# Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

13,392
questions

388
votes

80
answers

178k
views

### Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I ...

232
votes

36
answers

32k
views

### Conway's lesser-known results

John Horton Conway is known for many achievements:
Life, the three sporadic groups in the "Conway constellation," surreal numbers, his "Look-and-Say" sequence analysis, the Conway-Schneeberger $15$-...

224
votes

12
answers

36k
views

### Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain.
I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...

204
votes

50
answers

74k
views

### Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?
Please give a new way in each answer, and if possible give reference. I start by giving two:
...

186
votes

18
answers

13k
views

### Great graduate courses that went online recently

In 09.2020 by pure chance I discovered the YouTube channel of Richard Borcherds where he gives graduate courses in Group Theory, Algebraic Geometry, Schemes, Commutative Algebra, Galois Theory, Lie ...

175
votes

11
answers

27k
views

### Do you know important theorems that remain unknown?

Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost
nobody knows about them. If you provide an ...

139
votes

4
answers

60k
views

### What are "perfectoid spaces"?

This talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them?
Edit: A bit ...

129
votes

15
answers

31k
views

### Statistics for mathematicians

I'm looking for an overview of statistics suitable for the mathematically mature reader: someone familiar with measure theoretic probability at say Billingsley level, but almost completely ignorant of ...

125
votes

22
answers

8k
views

### Books that teach other subjects, written for a mathematician

Say I am a mathematician who doesn't know any chemistry but would like to learn it. What books should I read?
Or say I want to learn about Einstein's theory of relativity, but I don't even know much ...

119
votes

7
answers

14k
views

### Topology and the 2016 Nobel Prize in Physics

I was very happy to learn that the work which led to the award of the 2016 Nobel Prize in Physics (shared between David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz) uses Topology. In ...

112
votes

11
answers

17k
views

### On mathematical arguments against Quantum computing

Quantum computing is a very active and rapidly expanding field of research. Many companies and research institutes are spending a lot on this futuristic and potentially game-changing technology. Some ...

98
votes

30
answers

25k
views

### Errata for Atiyah–Macdonald

Is there a good list of errata for Atiyah–Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists ...

97
votes

9
answers

10k
views

### Theoretical physics: Why not just $\mathbb{R}^4$?

You and I are having a conversation:
"Okay," I say, "I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles."
"...

91
votes

36
answers

15k
views

### The concept of duality

I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come....

91
votes

9
answers

35k
views

### Is Mac Lane still the best place to learn category theory?

For a student embarking on a study of algebraic topology, requiring a knowledge of basic category theory, with a long-term view toward higher/stable/derived category theory, ...
Is Mac Lane still ...

90
votes

13
answers

12k
views

### Deep learning / Deep neural nets for mathematician

I am interested in finding out the math ideas behind the technologies that are under the umbrella of "Deep Learning" or "Deep neural nets".
Most of the papers/books that are often quoted in papers/...

86
votes

0
answers

15k
views

### Hironaka's proof of resolution of singularities in positive characteristics

Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier...
Unlike Atiyah's paper, Hironaka's paper does not have a ...

85
votes

6
answers

16k
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 (...

83
votes

4
answers

13k
views

### How to find ICM talks?

I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in ...

82
votes

17
answers

10k
views

### Examples of algorithms requiring deep mathematics to prove correctness

I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course).
I hope this is not too broad.

79
votes

18
answers

24k
views

### What programming language should a professional mathematician know? [closed]

More and more I am becoming convinced that one should know at least one programming language very well as a mathematician of this century. Is my conviction justified, or not applicable?
If I am right,...

79
votes

2
answers

6k
views

### Vladimir Voevodsky's works

Vladimir Voevodsky has made several contributions in abstract algebraic geometry, focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and ...

78
votes

11
answers

11k
views

### What are examples of (collections of) papers which "close" a field?

There is sometimes talk of fields of mathematics being "closed", "ended", or "completed" by a paper or collection of papers. It seems as though this could happen in two ways:
A total characterisation,...

76
votes

15
answers

8k
views

### Sophisticated treatments of topics in school mathematics

Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples ...

74
votes

31
answers

5k
views

### Atlas-like websites on specific areas of mathematics

In this post, we look for the existing atlas-like websites providing well-presented classifications or database about some specific areas of mathematics. Here are some examples:
GroupNames: https://...

74
votes

22
answers

15k
views

### Essays and thoughts on mathematics

Many distinguished mathematicians, at some point of their career,
collected their thoughts on mathematics (its aesthetic, purposes,
methods, etc.) and on the work of a mathematician in written ...

73
votes

9
answers

11k
views

### What is the significance of non-commutative geometry in mathematics?

This is a question that has been winding around my head for a long time and I have not found a convincing answer. The title says everything, but I am going to enrich my question by little more ...

72
votes

17
answers

9k
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 ...

71
votes

2
answers

4k
views

### Does iterating the derivative infinitely many times give a smooth function whenever it converges?

I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let $d = \frac{d}{dx}$, and let $f \in C^{\infty}(\mathbb{R})$ be such that for every real $x$, $$g(...

71
votes

3
answers

9k
views

### Has the mathematical content of Grothendieck's "Récoltes et Semailles" been used?

This question is partly motivated by Never appeared forthcoming papers.
Motivation
Grothendieck's "Récoltes et Semailles" has been cited on various occasions on this forum. See for instance ...

70
votes

8
answers

9k
views

### Category theory and set theory: just a different language, or different foundation of mathematics?

This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics.
I am asking for a reference. In order to make the reference request as ...

70
votes

9
answers

6k
views

### How does one find out what's happening in contemporary mathematics research?

How does one find out what's happening in contemporary mathematics research?
EDIT: I should have mentioned that I am looking for open access online sources. It so happens that I have been outside ...

68
votes

4
answers

11k
views

### Nelson's program to show inconsistency of ZF

At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say:
Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel ...

68
votes

9
answers

8k
views

### What are "classical groups"?

Unlike many other terms in mathematics which have a universally understood meaning (for instance, "group"), the term classical group seems to have a fuzzier definition. Apparently it originates with ...

66
votes

5
answers

16k
views

### What is a chess piece mathematically?

Historically, the current "standard" set of chess pieces wasn't the only existing alternative or even the standard one. For instance, the famous Al-Suli's Diamond Problem (which remained ...

66
votes

4
answers

10k
views

### $C^1$ isometric embedding of flat torus into $\mathbb{R}^3$

I read (in a paper by Emil Saucan) that the flat torus may be isometrically embedded
in $\mathbb{R}^3$ with a $C^1$ map by the Kuiper extension of the Nash Embedding Theorem,
a claim repeated in this ...

66
votes

5
answers

8k
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 ...

65
votes

13
answers

19k
views

### Video lectures for algebraic geometry

Are there any good video lectures for studying algebraic geometry?

64
votes

17
answers

15k
views

### Good introductory references on algebraic stacks?

Are there any good introductory texts on algebraic stacks?
I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also ...

64
votes

3
answers

10k
views

### Is this differential identity known?

Recently I discovered the differential identity
$$ \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = \frac{(1 \times 3 \times \dots \times k)^2}{(1+x^2)^{(k+2)/2}}$$
valid for any odd natural number $k$; for ...

63
votes

10
answers

5k
views

### Fascinating moments: equivalent mathematical discoveries

One of the delights in mathematical research is that some (mostly deep) results in one area remain unknown to mathematicians in other areas, but later, these discoveries turn out to be equivalent!
...

63
votes

6
answers

4k
views

### Shortest closed curve to inspect a sphere

Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and
exterior to $S$
which has the property that every point $x$ on $S$ is visible to some point $y$ of $...

63
votes

6
answers

7k
views

### The logic of Buddha: a formal approach

Buddhist logic is a branch of Indian logic (see also Nyaya), one of the three original traditions of logic, alongside the Greek and the Chinese logic. It seems Buddha himself used some of the features ...

62
votes

14
answers

20k
views

### A reading list for topological quantum field theory?

Can you suggest a reading list, or at least a few papers that you think would be useful, for a beginner in topological quantum field theory? I know what the curvature of a connection is, know basic ...

60
votes

19
answers

87k
views

### Suggestions for a good Measure Theory book

I have taken analysis and have looked at different measures, but I am currently looking at realizing a certain problem in a different light and feel that I need a better background in various measures ...

60
votes

9
answers

8k
views

### Fundamental groups of noncompact surfaces

I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...

60
votes

12
answers

17k
views

### Advanced Differential Geometry Textbook

I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.
In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...

60
votes

1
answer

5k
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 ...

59
votes

6
answers

11k
views

### Why isn't integral defined as the area under the graph of function?

In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also ...

59
votes

8
answers

6k
views

### Natural transformations as categorical homotopies

Every text book I've ever read about Category Theory gives the definition of natural transformation as a collection of morphisms which make the well known diagrams commute.
There is another possible ...