**11**

votes

**1**answer

546 views

### Where did the term “additive energy” originate?

A fundamental object in modern additive combinatorics and harmonic analysis is additive energy. Given a subset $A$ of (say) an abelian group $G$ the additive energy of $A$ is defined to be the ...

**0**

votes

**1**answer

139 views

### Congruent numbers and elliptic curves

Who first explicitly stated the link between $N$ being a congruent number and the existence of rational points of infinite order on $y^2=x(x^2-N^2)$?

**4**

votes

**1**answer

186 views

### Motivation for cyclotomic units

I am wondering the original motivation for considering cyclotomic units. Maybe one can rephrase the question as:
Why did people initially consider such units in $\mathbb{Q}(\zeta_p)$ specially?
...

**6**

votes

**1**answer

123 views

### How to divide a square into three similar rectangles

Preparing some exercises for my High School pupils I came across this question: How can you tile a square into three similar (ie., same shape, different size) rectangles?
With a bit of algebra it can ...

**9**

votes

**1**answer

352 views

### Examples of abstractions that did *not* turn out to be useful [closed]

I’ve read (but cannot find any reference now) that new abstract mathematical concepts like set theory and – not too long ago – category theory were in their time often considered too abstract to be ...

**63**

votes

**28**answers

9k views

### What are some very important papers published in non-top journals?

There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here.
My concern in this question is slightly ...

**52**

votes

**3**answers

4k views

### What was Hilbert's view of Gödel's Incompleteness Theorems?

According to Solomon Feferman, in his slide presentation "Three Problems for Mathematics", Hilbert wrote (in regards to Gödel's second incompleteness theorem):
...the end goal [is] to establish as ...

**11**

votes

**1**answer

403 views

### Two Vinogradovs? Is one the son of the other? [closed]

Forgive me for my ignorance, but I'm very surprised to learn that there are two Vinogradovs, both famous in the field of analytic number theory. Guessing from their names and the Russian naming ...

**16**

votes

**2**answers

369 views

### Hahn-Banach theorem with convex majorant

At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and ...

**11**

votes

**2**answers

535 views

### First formulation of the Dedekind and Hasse-Weil conjectures

I'm looking for the original statement of two important conjectures in number theory concerning L-functions. I'm particularly interested in pinning down the year in which they were first formulated:
...

**35**

votes

**4**answers

2k views

### Hilbert's (cancelled) 24th problem

Hilbert's 23 problems, ten of which were presented at the 1900 ICM in Paris, are too famous for any mathematician to not know. If one reads the descriptions of the problems in Hilbert's paper, one ...

**2**

votes

**0**answers

115 views

### When did mathematicians begin to use the letter x to denote unknown values? [duplicate]

When did mathematicians begin to use the letter x to denote unknown values ?
Gérard Lang

**22**

votes

**1**answer

676 views

### How and why did mathematicians develop spin-manifolds in differential geometry?

First of all, I am neither a physicist nor a mathematician. And I am afraid that mathoverflow is not a suitable place for my question, but having asked similar questions on math SE it is obvious that ...

**11**

votes

**1**answer

288 views

### 'Updated' book in the same spirit as Dieudonné's Panorama des mathématiques pures

Today a colleague of mine asked me if I knew of any "more modern version" of J. Dieudonné's Panorama des mathématiques pures. Le choix bourbachique.
The very first thing that instantly came to my ...

**6**

votes

**1**answer

135 views

### History of spectral methods to the study of real analytic $GL_2$-Eisenstein series

I'm trying to sort out the history of spectral methods in the study of real analytic $GL_2$-Eisenstein series. From what I read so far, I would say that the subject was really kicked off by the ...

**18**

votes

**2**answers

520 views

### History of set-class distinction

I have two questions concerning the history of set theory, both related to the distinction between the notion of a set and the notion of a class:
Who was the first mathematician to make this ...

**1**

vote

**0**answers

208 views

### Why do some people adamantly insist on 'toposes' instead of 'topoi'? [closed]

I've heard that several category and topos theorists, first and foremost Johnstone (see the comments to this question) adamantly insist on 'toposes' as the plural of 'topos'. I was wondering whether ...

**6**

votes

**1**answer

730 views

### Windows into new mathematical worlds [closed]

Yitang Zhang's Annals of Mathematics primes-gap result
opened a new window, which
Polymath's reduction from $70\times 10^6$ to $246$ attests.
Perhaps
Harald Helfgott's
celebrated proof of the odd ...

**11**

votes

**2**answers

749 views

### Can I find Fermat's complete works anywhere?

I admire the mathematician very much and want to look at his writings. Is there anywhere in book or web form that has a collection of his writings?

**8**

votes

**0**answers

210 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 ...

**1**

vote

**0**answers

124 views

### Was this particular case of the tube formula known before Weyl and Hotelling?

The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 ...

**1**

vote

**0**answers

120 views

### History of categorical localization sans calculi of fractions

This question arises from a paper which I've just found and skimmed:
FW Bauer, J Dugundji. Categorical homotopy and fibrations. Transactions of the American Mathematical Society, 1969
With 28 ...

**42**

votes

**1**answer

3k views

### What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?

Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...

**6**

votes

**0**answers

180 views

### $\alpha$-minimal degrees for singular $\alpha$

An important question in $\alpha$-recursion theory is whether there is a minimal $\alpha$-degree at $\alpha=\aleph_\omega.$
Question 1. Who first introduced the above question, and where can I find ...

**0**

votes

**2**answers

307 views

### Linear Algebra classic books [closed]

I'm learning linear algebra at the moment, so I'm looking for some great old classic books. Something like Fermat's or Gauss books of some great mathematians.
I don't really like the nowadays books ...

**7**

votes

**1**answer

433 views

### Why is the set-theoretic principle $\diamondsuit$ called $\diamondsuit$?

A shallow answer would just point to theorem 6.2 in Jensen's 1972 paper "The fine structure of the constructible hierarchy", where Jensen introduces this property. Or was this symbol used already ...

**19**

votes

**2**answers

1k views

### History of Geometric Analogies in Number Theory

My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens?
For example, was it before Grothendieck introduced schemes to ...

**22**

votes

**0**answers

472 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 ...

**15**

votes

**1**answer

1k views

### The list of problems for Grothendieck's thesis

Is the list of open problems which were given by Dieudonne and Schwartz to Grothendieck for his thesis published somewhere? I know a quotation of Dieudonne that the problems concerned duality theory ...

**4**

votes

**1**answer

175 views

### Blow-up as polar coordinates?

While doing some explicit calculations involving a blow-up of the plane in a point, I realised what I was doing was basically writing things in polar coordinates. Somewhat astonished that I hadn't ...

**20**

votes

**1**answer

959 views

### What is $\infty^6$?

The title of this question may make you want to close it immediately, but bear with me a moment. In several older mathematics papers (early 20th century) I have seen statements such as
The ...

**1**

vote

**0**answers

123 views

### First to note the relation between Stasheff polytopes (associahedra) and compositional inversion?

In my answer to MO-Q: Enumerative geometry and nonlinear waves, I outline the relation between the refined face polynomials of the Stasheff polytopes (associahedra) and the partition polynomials for ...

**14**

votes

**2**answers

799 views

### Origin of the term “Diophantine equation”

It seems that the term "Diophantine equation" has been around at least since the second half of the 19th century, since the historian Hermann Hankel writes (polemically) in the chapter on Diophantus ...

**44**

votes

**4**answers

2k views

### How did Cole factor $2^{67}-1$ in 1903

I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193707721 \times 761838257287$. There doesn't seem to be a historical ...

**16**

votes

**1**answer

378 views

### What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their models of non-Euclidean Geometry?

As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now ...

**1**

vote

**1**answer

168 views

### Non-Pythagorean proof for the square root of 2 and solution to YBC7289 [closed]

My name is J. Frederic Teubner I am an independent researcher. I wish to publish a proof for the non-Pythagorean solution to the Babylonian tablet YBC7289 and am currently inquiring as to whether or ...

**6**

votes

**1**answer

394 views

### Who introduced the concept of topological mixing?

I am writing an introduction and I want to know who introduced the concept of topological mixing?

**2**

votes

**1**answer

183 views

### What kind of role has Functional Analysis played in Signal Processing? [closed]

Does it serve mainly as a narration or is there any substantive consequence which might not be derived without tools of functional analysis?

**12**

votes

**2**answers

800 views

### Describe the desired features of a “Mathematics Colloquium”?

I'm now a member of my department's colloquium committee. Our task is to make a great colloquium series. I thought that the first step would be to come up with an appropriate definition of ...

**5**

votes

**0**answers

154 views

### Reference to forcing with a sigma ideal $\cong$ Cohen forcing

This is a historical question: Who was the first person to notice the following?
If $V \models \kappa$ is measurable and $P$ adds $\kappa$ Cohen reals, then in $V^P$, letting $\hat{I}$ to be the ...

**19**

votes

**1**answer

783 views

### Steinhaus's Easter Egg Problem

The following is the text of Steinhaus's so-called Easter egg problem. According to this article of Roman Duda, this was recorded in the New Scottish Book around Easter 1955 and "Steinhaus offered an ...

**13**

votes

**4**answers

1k views

### “Epicycles” (Ptolemy style) in math theory?

By analogy:
The epicycles of Ptolemy explained the known facts in the sun system and in this sense were not "wrong". But they distracted from a better insight. From another viewpoint, everything fell ...

**17**

votes

**7**answers

2k views

### Are there any Algebraic Geometry Theorems that were proved using Combinatorics?

I'm collaborating with some algebraic geometers in a paper, and when writing the introduction I mentioned the interaction of Combinatorics and Algebraic Geometry, and gave some examples like the ...

**21**

votes

**1**answer

1k views

### Homeomorphism historically: When did it reach its modern formulation?

Q. When did the notion of homeomorphism reach its
modern formulation as a bicontinuous bijection, i.e., a
continuous bijection
between topological spaces whose inverse is also continuous?
...

**38**

votes

**1**answer

2k views

### Did Bourbaki write a text on algebraic geometry?

Certainly Bourbaki never wrote an introduction to algebraic geometry: we would have heard about it, right?

**2**

votes

**1**answer

2k views

### Famous examples of PhD advisors younger than their student [closed]

What are the most famous examples of PhD advisors in mathematics, younger than their student?
(if possible put the date of birth and/or the difference in age).

**6**

votes

**1**answer

731 views

### Cricket and the Hardy-Littlewod maximal function

I'v read somewhere that one motivation for Hardy to define his maximal function is the game of cricket. But I can't see how they are related. Could anyone provide some more information on their ...

**0**

votes

**1**answer

1k views

### The most cited paper in Mathematics [closed]

I am wondering about the most cited papers/books in Mathematics. I always had the impression that the number of citations in the mathematical community is several orders of magnitude below the number ...

**12**

votes

**4**answers

780 views

### Brandt's definition of groupoids (1926)

The definition of a category is usually attributed to Mac Lane and Eilenberg (1945). What seems to be less known is that the german mathematician Heinrich Brandt has developed the notion of a groupoid ...

**14**

votes

**2**answers

635 views

### Scott-Solovay unpublished paper on ``Boolean valued models of set theory''

I have read some papers from 1970$^{th}$, and in some of them, the paper of Scott and Solovay on ``Boolean valued models of set theory'' is given as a main reference, with many references to the ...