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Grothendieck's Fine Topology in Esquisse d’un programme

I would like to clarify a couple points in the following excerpt from these notes (page 3) discussing Grothendieck's seminal Esquisse d’un programme pointing out the importance to reformalize the ...
3 votes
3 answers
286 views

History of cohomology theory

I saw this post. And I already posted it on Math stack exchange, but since someone recommended this site, I'm refining it and posting it again. And I understand that the mathematical object called ...
2 votes
1 answer
179 views

What is the origin (and perhaps original) of this quote by André Weil

I found this quote by André Weil in a few places online: As every mathematician knows, nothing is more fruitful than these obscure analogies, these indistinct reflections of one theory into another, ...
4 votes
1 answer
208 views

What is the history of vector bundles and their characteristic classes?

The theory of vector bundles (and their characteristic classes) appears to have been standardized in the 20th century by all of the familiar names. Considering its substantial importance throughout ...
0 votes
0 answers
39 views

De Branges's theorem and perspective on Sheaf theory

As far as I know, there is an intuitive and easy way to think about sheaf theory through the Taylor series. For example, in the case of stalk, which is one of the important concepts in sheaf, we can ...
2 votes
1 answer
160 views

The reason why sheaf theory emerged

Motivation: In any history, there is a cause-and-effect relationship. So I became curious about the situation in which the sheaf theory came to appear. In other words, I'm curious about what problem ...
0 votes
0 answers
166 views

Why Serre need to develop the concept of "sheaf theory" in algebraic geometry? [duplicate]

I read Edward Frenkel's Love and Math. But reanding this book made me wonder about origin of the concept of the sheaf used in algebraic geometry. I think the conclusion that I came to in the process ...
5 votes
0 answers
98 views

History of Algebraic Geometry: Morphisms and Birational Geometry

Good people, I'm trying to get my head around the history of algebraic geometry, and while Dieudonné's tome is a very good source (very often the only source), it can from time to time be very ...
2 votes
0 answers
203 views

The origin of $∂^2=0$ and $d^2=0$

I know that formula $∂^2=0$ and $d^2=0$ very important in the homology and cohomology theory. And I understand that this formula was generated from the process of finding a solution to the partial ...
3 votes
0 answers
149 views

History of right hand rule

I am curious to know when the right-hand-rule for vector product was established and used consistently in mathematics. I read here Who gave right hand thumb rule for circular loop of current ...
9 votes
3 answers
381 views

When was the first time/s that sheaves entered algebra and algebraic geometry?

I'm interested in knowing about the first published texts in which sheaf-theoretic methods were used in algebra and/or in algebraic geometry. The oldest instance I am aware of is J.-P. Serre, ...
2 votes
0 answers
80 views

History behind Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$ for a commutative Noetherian ring

In 033Q we find defined what some sources call “Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$” (if you don't know what a scheme is, you can read the definition for a commutative Noetherian ring ...
0 votes
0 answers
41 views

Lefschetz historical proof of Hyperplane Theorem

I would like to understand the the idea behind the historically original proof by Lefschetz of his Hyperplane theorem sketched roughly Here. The basic setup: Let $X$ be an $n$ -dimensional complex ...
2 votes
1 answer
135 views

When did the notion of and term "weight" arise in geometry?

The notion of weight of a Hodge structure and its avatars on $\ell$-adic and $p$-adic cohomologies (via comparison theorems modeled on the Hodge decomposition of de Rham or singular cohomology with ...
3 votes
0 answers
101 views

When did modular forms start to get studied via algebraic geometry?

I'm looking, for instance, as to when people started studying modular curves and modular forms as sections of line bundles on them, as opposed to the point of view of modular forms being holomorphic ...
2 votes
0 answers
179 views

How did Grothendieck come in contact with Category theory?

Category theory was formalized around 1950s, and Grothendieck made his breakthrough papers about 10-20 years from that time. I wish to know, how was it possible the ideas of Category Theory were so ...
1 vote
2 answers
133 views

To what extent were Riemann surfaces a precursor to algebraic geometry?

I read that Riemann started studying the so-called Riemann's surfaces in the second half of the 19th century, introducing tools like meromorphic functions and meromorphic 1-forms. The culmination of ...
0 votes
0 answers
163 views

Why a second edition of EGA was never published?

Grothendieck's Éléments de géométrie algébrique, also known as EGA, was originally devised to consist of thirteen volumes (as stated in the introduction of the first volume), from which Grothendieck ...
5 votes
1 answer
246 views

Continuation on Galois’ lost memoir(s): was Poisson right in his review of Galois’ memoir?

A recent question asked whether Galois’ “lost” memoir has been since recovered (similar to Abel’s lost memoir). The memoir referenced in the question is the one Galois submitted to the French “...
10 votes
3 answers
2k views

What is the history behind the concept of "schemes" in algebraic geometry?

Let me start by saying that I've just started studying algebraic geometry. I have a habit of trying to find out something about the motivation behind new concepts that I'm studying, in this case that ...
1 vote
0 answers
303 views

Katz's symbol 兄 for Gauss-Manin connections

In his famous 1970 paper [1], Nicholas Katz used the character 兄 for the Gauss-Manin connection. I have always been curious about the history behind this symbol. Question: What motivated Katz to use ...
2 votes
2 answers
293 views

Omar Khayyam is well known as a mystical poet (Quatrains). He is also known as a mathematician. Are these the same?

Omar Khayyam is well known as a mystical poet (his famous Quatrains). He is also somehow known as a mathematician (Equations of degree 3 ?). Are these the same person? A colleague in Arithmetical and ...
15 votes
3 answers
2k views

How did Grothendieck encounter and adopt the categorical language?

Apparently the first mathematical publication of Grothendieck where he uses the terms “functor” and “category” in the technical sense is the Kansas report. From where Grothendieck had knowledge of ...
3 votes
1 answer
526 views

Grothendieck and Fields medal 1962

We can read as a mathunion excerpt that Grothendieck won the Fields medal in 1966 Built on work of Weil and Zariski and effected fundamental advances in algebraic geometry. He introduced the idea of ...
45 votes
1 answer
10k views

Why did algebraic geometry need Alexander Grothendieck?

Grothendieck is arguably the most brilliant mathematician of the 20th century, with his influence felt the most in algebraic geometry, which he transformed. Some time ago the story used to be told was ...
2 votes
0 answers
97 views

The Picard Group: Origin and History

I've come to the notion of the Picard Group. I was recently linked to this paper paper, which contains the line: The problem of computing the Picard groups of surfaces $S \subset \mathbb{P}_{\mathbb{...
2 votes
0 answers
131 views

Did the ancients construct higher genus curves?

I know that the ancients had several ways to construct geometric shapes. I know two of them: the biggest one, ruler and compass, with which you can build some polygons, bisect an angle, etc, but not ...
5 votes
1 answer
242 views

Why is the term "isotropic" used to describe a quadratic form and a vector?

A quadratic form $q$ on a vector space $V$ is isotropic if $q(v) = 0$ admits a nonzero solution. What was this terminology originally intended to evoke? There is some prior discussion on Math SE, ...
21 votes
3 answers
4k views

Grothendieck's approach to solving problems

Alexander Grothendieck is known to have revolutionized several areas of mathematics. His insights were very deep, original and revolutionary. Time after time he showed that he could see in ways that ...
6 votes
1 answer
405 views

Grothendieck and elementary topos

I would like to know some references (if any) for the claim that Grothendieck didn't like the idea of elementary topos.
0 votes
1 answer
87 views

Who first started parameterizing hyperbolas with hyperbolic functions?

Anyone know the history of using hyperbolic functions to parameterize the parabola? Wolfram hasn't really helped. Anyone know where to look?
3 votes
1 answer
351 views

Origins of Zariski topology

Why did Zariski feel the need to define his famous topology? Was this notion used in one form or another prior to him in algebraic geometry?
9 votes
0 answers
269 views

First use of term "Hilbert's Nullstellensatz"

This year (2021) marks the 100th anniversary of Emmy Noether's 1921 paper in which she introduced Noetherian rings and proved the primary ideal decomposition for them. The original version of her ...
23 votes
4 answers
4k views

Why are étale morphisms called "étale"?

Alexander Grothendieck developed the theory of "locally trivial coverings spaces for rings/schemes" in SGAI as an analog to the theory of covering spaces in algebraic topology. He called such ...
4 votes
1 answer
169 views

Complete list of publications of Rebecca Barlow

Rebecca Barlow is the discoverer of an interesting surface in algebraic geometry. Is anybody aware of a full list of her contributions? Has she continued working in mathematics in the 21st century?
9 votes
1 answer
929 views

What is the etymology of the mathematical terms "sheaf, stalk, germ"?

The peculiar agricultural terminology commonly used in algebraic geometry and category theory, "sheaf", "stalk", "germ", is quite well-known. A sheaf is pictured as something like a bundle of stalks, ...
6 votes
1 answer
595 views

What was the motive for inventing Gröbner bases?

How did professor Buchberger discover Gröbner (Groebner) bases for polynomial ideals? What was the problem(s) that lead to such a discovery?
11 votes
2 answers
1k views

Where was the word "pencil" first used in (projective) geometry and what is the reason behind this curious name?

The title is pretty self-explanatory: A pencil in projective (or algebraic) geometry is the family of all lines through a point. The above-linked website tells me that Cremona, on page x of Elements ...
6 votes
2 answers
441 views

Who first described the fundamental group as the group of deck transformations?

Grothendieck developed the theory of the fundamental group of a scheme in SGA 1. In order to do so he used the fact that the fundamental group of a topological space is isomorphic to the group of deck ...
9 votes
2 answers
335 views

(Co)Homology: From topology to the rest of mathematics?

I can appreciate how (co)homology arose in the context of topology/geometry. Trying to get a handle on the handles of spaces leads one to this idea. It's not obvious, but I can see how this would lead ...
16 votes
1 answer
338 views

Did Dedekind show any evidence of pictorial geometric sense?

The most pictorial geometric thinking I find in Dedekind is the intuitive-geometric idea of a continuum which he criticizes as too vague before he gives his account of the continuum based on cuts. ...
2 votes
1 answer
140 views

Material models of Riemann surfaces

It is known that during the last quarter of the 19th century there was a flourishing of the production of material models (from plaster, strings, card-board etc) of curves and surfaces in Germany (but ...
3 votes
3 answers
261 views

Who first had the idea to study surfaces via rings of functions, as in algebraic geometry?

This idea provides the foundations of algebraic geometry now; and they have certainly gone down the rabbit hole with it. As a student studying this subject, I have always found it such a great leap to ...
3 votes
2 answers
231 views

When did mathematician start to draw figures from equation?

I know that when solving geometric problem, Descartes used variables $x,y$ and derived equation such as $y^2=cy-\frac{cxy}{b}+ay-ac$. Conversely, in algebraic geometry, an arbitrary polynomial $F(X_1,....
6 votes
0 answers
246 views

History of Algebraic Geometry for a general readership?

I'm looking for a "pop maths" book on the subject. Something much more accessible than Dieudonné: "light reading" with emphasis on the history, personalities and general ideas and minimal ...
5 votes
1 answer
192 views

Origins and history of branched covering

During my research on branched coverings of the projective plane, I am interested to know the origins and history of branched coverings of the projective plane and the projective line, together with ...
7 votes
1 answer
1k views

Visualizing algebra before Descartes

The Cartesian coordinate system is what I have been told provided the first link between algebra and geometry. However, I also have learned that, for instance, Omar Khayyam solved cubic equations as ...
19 votes
1 answer
1k views

Who first introduced the notation $\mathcal{O}$ in algebraic geometry or algebraic number theory

This is my first question for HSM. If it is consider too specialized for HSM, perhaps it can be migrated to MathOverflow. In algebraic number theory, one frequently denotes the ring of algebraic ...