Questions tagged [algebraic-geometry]

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5
votes
2answers
196 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 ...
8
votes
2answers
228 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 ...
2
votes
1answer
112 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 ...
2
votes
3answers
163 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 ...
12
votes
3answers
860 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 ...
3
votes
2answers
181 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,....
10
votes
2answers
439 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 ...
8
votes
1answer
446 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 ...
5
votes
1answer
112 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 ...
5
votes
1answer
200 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.
6
votes
1answer
407 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 ...
14
votes
3answers
1k 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 ...
11
votes
1answer
354 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 ...
5
votes
0answers
145 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 ...
9
votes
3answers
995 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 ...
16
votes
1answer
257 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. ...
32
votes
1answer
5k 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 ...