20
votes
Why are étale morphisms called "étale"?
From Milne's site:
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 ...
- 201
15
votes
Accepted
How did Grothendieck encounter and adopt the categorical language?
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 ...
- 70.9k
15
votes
Grothendieck's approach to solving problems
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 ...
- 8,293
14
votes
Accepted
Who first introduced the notation $\mathcal{O}$ in algebraic geometry or algebraic number theory
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 ...
- 4,747
11
votes
Accepted
What was the motive for inventing Gröbner bases?
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 ...
- 70.9k
10
votes
Accepted
(Co)Homology: From topology to the rest of mathematics?
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 $Y$ ...
- 7,614
9
votes
Accepted
Why are étale morphisms called "étale"?
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 ...
- 70.9k
9
votes
Accepted
When did mathematician start to draw figures from equation?
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 ...
- 1,496
9
votes
Origins and history of branched covering
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 ...
- 1,496
9
votes
Accepted
Visualizing algebra before Descartes
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 ...
- 70.9k
8
votes
Origins of Zariski topology
Zariski introduced his topology in this paper: The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc., 50 (1944), 683-691. You can read it online ...
- 4,747
7
votes
What is the history behind the concept of "schemes" in algebraic geometry?
“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 ...
- 582
7
votes
(Co)Homology: From topology to the rest of mathematics?
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 ...
- 5,678
6
votes
Accepted
Who first had the idea to study surfaces via rings of functions, as in algebraic geometry?
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 ...
- 70.9k
6
votes
Accepted
Where was the word "pencil" first used in (projective) geometry and what is the reason behind this curious name?
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 [...
- 6,504
6
votes
Grothendieck's approach to solving problems
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 ...
- 1,687
6
votes
Accepted
When was the first time/s that sheaves entered algebra and algebraic geometry?
Here is what Dieudonné has to say on contributions other than Serre's in History of Algebraic Geometry:
"As early as 1909, Severi, while defining the arithmetic genus of a nonsingular ...
- 70.9k
5
votes
Grothendieck's approach to solving problems
"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 ...
- 53
5
votes
Who first described the fundamental group as the group of deck transformations?
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 ...
- 7,614
5
votes
Accepted
Who first described the fundamental group as the group of deck transformations?
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 ...
- 70.9k
4
votes
Who first had the idea to study surfaces via rings of functions, as in algebraic geometry?
The idea is usually attributed to Dedekind and Weber in Theorie der algebraischen Functionen einer Veränderlichen (1882): [1, 2, 3, 4, 5,...].
- 7,614
4
votes
What is the history behind the concept of "schemes" in algebraic geometry?
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 ...
- 46.4k
4
votes
Why are étale morphisms called "étale"?
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'...
- 141
4
votes
Omar Khayyam is well known as a mystical poet (Quatrains). He is also known as a mathematician. Are these the same?
According to the wiki page, he is a mathematician and a famous poet.Many of the books and magazines we read when we were students stated that Omar Khayyam (we say Ömer Hayyam) was a mathematician and ...
- 173
3
votes
How did Grothendieck encounter and adopt the categorical language?
In the "Esquisse Thématique des Principaux Travaux Mathématique 4.a Algèbre catégorique", Grothendieck says:
En fait, de facon continuelle depuis 1953, je me suis senti dans l'obligation, ...
- 356
3
votes
Where was the word "pencil" first used in (projective) geometry and what is the reason behind this curious name?
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 ...
- 3,415
3
votes
What is the history behind the concept of "schemes" in algebraic geometry?
"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 ...
- 582
3
votes
Material models of Riemann surfaces
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 ...
- 46.4k
3
votes
Accepted
Why is the term "isotropic" used to describe a quadratic form and a vector?
This is an example of how a term migrates from the original context by broken telephone through various generalizations and transfers. It started with Poncelet introducing "imaginaries", i....
- 70.9k
3
votes
Grothendieck and Fields medal 1962
There is a very interesting [resource], let me sum up my conclusions.
The beginnings of the Fields medal were not about honoring the best mathematicians, but "The committees used the medal as a ...
- 356
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