**0**

votes

**0**answers

173 views

### Shuffle (co-)multiplication and generalized Leibniz formula in tensor calculus

The headline already says it: Is anybody (except me, UPDATE: plus Gavrilov) aware of this formula for higher total covariant derivatives of tensor products?
It is the simplest application of the ...

**14**

votes

**2**answers

1k views

### What might extraterrestrial mathematics look like? [closed]

In an extensive anthropological joint research project concerning the necessities in the development of life and civilisation my group is concerned with mathematics. This forum seems to be extremely ...

**56**

votes

**22**answers

8k views

### Examples of major theorems with very hard proofs that have NOT dramatically improved over time

This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time.
I am looking for a list of
Major theorems in mathematics whose proofs ...

**60**

votes

**29**answers

6k views

### The half-life of a theorem, or Arnold's principle at work

Suppose you prove a theorem, and then sleep well at night knowing that future generations will remember your name in conjunction with the great advance in human wisdom. In fact, sadly, it seems that ...

**16**

votes

**4**answers

605 views

### History of powers beyond squares and cubes

The ancient Babylonians understood squares:
Plimpton 322
The ancient Athenians understood cubes, if we can take
doubling the cube, i.e., the Delian problem, as evidence.
My ...

**-5**

votes

**0**answers

112 views

### Mathematics textbooks with long title [closed]

L'Hôpital's book has the long title
Analyse des infiniment petits pour l'intelligence des lignes courbes,
and Ya. Perelman's Algebra Can be Fun mentions an old Russian Mathematics textbook with the ...

**231**

votes

**72**answers

89k views

### Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...

**18**

votes

**2**answers

494 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 ...

**64**

votes

**66**answers

10k views

### Old books still used

It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like ...

**136**

votes

**134**answers

29k views

### Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)
I'd love to learn about ...

**1**

vote

**0**answers

193 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 ...

**19**

votes

**9**answers

2k views

### Do there exist modern expositions of Klein's Icosahedron?

Reading Serre's letter to Gray
, I wonder if now modern expositions of the themes in Klein's book
exist. Do you know any?

**6**

votes

**1**answer

712 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 ...

**56**

votes

**8**answers

5k views

### Have you solved problems in your sleep?

I have hit upon major (for me—relative to my trivial accomplishments)
insights in my research
in various sleep-deprived altered states of consciousness,
e.g., long solo car-drives extending ...

**90**

votes

**94**answers

11k views

### What would you want to see at the Museum of Mathematics?

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...

**11**

votes

**2**answers

734 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

201 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 ...

**44**

votes

**6**answers

6k views

### Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th century?

According to Steven Krantz's Mathematical Apocrypha (pg. 186):
As was custom, Weil often attended tea
at [Princeton] University . Graduate
student Steven Weintrab one day went
about the room ...

**1**

vote

**0**answers

122 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

634 views

### Area Under Generalized Parabolas and Hyperbolas without Calculus

This is shorter and more specific version of certain questions about a rather simple quadrature method. The answers I got were great but not what I asked. The terms in the title for $y=x^p$ look ...

**10**

votes

**4**answers

1k views

### Integrating Powers without much Calculus

I'll jump into the question and then back off into qualifications and context
Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) ...

**51**

votes

**27**answers

8k views

### Writing papers in pre-LaTeX era?

I wonder how people wrote papers in the pre-LaTeX era? I mean, when typewriters and simple computers were (60th-70th?). Did they indeed put formulas by hand in the already printed articles?

**1**

vote

**0**answers

119 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 ...

**13**

votes

**1**answer

1k views

### Did Gauss know Dirichlet's class number formula in 1801?

Let $h_d$ be the number of $SL_{2}(\mathbb{Z})$ classes of primitive binary quadratic forms of discriminant $d$. It's natural to impose the hypothesis that $d$ is not at square, as we do below.
In ...

**6**

votes

**0**answers

178 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 ...

**13**

votes

**4**answers

1k views

### Notation in Frege's Grundgesetze der Arithmetik: The U with a flourish

In the Grundgesetze der Arithmetik, Frege used a number of strange characters for notation. I would be most interested to know anything about the typography (origin, usage and so on) of the strange U ...

**0**

votes

**2**answers

268 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 ...

**11**

votes

**1**answer

333 views

### What's the role of $H^{p}(\mathbb{R}^{n})$ in modern (harmonic) analysis?

The classical theory of $H^p$,due to it's heavy dependence on the complex function theory(such as Blaschke products), seemed to have an insurmountable obstacle barrying its extension to several ...

**7**

votes

**1**answer

412 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 ...

**20**

votes

**3**answers

1k views

### Why is the identity element of a group denoted by $e$?

The question was asked by a student, and I did not have a ready answer. I can think of the German word ``Einheit'', but since in German that is not how the identity element of a group is called, I ...

**27**

votes

**22**answers

7k views

### Titles composed entirely of math symbols

I apologize for burdening MO with such a vapid, nonresearch question, but
I have been curious ever since
Suvrit's popular October 2010
Most memorable titles MO question
if there were any ...

**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 ...

**69**

votes

**22**answers

7k views

### Fields of mathematics that were dormant for a long time until someone revitalized them

I thought that the closed question here could be modified to a very interesting question (at least as far as big-list type questions go).
Can people name examples of fields of mathematics that were ...

**22**

votes

**0**answers

452 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 ...

**12**

votes

**2**answers

792 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 ...

**60**

votes

**15**answers

6k views

### Mathematical research published in the form of poems

The article
Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen,
Math. Z. 127 (1972), no. 1, 10-16
is written in the form of a lengthy poem, in a style similar to that
of the ...

**4**

votes

**1**answer

167 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 ...

**45**

votes

**19**answers

7k views

### Mathematicians whose works were criticized by contemporaries but became widely accepted later

Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's ...

**150**

votes

**36**answers

41k views

### Widely accepted mathematical results that were later shown wrong?

I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ...

**20**

votes

**1**answer

943 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 ...

**12**

votes

**4**answers

2k views

### What are hypergroups and hyperrings good for?

I came across the concept of a hyperring in two recent papers by Connes and Consani (From monoids to hyperstructures: in search of an absolute arithmetic and The hyperring of adèle classes). It's a ...

**0**

votes

**0**answers

102 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

780 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 ...

**11**

votes

**0**answers

283 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 ...

**43**

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 ...

**54**

votes

**5**answers

5k views

### Are there any serious investigations of whether “mathematicians do their best work when they're young”?

There is no shortage of anecdotes and conjectures on both sides of this widespread belief, but good supporting data either way is harder to find. Can anyone provide any references for serious ...

**19**

votes

**18**answers

4k views

### Examples of conjectures that were widely believed to be true but later proved false

It seems to me that almost all conjectures (hypotheses) that were widely believed by mathematicians to be true were proved true later, if they ever got proved. Are there any notable exceptions?

**18**

votes

**5**answers

923 views

### Historical use of figures in geometry

I was surprised to learn from John Stillwell's comment in answer to the
question,
"Can the unsolvability of quintics be seen in the geometry of the icosahedron?",
that
There is not a single ...