Questions tagged [ho.history-overview]
History and philosophy of mathematics, biographies of mathematicians, mathematics education, recreational mathematics, communication of mathematics.
185
questions with no upvoted or accepted answers
33
votes
0
answers
2k
views
History of the Proj construction in algebraic geometry
Projective geometry was introduced by fifteenth century Renaissance painters (like Alberti, da Vinci and Dürer) in the guise of perspective theory, although one could argue that Pappus was already ...
32
votes
0
answers
2k
views
Next steps on formal proof of classification of finite simple groups
While people are steaming ahead on finessing the proof of the classification of finite simple groups (CFSG), we have a formal proof in Coq of one of the first major components: the Feit-Thompson odd-...
24
votes
0
answers
2k
views
How did Gauss find the units of the cubic field $\mathbb Q[n^{1/3}]$?
Recently I read the National Mathematics Magazine article "Bell - Gauss and the Early Development of Algebraic Numbers", which gives a good description of the genesis of Gauss's ideas ...
21
votes
0
answers
3k
views
Origins of the Nerve Theorem
Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question:
Who was the first to prove the Nerve Theorem?
19
votes
0
answers
697
views
Eckmann-Hilton argument / Grothendieck
In terse terms, the “Eckmann-Hilton argument” says that a monoid in the category of monoids is commutative. It is essentially used, for example, to prove that homotopy groups of topological groups are ...
18
votes
0
answers
1k
views
Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?
The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\...
18
votes
0
answers
2k
views
History of the functor of points
Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar.
However, in this note by Lawvere the author writes:
"I myself had learned the ...
17
votes
0
answers
594
views
Who first noticed the duality for finite groups?
A.A.Kirillov in section 12.3 of his "Elements of the Theory of Representations" writes that the first "symmetric" duality theory for non-commutative groups was the theory for finite groups. In short ...
17
votes
0
answers
2k
views
Why did Bourbaki not use universal algebra?
I have seen a discussion about Bourbaki’s usage of categories before. So let me ask a different question: why did he not use universal algebra?
Well, universal algebra is not much older than category ...
16
votes
0
answers
514
views
Reference request for Grothendieck's work on "Integration with values in a topological group"
Disclaimer. This question was already asked in Mathematics Stack Exchange (see the link here). I wanted the question to be migrated here but I was told by a moderator that a question that old is ...
16
votes
0
answers
7k
views
Story of "Grothendieck's prime number" 57
I asked this question earlier, at hsm.stackexchange.com without much luck. Maybe somebody can answer it here.
There is a story about Alexander Grothendieck and the "Grothendieck Prime" 57, which ...
16
votes
0
answers
1k
views
Galois theory timeline (II)
This question is a sequel. I structured the previous one around Emil Artin's classic treatment of Galois theory from the 1940s, though making clear some reservations of my own about whether Artin ...
14
votes
0
answers
632
views
Who stated and proved the "Hopf lemma" on bilinear maps?
If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$.
Nondegenerate here means that ...
13
votes
0
answers
644
views
List of problems that Erdős offered money for?
Is there a list somewhere of all the problems that Erdős offered cash awards for, including both solved and unsolved problems? One would think that the answer is yes, but so far I have had no luck ...
13
votes
0
answers
232
views
Galois group of polynomials related to Fibonacci and Catalan numbers
Let $F_n$ be the Fibonacci and $C_n$ the Catalan numbers.
Define a polynomial by $F_n(x):=\sum\limits_{k=1}^{n}{F_k x^{n-k}}$.
For example $F_8(x)=x^7+x^6+2x^5+3x^4+5x^3+8x^2+13x+21$.
And another ...
13
votes
0
answers
619
views
Unpublished Lecture Notes
Hi, Overflowers
There was a time (not so long ago) where lecture notes were not published, not commonly at least, and their reproduction was expensive. In my case, that was precisely the time when ...
12
votes
0
answers
337
views
What was the "stormy discussion" about differential Galois theory at IHES?
In Kazuo Okamoto and Yousuke Ohyama's paper "Mathematical works of Hiroshi Umemura", Annales de la faculté des sciences de Toulouse Mathématiques, XXIX, no. 5 (2020) pp. 1053-1062, there is ...
12
votes
0
answers
408
views
History of use of "=" symbol to mean "is canonically isomorphic to"
Let $A$ be a commutative ring, and let $f$ and $g$ denote elements of $A$ such that the prime ideals of $A$ containing $f$ are precisely the prime ideals containing $g$ (a not completely trivial ...
12
votes
0
answers
988
views
What is the reason that $\sigma$-algebra replaced $\sigma$-ring in introductory measure theory?
May I ask what is the (historical) reason we adopted the $\sigma$-algebra rhetoric instead of $\sigma$-rings (like used in Halmos)? To my knowledge almost all modern measure theory or real analysis ...
12
votes
0
answers
286
views
Who first showed that $SL(n,O_K)$ is a lattice for a number ring $O_K$?
Let $O_K$ be the ring of integers in an algebraic number field $K$. Assume that $K$ has $r$ real embeddings and $s$ pairs of complex conjugate complex embeddings. There is then an injective ...
12
votes
0
answers
1k
views
Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
12
votes
0
answers
450
views
What is the history of the notion of subdivision of categories?
A recent answer by Peter May prompts me to ask a question which I have been considering to ask for several months. (The reason why I have not asked it before is that it is not directly related to my ...
11
votes
0
answers
357
views
Atiyah's remark on Tate-Schafarevich & Poincare conj.?
Peter Woit quoted Atiyah in his blog ( http://scilogs.spektrum.de/hlf/sir-michael-atiyah-unity-mathematics-physics/ ) : "Tate-Shafarevich conjecture might have something to do with the 4-dimensional ...
11
votes
0
answers
847
views
Reference/quote request: "All of combinatorics is the representation theory of $S_n$"
I think I remember reading somewhere a glib (or is it deep?) quote, perhaps due to Rota?, which was something like
"All of combinatorics is essentially [or can be reduced to?] the representation ...
11
votes
0
answers
1k
views
Original references for the homotopy groups $\pi_5(SU(3))$ and $\pi_4(SU(2))$?
For revision of a paper (http://arxiv.org/abs/1008.1189), I'd like to correct my references to the original work on aspects of the homotopy groups $\pi_5(SU(3))$ and $\pi_4(SU(2))$. I'm not a ...
10
votes
0
answers
255
views
History of preservation theorems in forcing theory
For my honours thesis, I am studying a general preservation theorem using a framework provided by Shelah. I am mainly concerned about revised countable support iteration of $\dot{S}$-semiproper ...
10
votes
0
answers
732
views
What is Quillen's contribution to index theorem?
In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a ...
9
votes
0
answers
154
views
Who was the first to prove that the automorphism group of a finite field is cyclic and is generated by the Frobenius automorphism?
$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\...
9
votes
0
answers
317
views
History behind Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$ for a commutative Noetherian ring
This is a repost. So far, I've received no answers on HSM Stack Exchange; maybe I do in MO.
In 033Q we find defined what some sources call “Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$” (for a ...
9
votes
0
answers
275
views
Grothendiecks's lectures on Kohärente Garben und verallgemeinerte Riemann-Roch-Hirzebruch Formel
In his biography of Hirzebruch in Jahresber Dtsch Math-Ver (2015) 117:93–132, Zagier says that
[T]he dominating event [of the first Arbeitstagung in 1957] was unquestionably Grothendieck’s lecture ...
9
votes
0
answers
305
views
Thurston on the Robertson-Seymour theorem
Danny Calegari recounts here that Bill Thurston "gave one talk explaining his idea of a new proof of (some version of) the Robertson-Seymour theorem" at MSRI in 1996-1997. I could not find it in the ...
9
votes
0
answers
386
views
History of the definition of smooth manifold with boundary
I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it ...
9
votes
0
answers
320
views
Why does Loday call the permutohedra "zylchgons"?
Today I was reading Jean-Louis Loday's classic paper, "Realization of the Stasheff polytope", in which he produces a simple and very pretty realization of the associahedra as convex polytopes. He ...
9
votes
0
answers
1k
views
A quote by Lev Landau about prime numbers
I was talking with a student of mine about Goldbach's conjecture, and a certain point he asked why this apparently simple statement is so difficult to prove.
Half-joking, I answered "well, because ...
9
votes
0
answers
471
views
"A remarkable Moufang loop"
The 1985 paper A simple construction of the Fischer-Griess monster group by Conway refers to an "in press" article, A remarkable Moufang loop, with an application to the Fischer group $Fi_{24}$, by ...
9
votes
0
answers
326
views
Is the perfectness of Fourier-Mukai kernels proved by Toen?
In Toen's paper The homotopy theory of dg-algebras and derived Morita theory, Theorem 8.15, he essentially proved the following result.
Let $X$ and $Y$ be two smooth and proper schemes over $k$. ...
9
votes
0
answers
293
views
On an unpublished result of Magidor
In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$
is strong limit and $2^{\aleph_\omega}=\aleph_{\...
9
votes
0
answers
364
views
Filmed lectures by Hassler Whitney
Are there any filmed lectures by outstanding American mathematician Hassler Whitney, besides the two Einstein Chair lectures below (links updated)?
Old lectures, from the 1940s onwards, would be ...
9
votes
0
answers
714
views
Reference request : Grothendieck's topological space valued integral
As I am learning the different kind of Banach space valued integrals (Pettis, Bochner), I know that Grothendieck made a "mémoire" in his youth about this topic, but I don't know if it is available ...
9
votes
0
answers
314
views
From Frege to Gödel - German equivalent?
I know this question does not quite fit here, but I felt it could best be answered here. I recently stumbled upon the book From Frege to Gödel, which is a sourcebook containing some of the most ...
9
votes
0
answers
939
views
Has anyone pursued Frege's idea of numbers as second-order concepts?
Gottlob Frege was a pivotal figure in the history of mathematical logic. He gave an analysis of numbers that proceeded along roughly the following lines, in his books "The Foundations of Arithmetic" (...
9
votes
0
answers
595
views
Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes
Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...
8
votes
0
answers
320
views
Who introduced the notion of ringed spaces?
My question is very concise, please forgive it.
Who introduced the concept of ringed space?
My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...
8
votes
0
answers
498
views
Landau's century-old problems: Anything comparable?
Landau's four problems
are now over a century old (1912), and each still unsolved.
This seems remarkable, even though he was not the originating author all four
(maybe only the 4th?). Still, he ...
8
votes
0
answers
407
views
Who realized the finite fields $\mathbb F_{p^n}$ first? Gauss or Galois?
Let $p$ be a prime, and let $n$ be a positive integer. The finite field $\mathbb F_{p^n}$ is often called a Galois field and denoted by $\mathrm{GF}(p^n)$ by researchers on coding theory.
On the other ...
8
votes
0
answers
391
views
Bounding eigenvalues by taking high powers of matrices: history?
Let $A$ be real symmetric matrix. It is a well-known observation that we can bound any eigenvalue $\lambda$ of $A$ by using the fact that
$$\lambda^{2 k} \leq \textrm{Tr} A^{2 k}$$
for any $k\geq 1$. ...
8
votes
0
answers
183
views
History of the Lewis-Stegall theorem on factorization of representable operators
The following questions are about the history of a particular result in functional analysis, hence not "mathematical questions" per se; but I think they are relevant to the business of ...
8
votes
0
answers
117
views
Literature and history for: lifting matrix units modulo various kinds of ideal
This is not so much a mathematics question as a cross between a "history of mathematics" question and a reference request.
My PhD student has been working on some problems concerning ...
8
votes
0
answers
250
views
Did Euler ever use anything similar to Cauchy's inequality?
This could be asked more provocatively, indeed how it arose, as "how did Euler do so much mathematics without using and/or knowing Cauchy's inequality?", something that came up in the ...
8
votes
0
answers
376
views
Silver's unpublished work on reverse Easton iteration
Silver was the first person who used the method of reverse Easton iterations in connection with large cardinals, and used it to force the failure of $GCH$ at some measurable cardinal.
At most papers ...