**10**

votes

**0**answers

375 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### Unpublished Lecture Notes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 ...

**18**

votes

**1**answer

356 views

### Biholomophic non-Algebraically Isomorphic Varieties

Recently, when writing a review for MathSciNet, the following question arose:
Is it true that two smooth complex varieties that are biholomorphic are algebraically isomorphic? The converse is true ...

**10**

votes

**0**answers

400 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 ...

**19**

votes

**5**answers

2k views

### Why did Bourbaki ignore the theory of categories? [closed]

QUESTION
They had plenty of time to adopt the theory of categories. They had Eilenberg, then Cartan, then Grothendieck. Did they feel that they have established their approach already, that it's too ...

**5**

votes

**1**answer

333 views

### Generalization of the Lefschetz fixed point theorem

I have encountered a certain generalization of the Lefschetz fixed point theorem as folklore, and I am hoping that someone out there knows its provenance or can otherwise refer me to a source where it ...

**8**

votes

**0**answers

389 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 ...

**6**

votes

**1**answer

211 views

### Did Smith correctly state the mass formula?

Did Smith correctly state the mass formula?
H.J.S. "normal form" Smith was the first, in 1867, to state the mass formula for integral quadratic forms in a genus of 4 or more variables. This was ...

**1**

vote

**0**answers

128 views

### Reference for original paper (but translated to English) of Matiyasevich's proof of Fibonacci relation being Diophantine?

Hello. I am a maths undergraduate. I am doing a project about history of mathematics. I am looking for the original solution to Hilbert's 10th problem, or at least the theorems that is accessible to ...

**11**

votes

**3**answers

657 views

### The role of the Automatic Groups in the history of Geometric Group Theory

What is the role of the theory of Automatic Groups in the history of Geometric Group Theory?
Motivation:
When I read through the "Word Processing in Groups" I was amazed by the supreme beauty and ...

**9**

votes

**0**answers

398 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 ...

**58**

votes

**24**answers

5k 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 ...

**2**

votes

**0**answers

260 views

### What is the oldest known evidence of application of mathematics?

According to Wikipedia the Lebombo bone (age 35 KY) and the Ishango bone (age at least 20 KY) presently are believed to show the first evidence for application of mathematics by humans. (Possibly ...

**9**

votes

**1**answer

310 views

### Discrete Morse theory and chess

There are many mathematical objects that are similar to groups and Cayley graphs of groups but lack homogeneity in some sense. Graphs of webpages with edges corresponding to links are one example. ...

**21**

votes

**2**answers

1k views

### Hahn's Embedding Theorem and the oldest open question in set theory

Hans Hahn is often credited with creating the modern theory of ordered algebraic systems with the publication of his paper Über die nichtarchimedischen Grössensysteme (Sitzungsberichte der ...

**8**

votes

**4**answers

832 views

### History of the high-dimensional volume paradox

Inscribe an $n$-ball in an $n$-dimensional hypercube of side equal to 1, and let $n \rightarrow \infty$. The hypercube will always have volume 1, while it is a fun folk fact (FFF) that the volume of ...

**11**

votes

**1**answer

198 views

### What is flexible about flexible algebras?

A possibly non-associative algebra is flexible if it satisfies the identity $$(xy)x=x(yx).$$ This is clearly a very weak form of associativity —and obviously an associative algebra is flexible— but it ...

**7**

votes

**1**answer

130 views

### Who first observed that Conjugate Gradient for Symmetric Positive Definite linear systems is a Krylov method?

Conjugate gradient was originally presented in the 50's before the modern understanding of Krylov subspaces (and the resulting iterative methods) was fully realized. As such, the method was derived ...

**28**

votes

**13**answers

1k views

### Great mathematics books by pre-modern authors

Last summer, I read Euclid's Elements, and it was an eye-opening experience; I had assumed that three thousand years' difference would make the notation incomprehensible and the reasoning alien, but ...

**5**

votes

**1**answer

631 views

### Why do mathematicians prefer one definition over the other when they both define the same concept?

Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of ...

**1**

vote

**1**answer

299 views

### Newton integration without integration

Consider a function f continuous on a compact interval.
Approximate it by a sequence of polygonal functions (you can).
Then consider a sequence of primitives of the polygonal functions (you can).
...

**6**

votes

**1**answer

207 views

### Does the vanishing of the Poisson bracket on $S(\mathfrak{g})^{\mathfrak{g}}$ inspire the disover of Duflo's isomorphism theorem?

For any finite dimensional Lie algebra $\mathfrak{g}$, we know that the universal enveloping algebra $U(\mathfrak{g})$ is a deformation of the symmetric algebra $S(\mathfrak{g})$. In fact let's define
...

**4**

votes

**2**answers

235 views

### Was Desargues more an Euclid or an Eudoxos?

In the course of preparing lessons on projective geometry I want to give an account on the historical development. It is easy to obtain an overview of the history starting with G. Desargues. And with ...

**13**

votes

**1**answer

1k views

### Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...

**3**

votes

**0**answers

119 views

### Citation for subset complement result

Let $S=\lbrace s_1,\ldots,s_n \rbrace \subset \lbrace1,\ldots,2n\rbrace$. Consider two operations on $S$: the complement $C(S)=\lbrace 1,\ldots,2n \rbrace \setminus S$ and a reflection* ...

**0**

votes

**1**answer

377 views

### Has the controversy about *fiducial distribution* been settled? [closed]

Has the controversy about the correct meaning of Fisher's notion fiducial distribution meanwhile been settled? And are there newer applications than quoted in the following literature?
G.P. Klimov: ...

**2**

votes

**3**answers

689 views

### Lapses of “the early proponents of the doctrine of limits”

I have a question that I have been wondering about for a long time without finding any answer. Concerning the period around 1900, Robinson commented in his 1966 book that "there is in the writings of ...

**8**

votes

**3**answers

748 views

### who invented projective space $\mathbb{P}^n$?

Who invented projective space $\mathbb{P}^n$ as an extension of the usual affine space $\mathbb{A}^n$?
Who was the first person to consider projective closure of
plane affine algebraic curves (curves ...

**17**

votes

**9**answers

3k views

### Was the early calculus inconsistent?

This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY.
George Berkeley wrote in 1734 with reference to the early calculus that ...

**26**

votes

**3**answers

2k views

### What is the difference between an automorphic form and a modular form?

This is more of a question about terminology than about math.
The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it ...

**3**

votes

**0**answers

182 views

### Where was the arithmetic zeta function of a scheme first defined?

Let $X$ be an arithmetic scheme, that is, a scheme of finite type over the integers. We denote the set of closed points of $X$ by $|X|$. For every $x\in|X|$, write $N(x)$ for the cardinality of the ...

**4**

votes

**0**answers

194 views

### origin of the notion of “network” in graph theory

In current graph theory, a "network" is a precisely defined object: It is a directed graph associated with a function, the "capacity", which is defined on the edge set and has certain specific ...

**18**

votes

**2**answers

1k views

### Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated ...

**2**

votes

**2**answers

157 views

### local subrings of matrix ring

When is the subring (containing 1) of a matrix ring $M_n(k)$ over a field $k$
is local?
I would be grateful for every reference concerning this matter,
Thank you!

**19**

votes

**4**answers

891 views

### The role of ANR in modern topology

Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ ...

**20**

votes

**7**answers

2k views

### Modern developments in finite-dimensional linear algebra

Are there any major fundamental results in finite-dimensional linear algebra discovered after early XX century? Fundamental in the sense of non-numerical (numerical results, of course, are still ...

**5**

votes

**1**answer

302 views

### First mention of the fundamental bigroupoid of a space?

The fundamental bigroupoid $\Pi_2(X)$ of a space $X$ was independently described by Hardie, Kamps and Kieboom (paywall) and Stevenson (arXiv) around the year 2000. HKK cite Baez-Dolan's seminal HDA0, ...

**15**

votes

**1**answer

527 views

### Mendelson's *Mathematical Logic* and the missing Appendix on the consistency of PA

A very soft question, but I hope not out of order here.
In the first edition of Elliott Mendelson's classic Introduction to Mathematical Logic (1964) there is an appendix, giving a version of ...

**4**

votes

**0**answers

341 views

### dreams of mathematics(ramannujan) others? [closed]

"Ramanujan credited his acumen to his family Goddess, Namagiri of Namakkal. He looked to her for inspiration in his work,[84] and claimed to dream of blood drops that symbolised her male consort, ...

**34**

votes

**19**answers

5k 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 ...

**4**

votes

**3**answers

381 views

### Biographical Information Concerning Henry Sherwin

Henry Sherwin's Mathematical Tables achieved some popularity. The first edition was published ~1706 and the fifth and last in 1771. Some editions were more erroneous than others and the error rate was ...

**4**

votes

**1**answer

1k views

### Algebraic number theory: building and simplifying

This is a somewhat subjective question, about the past, present and especially future of algebraic number theory. I'm not at all in this area, but I'd be interested in an answer.
As we all know, ...

**4**

votes

**1**answer

543 views

### Ancient method to study Archimedean spiral

It is well-known the properties of Archimedean spiral ($\rho = k\phi$) which is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant ...

**1**

vote

**3**answers

919 views

### Who invented the expression “pairwise different” and what is its advantage over “different”

There are many applications of "pairwise", for instance different, disjunct, orthogonal, independent, intersecting, connected, and many more. Some of them like "pairwise intersecting" or "pairwise ...

**32**

votes

**31**answers

5k views

### Trichotomies in mathematics

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...

**21**

votes

**6**answers

904 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, ...

**1**

vote

**2**answers

233 views

### What are Normal Sets (Fréchet)?

In 1913, LEJ Brouwer started a new approach to give a topologist's definition of the notion dimension ("Über den natürlichen Dimensionsbegriff", Journal für die reine und angewandte Mathematik, 142, ...

**3**

votes

**1**answer

644 views

### Elements of the history of mathematics

Is it known who actually wrote Bourbaki's Elements of the History of Mathematics?

**7**

votes

**3**answers

1k views

### Retracted Mathematics Papers [closed]

Can anyone cite an example of a mathematics paper that has been retracted?
It is said that on the order of 100,000 new theorems enter the mathematics literature every year. For a number of reasons ...

**15**

votes

**2**answers

968 views

### What is the history of $\sqrt{}$

Why we use the symbol $\sqrt{}$ when we take square roots ? Anybody knows the history ?

**27**

votes

**4**answers

2k 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, ...