All Questions
1,708 questions
13
votes
4
answers
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The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
6
votes
0
answers
278
views
The history of Riemann-Roch theorem [closed]
I became interested in the history of the Riemann-Roch theorem, so I searched various materials.
So, I read Über die Wechselwirkungen zwischen der französischen Schule, Riemann und Weierstraß. Eine ...
- 245
6
votes
2
answers
468
views
About Grothendieck and special cases
I was recently reminded of a quote about (not by!) Alexander Grothendieck that I had read many years ago, I think in the 1990s or 2000s.
The quote was about the way in which Grothendieck solved ...
- 887
4
votes
0
answers
276
views
Derek the Differentiable Dinosaur
I’ve come across several fond references to some semi-published lecture notes from Warwick in the 80s, by Bill Breckon (and, in some mentions, I. Harrison), Differentiating functions of lots of ...
- 21.3k
12
votes
0
answers
271
views
What is known about G. A. Croes
G. A. Croes is the author of the first description of the 2-opt moves heuristic for improving non-optimal traveling salesman tours:
Croes, G. A. “A Method for Solving Traveling-Salesman Problems.” ...
- 13.2k
16
votes
1
answer
977
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
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5
votes
2
answers
670
views
Recent breakthroughs with applied origins
Historically, the boundary between pure mathematics and its applications was much less defined. However, with the increasing complexity of modern mathematics and the resulting need for specialization, ...
3
votes
0
answers
101
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Historical appearance of using $\operatorname{SO}_3$-representation theory for spherical harmonics
$\DeclareMathOperator\SO{SO}$The spherical Laplace equation and the spherical harmonics are a beautiful example of a differential equation dominated by the representation theory of the Lie group of ...
- 1,846
1
vote
4
answers
799
views
Examples of long running and consecutively numbered international meetings [closed]
I just saw a poster at the next office's door announcing the
95th Annual Meeting of the International Association of Applied Mathematics and Mechanics.
Here is another example of a meeting I will ...
0
votes
0
answers
70
views
Evaluating the coprimality in a bivariate polynomial equation
Given a prime $p>2$, let $x$ and $y$ be real numbers such that $x>y>0$ and
$$ \begin{equation} x^p-y^p=(x-y)^p+pxy(x-y)R \tag{1} \label{eq:one} \end{equation} $$
where $R$ is a bivariate ...
- 125
45
votes
10
answers
10k
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Has the mathematics research community ever been led astray by a dumb mistake?
This is a highly subjective question, but here goes.
Has anyone ever published a result that was "taken seriously" by the research community, but was then discovered to be incorrect because ...
18
votes
2
answers
2k
views
When did the distinction between "pure" and "applied" mathematics become common?
Some ages ago, there was no difference between chemistry, physics, mathematics, and perhaps even philosophy. These were not further distinguished and largely practiced by the same people.
Obviously, ...
- 5,327
7
votes
0
answers
265
views
Herbrand's consistency proof
Jacques Herbrand's thesis "Investigations in proof theory: The properties of true propositions" (or in the original French "Recherches sur la théorie de la démonstration", with the ...
- 161
1
vote
0
answers
106
views
The proposition associated with a set
Given a set $U$ and a set $A \subseteq U$, is there an accepted symbol for the proposition $p$ over the universe $U$ such that for each $x \in U$, $p(x)$ is the assertion that $x \in A$? (The symbol $...
- 19.7k
0
votes
1
answer
807
views
What is John Charles Martin Nash known for? [closed]
John Nash and his wife Alicia tragically passed away in 2015. According to Sylvia Nasar's book "A Beautiful Mind", their son is apparently a good mathematician. What works is he known for?
user538849
37
votes
6
answers
6k
views
What is the oldest open math problem outside of number theory?
The question of "What is the oldest open problem in mathematics?" comes up from time to time, and there seems to be consensus that the answer is "Are there any odd perfect numbers?"...
- 13k
7
votes
0
answers
314
views
Did Lebesgue like non-measurable set or not?
I was surprised by the following paragraph in Bressoud's A radical approach to Lebesgue's theory of integration, quoted by Caicedo's in his comment to this question:
Vitali's nonmeasurable set, ...
- 877
6
votes
2
answers
607
views
Whence “uniform distribution”?
The “Earliest Uses” site suggests that the expression “uniform distribution” first appeared in Uspensky (1937), and “uniformly distributed” in Sakamoto (1943). Is that true?
- 31.5k
12
votes
3
answers
797
views
The orders of the exceptional Weyl groups
Who first calculated the orders of the Weyl groups $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$? How were these orders calculated?
- 415
6
votes
0
answers
141
views
Historical background of finding the roots of cubic equations using continued fractions
I came across an algebra problem book written in 1899 for students of Dar al-Fonun ([dɒːɾolfʊˈnuːn], meaning, "polytechnic college",) the only modern educational institute in Iran at the ...
- 2,437
7
votes
0
answers
335
views
Gauss, Cantor, and infinite confusion
There is an interesting comment by Gauss on "infinite magnitude as a
complete thing" that has invited varying interpretations. In a
well-known passage, Gauss criticized the use of infinity ...
- 16.6k
9
votes
1
answer
435
views
On the origin of a fundamental theorem of additive number theory
Given $a, b \in \mathbb Z$, set $[\![a,b]\!] := \{x \in \mathbb Z: a \le x \le b\}$. A basic result in additive number theory goes as follows:
If $A$ is a finite subset of $\mathbb N$ with $0 \in A$ ...
- 10.5k
5
votes
1
answer
588
views
Was homology influenced by Euler's polyhedron formula?
First of all, Alama - Formal proofs and refutations contains information about Euler's Polyhedron Formula. If you look at pp. 47–49, you will see that Poincaré proved Euler's Polyhedron Formula, and ...
- 245
60
votes
1
answer
3k
views
Eisenstein's last theorem
Hopefully the following is appropriate for MathOverflow; it's possible the question (of a somewhat historical nature) is unanswerable, but I think there's some hope it can be answered, as I'll explain ...
- 23k
18
votes
2
answers
888
views
Serre’s comment on Hurwitz: connecting FLT to points of finite order on elliptic curves
In the paper Sur les représentations modulaire de degré 2 de Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$, Serre makes the following comment:
Remarque. La relation existant entre "solutions de l'...
- 6,734
7
votes
0
answers
77
views
Earliest historical work on Cauchy's infinitesimal delta functions?
As early as 1981, Hans Freudenthal briefly mentioned Cauchy's work on "singular integrals (i.e., integrals of infinitely large functions over infinitely small paths [$\delta$ functions])" on ...
- 16.6k
3
votes
2
answers
141
views
Accessible literature on fractional dimensions of subsets of $\mathbb R^n$
I am currently wondering whether it is realistically possible to choose the topic "Fractals and fractal dimensions" for a seminar aimed at undergraduate students in the 2nd semester, with ...
- 1,942
0
votes
0
answers
75
views
Logan's theorem in compressed sensing
In some research papers in the nuclear magnetic resonance field Ref:, Logan's theorem is used to provide a justification for randomized sampling of free induction decay curves which are converted to ...
- 879
12
votes
1
answer
598
views
Fermat last theorem : proof of a criterion by Cauchy
In 13 Lectures on Fermat's Last Theorem, Ribenboim states the following theorem (on page 7) attributed to Cauchy:
If the first case of Fermat's theorem fails for the exponent $p$, then the sum:
$$ 1^{...
- 121
2
votes
3
answers
1k
views
Concrete works by Alexandre Grothendieck, other than Dessin d'Enfants?
For me "Dessin d'Enfants" by Alexandre Grothendieck is the more concrete research work he has done. I would like to know if there are others.
When he was teaching at Montpellier University (...
- 85
14
votes
1
answer
295
views
When were Allegories first introduced?
I’m doing some bibliographic work for my PhD and I’m struggling to find the earliest resources on Allegories.
They were surely made famous by the 90s book “Categories, Allegories” by Freyd and Scedrov....
- 249
2
votes
1
answer
261
views
Usage and origin of the terms dictionary and atom in compressed sensing
In compressed sensing two terms or perhaps fancy word are frequently encountered. One is the dictionary and the other is atom. The dictionary is the matrix and its columns are called "atoms" ...
- 879
4
votes
1
answer
686
views
Who and when proved Artin's Theorem on alternative rings?
I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).
Question. When has Artin proved this theorem and where was it published for the first ...
- 42k
10
votes
1
answer
522
views
About Friedrichs historical contribution to QFT cited in Reed and Simon
In the Reed and Simon book, Appendix X.7, they mention that Friedrichs provided the first examples of inequivalent representations of the canonical commutation relations via the Weyl relations in the ...
- 424
8
votes
1
answer
2k
views
Mathematical emails
How should one archive mathematical emails (especially keeping in mind that one may move from one institution to another, or may forget precise wording later on so that searching becomes difficult)?
...
10
votes
1
answer
516
views
Earliest proof of Solovay's theorem for successor cardinals
Solovay's partition theorem states that a stationary set over a regular cardinal $\kappa$ can be partitioned into $\kappa$-many disjoint stationary sets. The full theorem was proven by Solovay in 1971 ...
- 576
2
votes
0
answers
337
views
Who contributed [GT13] to "Computers and Intractability"?
This is a followup to my question How does the complexity of calculating the Permanent imply the NP completeness of directed 3-cycle cover?
Question:
who contributed problem [GT13] PARTITION INTO ...
- 13.2k
13
votes
0
answers
2k
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Is there any correspondence between Gödel and Kreisel that supports Kreisel's observation that Gödel changed his mind about his 1938 set theory note?
At a conference in 1965 there were some interesting comments made by Kreisel and Mostowski asserting that Gödel later changed his mind regarding his1938 note on his set theory results (see Problems in ...
- 481
4
votes
1
answer
145
views
On the history of cone-3-manifolds
A cone-3-manifold (of constant curvature) is a geometric 3-manifold locally modelled either on the Euclidean/hyperbolic/spherical 3-space or on the respective metric cones over spherical cone-surfaces ...
- 353
11
votes
6
answers
2k
views
Hard problems with an easy-to-understand answer
I am very interested by problem in mathematics which are difficult (go at least 10 years without a resolution, say) but which have a solution that is short and elementary.
In this video Launay gave an ...
2
votes
0
answers
114
views
Robinson's views on Heyting's work?
Abraham Robinson and Arend Heyting had mutual respect (though holding differing philosophical views on the nature of mathematics). Heyting repeatedly expressed admiration for Robinson's work; see for ...
- 16.6k
4
votes
0
answers
207
views
Who first considered "Pascal Triangle"? [closed]
Arnold was used saying in his talks,
"Pascal’s triangle, so called, because it was by Chinese discovered"!
How much is he right?
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7
votes
1
answer
1k
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An unpublished calculation of Gauss and the icosahedral group
According to p. 68 of Paul Stackel's essay "Gauss as geometer" (which deals with "complex quantities with more than two units") , Gauss calculated the coordinates of the vertices ...
- 2,099
3
votes
1
answer
234
views
Independence of CH and permutation models?
Can independence of $\sf CH$ from $\sf ZFCA$ be established using $\sf FM $ permutation models? And if so, then historically did this came first or Cohen's forcing?
- 11.3k
2
votes
0
answers
207
views
History of bump functions
When were the standard bump function examples such as $e^{-1/(1-x^2)}$ first understood, and what was the context or motivation at the time?
As an upper bound I would guess that they must have been ...
- 2,247
4
votes
1
answer
183
views
Notation for weak derivatives
I remember that, as a student, I felt a bit uncomfortable because I had to use the same notation (say $f'$, $D^\alpha f$, $\frac{\partial f}{\partial x^j}$, $\nabla \cdot f$ etc...) for classical and ...
- 4,459
15
votes
1
answer
1k
views
A cipher proposed by Littlewood
In J. E. Littlewood's, "A Mathematicians Miscellany" there is the following passage about ciphers. I found it interesting for a couple of reasons.
First of all the "legend that every ...
12
votes
1
answer
450
views
Did Gödel possess a proof of the independence of $\mathsf{AC}$?
We all know Gödel proved the consistency of the Axiom of Choice with $\mathsf{ZF}$ using his constructible universe, and Cohen proved the consistency of $\neg \mathsf{AC}$ using his new method of ...
- 1,322
3
votes
0
answers
167
views
Suitability of formal type theory for mathematical thinking (vs. traditional set theory)
Type theory has advantages over set theory for the (computer) formalisation of mathematics, but has anybody who does mathematics with pen and paper found proof assistants or automated theorem provers, ...
4
votes
2
answers
275
views
Diagrammatic representation of sets as irregular plane figures
I am trying to find the earliest use of plane diagrams of various shapes for representing sets. For example, this is snapshot is from the book called Some Modern Mathematics for Physicists and Other ...
- 879