the ethics of scientific communities
(at least the maths) has decayed to the point that the theft said
among colleagues (especially at the expense of
those who are not in a position to defend themselves) has almost become the norm and
is, however, tolerated by all,
even in the most obvious and unfair. Under these conditions,
agree to participate in the game of the prizes and honors
would also give my approval to
a spirit and a tendency in the scientific world that I consider to be fundamentally unhealthy
and more condemned to disappear soon
, as such a spirit and so ruinous trend,
spiritually, intellectually and materially.
The third reason is for me by far the most important,
even if not intended in any way as a criticism
the Royal Academy and how it will manage its
century
totally unexpected change of events
completely our concept of "science" and its objectives and
spirit in which scientific work is done. Of course, at that time
the Royal Academy will be among the institutions and people
who will play an important role in this renewal
unprecedented, after a similar collapse of
civilization never before. Sorry
incident that may have caused her and the Royal Academy
my refusal to receive the Crafoord prize, especially for the fact that
award had already been advertised before the candidates had agreed
. However, I never gave up
to express my opinion on the scientific community and
"official science" today known as one community and especially
to my old friends and to my young students
mathematical world. What I think is in Récoltes et
Semailles, a long reflection on my life as a mathematician,
on creativity in general and scientific creativity
in particular, this paper has become an unexpectedly
portrait of the moral principles of the mathematical world
from 1950 until today. Pending published in book form,
provisional edition of two hundred copies have been sent
fellow mathematicians, mainly to algebraic geometers
(now I do honor Commemoration). In an envelope
aside, sending the two introductory sections for your information
staff. Again thank you and the Royal Swedish Academy
and offer my apologies for the inconvenience, unwanted. The
Please accept my warmest compliments.
A. Grothendieck
, which I received today, and the telegram. The Crafoord Prize
insignitomi with Pierre Deligne (who was my student
) this year by the Royal Swedish
accompanied by a large sum of money, I was very honored.
However, I regret to inform you that I do not want to accept this award
like no other, for the following reasons:
1) The salary of a professor and the board, starting from next October
, are more than adequate for my needs materials
and those of my employees, so I do not need
money. As for the honors conferred
to some of my work on the fundamentals, I am convinced that only time will
test the fertility of new ideas or visions. Fertility
measure with the result and not with recognition.
2) I note also that all high-quality research, to
such a prestigious award like that is directed Crafoord,
have a social status that gives them more and more material wealth
scientific prestige to what is necessary
with the power and privileges that this entails. Yet,
is not clear that the abundance of some is possible only at the cost
the needs of others?
3) The work that brought me to the kind attention of the Academy
finished five years ago, when I
the scientific community and essentially endorses
the spirit and values. I came from that environment
in 1970 and, although scientific research has continued to thrill
inwardly I retired from the increasingly
milieu of science. Meanwhile, the ethics of scientific communities
(at least the maths) has decayed to the point that the theft said
among colleagues (especially at the expense of
those who are not in a position to defend themselves) has almost become the norm and
is, however, tolerated by all,
even in the most obvious and unfair. Under these conditions,
agree to participate in the game of the prizes and honors
would also give my approval to
a spirit and a tendency in the scientific world that I consider to be fundamentally unhealthy
and more condemned to disappear soon
, as such a spirit and so ruinous trend,
spiritually, intellectually and materially.
The third reason is for me by far the most important,
even if not intended in any way as a criticism
the Royal Academy and how it will manage its
century
totally unexpected change of events
completely our concept of "science" and its objectives and
spirit in which scientific work is done. Of course, at that time
the Royal Academy will be among the institutions and people
who will play an important role in this renewal
unprecedented, after a similar collapse of
civilization never before. Sorry
incident that may have caused her and the Royal Academy
my refusal to receive the Crafoord prize, especially for the fact that
award had already been advertised before the candidates had agreed
. However, I never gave up
to express my opinion on the scientific community and
"official science" today known as one community and especially
to my old friends and to my young students
mathematical world. What I think is in Récoltes et
Semailles, a long reflection on my life as a mathematician,
on creativity in general and scientific creativity
in particular, this paper has become an unexpectedly
portrait of the moral principles of the mathematical world
from 1950 until today. Pending published in book form,
provisional edition of two hundred copies have been sent
fellow mathematicians, mainly to algebraic geometers
(now I do honor Commemoration). In an envelope
aside, sending the two introductory sections for your information
staff. Again thank you and the Royal Swedish Academy
and offer my apologies for the inconvenience, unwanted. The
Please accept my warmest compliments.
A. Grothendieck
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