**18**

votes

**2**answers

813 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 ...

**68**

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

388 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

943 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 ...

**11**

votes

**2**answers

764 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 ...

**59**

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 ...

**3**

votes

**1**answer

146 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 ...

**44**

votes

**19**answers

6k 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 ...

**146**

votes

**36**answers

39k 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

916 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 ...

**1**

vote

**0**answers

92 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

736 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

250 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 ...

**41**

votes

**4**answers

1k 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 ...

**52**

votes

**5**answers

4k 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

886 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 ...

**60**

votes

**63**answers

9k 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 ...

**10**

votes

**2**answers

458 views

### History of Tarski's problems on free groups

As is known, Tarski posed his questions about first-order theories of non-abelian free groups around 1945. However, the questions were not published in his papers or books.
What is the original ...

**66**

votes

**25**answers

7k views

### Modern Mathematical Achievements Accessible to Undergraduates

While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even ...

**6**

votes

**1**answer

355 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?

**7**

votes

**2**answers

696 views

### Who first introduced the functional definition of symmetry?

Who first introduced the definition of symmetry using functions explicitly? (That is, for instance, a symmetry of a subset $X$ of the plane is a function $F$ from the plane to the plane that preserves ...

**16**

votes

**1**answer

359 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 ...

**8**

votes

**1**answer

299 views

### Who first proved the fundamental theorem of projective geometry?

The following theorem is often called the fundamental theorem of projective geometry:
Let $k$ be a field and let $n \geq 3$. Let $X$ be the partially ordered set of nonzero proper subspaces of ...

**15**

votes

**2**answers

783 views

### Who was Hermann Künneth?

Question as in the title:
Who was Hermann Künneth? Where can I find some biographical information beyond what is available on Wikipedia?
The well-known Künneth formula, for example in the form of ...

**55**

votes

**13**answers

10k views

### Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic.
The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...

**1**

vote

**1**answer

156 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 ...

**3**

votes

**1**answer

143 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?

**59**

votes

**7**answers

9k views

### Is the boundary $\partial S$ analogous to a derivative?

Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...

**5**

votes

**0**answers

148 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 ...

**12**

votes

**2**answers

2k views

### When did the term “Lie group” first appear?

Does anyone know who was the first to coin the term "Lie group"?
The following thesis from 1928 suggests that the term was already in use by that time: "Systems of Two Differential Equations from the ...

**15**

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 ...

**36**

votes

**3**answers

2k views

### What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?

This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields ...

**4**

votes

**1**answer

467 views

### What is the advantage of inverting elliptic integrals?

In the case of the circle I can hardly make any conclusions from the integral $(1)$, most of the theorems come from geometrical considerations. It's not clear how to prove periodicity from this ...

**15**

votes

**5**answers

3k views

### Fermat numbers and the infinitude of primes

Wonder whether any of you guys know why it is that the proof of the infinitude of primes that is based on the coprimality of any pair of (distinct) Fermat numbers is commonly attributed to Pólya.
In ...

**96**

votes

**27**answers

13k views

### Extremely messy proofs

Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what ...

**130**

votes

**132**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 ...

**3**

votes

**0**answers

251 views

### Galois correspondence subgroups/subsystems

In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups:
Lemma 3.16. Let $G$ be a compact group and $Rep(G)$ the category of finite ...

**25**

votes

**21**answers

6k views

### What are some mathematical concepts that were (pretty much) created from scratch and do not owe a debt to previous work?

Almost any mathematical concept has antecedents; it builds on, or is related to, previously known concepts. But are there concepts that owe little or nothing to previous work?
The only example I know ...

**16**

votes

**2**answers

1k views

### Original manuscript of Archimedes' cattle problem

Wikipedia states that
[Archimedes' cattle problem] was discovered by Gotthold Ephraim Lessing in a Greek manuscript containing a poem of forty-four lines, in the Herzog August Library in ...

**47**

votes

**19**answers

7k views

### Rediscovery of lost mathematics

Archimedes (ca. 287-212BC) described what are now known as the 13
Archimedean solids
in a lost work, later mentioned by Pappus.
But it awaited Kepler (1619) for the 13 semiregular polyhedra to be
...

**12**

votes

**4**answers

928 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 ...

**19**

votes

**1**answer

764 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

**6**answers

1k views

### The origins of forcing in mathematical logic and other branches of mathematics

As everyone knows, forcing was created by Cohen to answer questions in set theory.
Question 1. What are the first applications of set theoretic forcing in other branches of mathematical logic, like ...

**16**

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 ...

**47**

votes

**4**answers

3k views

### Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative ...

**30**

votes

**4**answers

3k views

### What, precisely, does Klein's Erlangen Program state?

People write that the Erlangen Program is a "program" (like the "Langlands Program"), i.e. a series of related conjectures, which in this case were all solved. There are various intuitive accounts, ...

**25**

votes

**7**answers

1k views

### Concise model of modern fiat money and its non-conservation

A confession: I have never really understood the basic model of fiat money and central banking, by which a central bank controls the money supply. By the standards of someone trained in mathematics, ...

**23**

votes

**4**answers

868 views

### Communal problem books

A certain class of books is defined as follows: (1) the book was kept for years in a cafe or mathematics library; (2) the primary contents are research problems and comments, handwritten by resident ...