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Quelques idees maˆ itresses de l'œuvre de A. Grothendieck
Abstract
We try to explain four fundamental ideas invented by Grothendieck: schemes, topos, the six operations and motives. Dans Recoltes et Semailles (troisieme partie), Grothendieckecrit :
... It is in this stay when he wrote the main ideas of his work in abelian categories 10 and also he started to work in his 7 We include this brief note in order to give the reader an idea of Grothendieck's particular life and shed light on how his personality as mathematician, and more general as human being, was formed. 8 The years in Hamburg had a big impression in Grothendieck. We were informed by Winfried Scharlau [18] that Grothendieck visited Hamburg around 2006. ...
... As it is discussed in [8,20] Grothendieck's intrepid idea is to accept that for every commutative ring A (with unit) it is possible to define an affine scheme X = Spec(A). What is needed here is to define the structure sheafÃ. ...
In this essay we give a general picture about the evolution of Grohendieck's ideas regarding the notion of space. Starting with his fundamental work in algebraic geometry, where he introduces schemes and toposes as generalizations of classical notions of spaces, passing through tame topology and ending with the formulation of a geometry of forms, we show how the ideas of Grothendieck evolved from pure mathematical considerations to physical and philosophical questions about the nature and structure of space and its mathematical models.
... The second lecture was by I.R.Shafarevich and was given at the Stockholm Congress in 1962 [48]. Between these two events A.Grothendieck created the theory of schemes ([20] [14]). I think that Weil's lecture had some influence on Grothendieck. ...
... Between these two events A.Grothendieck created the theory of schemes ( [20,14]). I think that Weil's lecture had some influence on Grothendieck. ...
This is expanded text of a lecture delivered by the author at the conference "Mat\'eriaux pour l'Histoire des Math\'ematiques au XX\`eme si\`ecle", which took place in Nice in January 1996. The task was to describe one area in the development of arithmetical algebraic geometry in Moscow during the 1950s and 1960s. We shall begin by explaining the meaning of the analogy between numbers and functions, starting with the simplest concepts. In the second part we study a nontrivial example: the explicit formula for the law of reciprocity. In the third part we shall become acquainted with certain aspects of the "social" life of the Moscow school, in particular, with certain seminars, lectures, and books. In the final part we shall examine another example of this analogy: arithmetical surfaces and Arakelov theory. Comment: 36 pages, 3 pictures, published in:Bolibruch, A.A. (ed.) et al., Mathematical events of the twentieth century. Berlin: Springer; Moscow: PHASIS, 2006, 297-329
... While this holds of many geometers,[14] puts toposes among Grothendieck's best ideas. Only Deligne emphasizes what Grothendieck also knew: you can think with topos intuitions while officially using only small sites in proofs. ...
Grothendieck’s “vast unifying vision” provided new working and conceptual foundations for geometry, and even led him to logical foundations. While many pictures here illustrate the geometry, Grothendieck himself favored apt words and commutative diagrams over pictures and did not think of geometry pictorially.
... This measure also works in other toposes to give other cohomology theories and it was the key to producingétale cohomology. For one expert view of how far toposes can be eliminated frométale cohomology, and yet how they help in understanding it, see Deligne (1977Deligne ( , 1998. ...
Today's number theory solves classical problems by structural tools that violate standard philosophical expectations in ontology. The far-reaching practical demands of this mathematics require on one hand that the tools be fully explicit and more rigorous than many philosophical theories are, and on the other hand that they relate as directly and as concisely as possible to guiding intuitions. The standard tools grew from a century-long trend of unifying algebra, topology, and arithmetic, notably in the Weil conjectures; and they rely on devices that Kronecker produced for his idea of a pure arithmetic. Very large functors serve to organize individually simple kinds of data that can themselves even be depicted in simple pictures. Mathematicians and philosophers have debated issues of individuation and identity raised by these tools.
... According to Deligne [82], motives were first introduced by Grothendieck in an attempt to find an explanation for the "family likeness" 20 of the étale l-adic cohomology groups of an algebraic variety over a field of nonzero characteristic p, when l varies over the primes l = p. In characteristic zero, for a an algebraic variety X over a subfield k of C, an explanation would be given by comparison isomorphisms between the ("usual") integer valued topological cohomology groups H i (X(C), Z) and the -adic étale cohomology groups H í et (X, Z l ) : ...
RÉSUMÉ. Nous donnons un aperçu des développements de la théorie des modèles des nombres p-adiques depuis les travaux de Denef sur la rationalité de séries de Poin-caré, par une revue de la bibliographie. ABSTRACT. We survey the literature in the model theory of p-adic numbers since Denef's work on the rationality of Poincaré series.
The paper contrasts two ways of generalizing and gives examples: probably most people think of examples like generalizing Cartesian coordinate geometry to differential manifolds. One kind of structure is replaced by another more complicated but more flexible kind. Call this articulating generalization as it articulates some general assumptions behind an earlier concept. On the other hand, by unifying generalization, I mean simply dropping some assumptions from an earlier concept or theorem. Hilbert, Noether, and Grothendieck were all known for highly non-trivial unifying generalizations.
By providing quantifier-free axioms systems, without any form of induction, for a slight variation of Euclid’s proof and for the Goldbach proof for the existence of infinitely many primes, we highlight the fact that there are two distinct and very likely incompatible concepts of infiniteness that are part of the theorems proved. One of them is the concept of cofinality, the other is the concept of equinumerosity with the universe.
We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations \(A \rightarrow B\) into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and (what we shall call) the structure-semiotics duality (of which the syntax-semantics duality for propositional and predicate logic are particular cases). Whereas in classical algebraic geometry a certain kind of rings can be recovered by considering their representations with respect to a unique codomain B, Grothendieck’s theory of schemes permits to reconstruct general (commutative) rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized—by means of what we shall generally call Gelfand transform—as quantities on a topological space that parameterizes the relevant representations of A. As we shall argue, important dualities in different areas of mathematics (e.g. Stone duality, Gelfand duality, Pontryagin duality, Galois-Grothendieck duality, etc.) can be understood as particular cases of this general pattern. In the wake of Majid’s analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck’s notion of functor of points can be understood as a “relativization of the a priori” (Friedman) that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding (co)domains.
About 50 years ago, Éléments de Géométrie Algébrique (EGA) by A. Grothendieck and J. Dieudonné appeared, an encyclopedic work on the foundations of Grothendieck’s algebraic geometry. We sketch some of the most important concepts developed there, comparing it to the classical language, and mention a few results in algebraic and arithmetic geometry which have since been proved using the new framework.
Alexandre Grothendieck (1922-2014) foi um dos maiores matemáticos do século 20 e um dos mais atípicos. Nascido na Alemanha a um pai anarquista de origem russa, sua infância foi marcada pela militância política dos seus pais, assim passando por revoluções, guerras e sobrevivência. Descoberto por sua precocidade matemática por Henri Cartan, Grothendieck fez seu doutorado sob orientação de Laurent Schwartz e Jean Dieudonné. As principais contribuições dele são na área da topologia e na geometria algébrica, assim como na teoria das categorias. No final dos anos de 1960, ele se dedicou à militância política e ecológica, organizando a revista Survivre durante três anos. Em 1986, publicou um manuscrito autobiográfico de 1000 páginas, Récoltes et semailles, em que ele descreve sua experiência e sua prática da matemática, assim suas contribuições à comunidade matemática francesa. Pouco comentado na filosofia, as implicações dos seus descobrimentos fora mais recentemente discutidas por Alain Badiou na sua "fenômeno-lógica", em Logiques des mondes (2016) e Arkady Plonitsky, Mathgematics, Science and postclassical Theory (1997), pesquisa trata da semelhança entre os aspectos formais da filosofia de Gilles Deleuze e da topologia de Grothendieck.
Alexander Grothendieck was one of the greatest mathematicians of the twentieth century. His work revolutionized algebraic geometry and he made tremendous contributions to the concepts, content and techniques of this field, advancing it to an unprecedented level. Moreover, his work had enormous influence in other areas of mathematics like number theory, topology, homological algebra and functional analysis. His unifying vision provided the framework and tools for solving famous and long-standing problems in algebraic geometry and number theory: for example,Weil conjectures by Deligne, Mordell conjecture by Faltings and Modular conjecture–Fermat’s ‘last’ theorem by Wiles–Taylor.
THE LARGE STRUCTURES OF GROTHENDIECK FOUNDED ON FINITE ORDER ARITHMETIC - COLIN MCLARTY
Grothendieck preempted set theoretic issues in cohomology by positing
universes, where his version made these sets so large that Zermelo Fraenkel set
theory (ZFC) cannot prove they exist. We show the weak fragment of ZFC called
MacLane set theory (MC) suffices for existing applications in number theory. It
has the proof theoretic strength of simple type theory. Adding a version of Mac
Lane's axiom of one universe gives MC+U, also a weak fragment of ZFC yet
sufficient for the whole SGA.
Large-structure tools like toposes and derived categories in cohomology never
go far from arithmetic in practice, yet existing foundations for them are
stronger than ZFC. We formalize the practical insight by founding the entire
toolkit of EGA and SGA at the level of finite order arithmetic.
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