# Tag Info

37

Just prior to Grothendieck's entry to the subject, Weil had gotten terrific results in number theory by algebro-geometric arguments, and pointed the way to far more, but some of his methods went beyond existing rigorous foundations. He aimed to supply new foundations adequate to his ideas. Around the same time Zariski and van der Waerden were also ...

12

There are two different words in French, "étaler", which means spread out or displayed and is used in "éspace étalé", and "étale", which is rare except in poetry. According to Illusie, it is the second that Grothendieck chose for étale morphism. The Petit Larousse defines "mer étale" as "mer qui ne monte ni ...

11

Fortunately, Buchberger himself described the context of his discovery, see Historical background to Gröbner's paper by Abramson. The method, in general outline, was known to Gröbner long before the Buchberger's thesis (1965). In the 1950 paper Über die Eliminationstheorie (Abramson's English translation) it is applied to finding bases of integrals of ...

10

A person's life and behavior are always shaped by a number of factors, not (in general) just one. I think it highly unlikely that any one of the three points you bring up is responsible for Grothendieck's success. At the same time, all of them may have contributed to his life. The point that I find singular about Alexander Grothendieck is his overarching ...

10

Your guess is right: the notation $\mathfrak o$ goes back to Dedekind. If you get a copy of Dirichlet-Dedekind's Vorlesungen über Zahlentheorie and look in Dedekind's famous XI-th Supplement, which was the first systematic development of algebraic number theory, you'll see $\mathfrak o$ starting in section 170 when Dedekind defines Ordnung (= Order).

9

I'd recommend Weibel’s History of homological algebra (1999)(pdf). He describes many threads, such as roots of group cohomology in Hurewicz’s observation that cohomology of an aspherical space $Y$ depends only on (what we now call cohomology of) its fundamental group $\pi=\pi_1(Y)$: [Since] homology and cohomology groups of $Y$ (with coefficients in $A$) ...

9

Cartesian coordinates provided the first systematic way of converting geometric problems into algebraic ones and vice versa, but one can do that in elementary geometry without any coordinates simply by denoting sides and angles of polygons by letters and using trigonometry. This is more or less what ancient astronomers did when computing with their geometric ...

8

The theory of branched (or ramified) coverings has its origins in continuation of analytic functions and the attempts to find maximal analytic continuations of a given function. However, certain complex functions, e.g. $f(z) =z^{1/2}$ are multi-valued in certain subdomains of the complex plane, so when trying to continue along the closed curve one might ...

8

The coordinate method may be traced to antiquity, specifically to the works of Apollonius of Perga (c. 262 – c. 190 BC) The following quotation from Carl B. Boyer,"Apollonius of Perga" (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. pp. 156–157. ISBN 0-471-54397-7 (as referenced in the Wikipedia article on Apollonius) explains ...

7

As a complement to the answer provided by Francois Ziegler, I would add the first three paragraphs of Homological Algebra (1956), by Henri Cartan and Samuel Eilenberg: During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. The purpose of this book is to ...

7

Grothendieck's familiarity with the categories predates Kansas. In 1948-1949 he attended Séminaire Cartan at École Normale Supérieure, where he "took the liberty of speaking to Cartan, as if to his equal" (Cerf's obituary). That would be Henri Cartan, the agebraic topologist, he is the son of Eli Cartan known for his contributions to Lie group theory and ...

6

From the Earliest Known Uses of Some of the Words of Mathematics site : PENCIL OF LINES. Desargues coined the term ordonnance de lignes, which is translated an order of lines or a pencil of lines [James A. Landau]. Boyer's A History of Mathematics (1968, p. 396) also attributes this terminology to Desargues. From the section on Blaise Pascal, when ...

6

The use of étale predates SGA, and "spread out" fits Grothendieck’s idea of all-encompassing topos, "vast" and "slack", better than usual, as these things go. The name of étale morphisms derives from that. Also, French étaler comes from old French estal, which meant position/place, same as Greek topos, although it is unclear if that was intended too. See ...

5

The answer to the title question is Poincaré, in the very note Sur l’Analysis situs (1892) where he first introduced the fundamental group. Cf. the description by “Saint-Gervais”: Now Poincaré gives his definition of the fundamental group. To this end he starts by almost copying his 1883 article which we have just discussed. But now he introduces a group! ...

5

“A story says that in a Paris café around 1955 Grothendieck asked his friends “what is a scheme?” The very first time the word “schéma” was uttered, in Paris, at an official seminar talk, was during the Cartan seminar of 1955/56 on algebraic geometry. The lecturer was Claude Chevalley, and the date was december 12th 1955. http://www.neverendingbooks.org/...

5

A canonical reference on this is Dieudonne's History of Algebraic Geometry. An abridged version Historical Development of Algebraic Geometry is freely available, see also Easton's slides. Let me make a general comment first. When we wonder "however did someone first connect these two [modern ideas]?" we tacitly presuppose that they were always separately ...

4

The idea is usually attributed to Dedekind and Weber in Theorie der algebraischen Functionen einer Veränderlichen (1882): [1, 2, 3, 4, 5,...].

4

"The way to understand a mathematical problem is to express it in the mathematical world natural to it -that is, in the topos natural to it. Each topos has a natural cohomology, simply taking the category of abelian groups in that topos as the category of sheaves. The cohomology of that topos may solve the problem. In outline: 1) Find the natural world for ...

4

This citation, form Grothendieck himself, to me shows a little bit why, a part from him being exceptionally gifted, his approach to problems was radically different. He describes the process of solving a math problem as that of opening a nut: *The ... analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? ...

4

The history is simple:-) Schemes were invented by Grothendieck. The purpose was unification and simplification of the foundations of algebraic geometry. The general idea is to shift from considering points of a manifold to certain ideals of a ring of functions on this manifold. This general idea evolved gradually. An early predecessor was Gelfand's theory ...

4

The idea of a relation between fundamental groups and permutations of the universal cover long predates Grothendieck and SGA. It appears implicitly already in Riemann's work on complex surfaces in 19th century. In Cauchy and Puiseux: Two Precursors of Riemann Papadopoulos even mentions earlier authors, especially Puiseux: "Riemann defined these surfaces ...

3

Kharkiv University (Ukraine) subscribed to all models made M. Schilling, who probably was a student of Klein, and who run a company making and selling these models. Currently they photograph them and place on internet: http://touch-geometry.karazin.ua/list This page also has Schilling's catalog of his production. Recently I learned that they have some ...

3

"The term itself was coined by Chevalley, although accepted in a more restrictive sense than the term as used by Grothendieck. In Foundations of Algebraic Geometry, André Weil had introduced into algebraic geometry the methods used by his mentor, Élie Cartan, in differential geometry (following Carl Friedrich Gauss and Jean Darboux). But Weil’s method was by ...

2

This is a question about English terminology. As others on here have pointed out, the French terminology is different. The original meaning of the English word “pencil” is a fine brush; this is also the meaning of French pinceau (as opposed to French "crayon" = English "pencil"). According to the Oxford English Dictionary the earliest English attestation of ...

1

I don't think this was Riemann, or that Riemann knew that any ring of functions determines the surface. In fact, Riemann studied compact surfaces on which the ring of regular functions is trivial, and he studied the field of meromorphic functions instead. The idea that a ring of functions determines the space is of much later origin. It can be traced to ...

1

The mathematical terminology "étalé" [spread out] was used by Grothendieck in his 1957 Tohoku paper, and was preexisting at that time. Grothendieck, A. (1959). Technique de descente et théorèmes d'existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats. Séminaire Bourbaki, 5, 299-327. contains the passage ...\$f[:T \to ...

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