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106 votes
15 answers
35k views

Most striking applications of category theory?

What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples: Joyal's ...
41 votes
4 answers
8k views

Math papers where the only issue is that someone else could've done it but didn't

Do editors for top math journals ever read a submitted paper, agree that there are no mistakes and the result is new, yet still reject it on the basis that this is a top math journal and someone could'...
22 votes
1 answer
3k views

A gerrymandering problem - can you always turn a tie into a landslide victory?

Note: Here we use $|A|$ to denote the Lebesgue measure of a measurable subset $A$ of $\mathbb R^2$. Your party is running for election! In your country, voters are approximately uniformly distributed. ...
Nate River's user avatar
  • 4,822
24 votes
8 answers
39k views

A symbol to denote the set of prime numbers ?

It strikes me that there is no widely accepted symbol to denote the set of usual prime numbers in $\mathbb{N}$. Look: $$\zeta(s)=\prod_{p\in \mathrm{?}}\frac{1}{(1-p^{-s})}$$ Wouldn't it be nicer ...
34 votes
6 answers
4k views

Why study finite topological spaces?

In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage: … this means that some concepts that I use freely and naturally in my personal thinking are foreign to ...
Wahome's user avatar
  • 737
92 votes
2 answers
14k views

Coming out as transgender in the mathematical community

I don't know if MO is the right place to ask such a question, but anyway it's my only hope to get an answer, and it's very important for me (not to say 'vital'); so let's try. I'm at this time a Ph.D....
74 votes
22 answers
17k 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 ...
28 votes
5 answers
5k views

What is the motivation for infinity category theory?

To my understanding, most mathematical theories can be simply understood in the view point of Category theory and its derivative theories. But what exactly is the motivation to study infinity category ...
tryst with freedom's user avatar
45 votes
17 answers
46k views

Good algebraic number theory books

I have just finished a master's degree in mathematics and want to learn everything possible about algebraic number fields and especially applications to the generalized Pell equation (my thesis topic),...
86 votes
27 answers
19k views

Which popular games have been studied mathematically?

I'm planning out some research projects I could do with undergraduates, and it struck me that problems analyzing games might be appropriate. As an abstract homotopy theorist, I have no experience with ...
150 votes
45 answers
29k views

Nontrivial theorems with trivial proofs

A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because ...
28 votes
3 answers
7k views

Famous examples of "serendipity" in 20th century mathematics

The term "serendipity" is commonly used in the literature to refer to the historical evidence that very often researchers make unexpected and beneficial discoveries by chance, while they are ...
59 votes
1 answer
32k views

If I want to study Jacob Lurie's books "Higher Topoi Theory", "Derived AG", what prerequisites should I have?

I've been told that it's important to know modern physics, Differential Geometry and Algebraic Topology for understanding higher structures. Is there any other prerequisite for understanding Lurie's ...
Chuang 's user avatar
  • 609
218 votes
8 answers
33k views

How to memorise (understand) Nakayama's lemma and its corollaries?

Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...
aglearner's user avatar
  • 14k
62 votes
68 answers
15k views

Mathematicians with both “very abstract” and “very applied” achievements

Gödel had a cosmological model. Hamel, primarily a mechanician, gave any vector space a basis. Plücker, best known for line geometry, spent years on magnetism. What other mathematicians had so distant ...
28 votes
1 answer
6k views

What is the cofinality of the co-infinite subsets of ${\bf N}$?

Let ${\mathcal A}$ be the family of subsets $A$ of the natural numbers ${\mathbf N}$ which are co-infinite (i.e., their complement is infinite). We partially order this family by set inclusion. A ...
Terry Tao's user avatar
  • 108k
24 votes
4 answers
35k views

Finding a cycle of fixed length

Is there any result about the time complexity of finding a cycle of fixed length $k$ in a general graph? All I know is that Alon, Yuster and Zwick use a technique called "color-coding", which has a ...
Hsien-Chih Chang 張顯之's user avatar
189 votes
79 answers
42k views

Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...
19 votes
6 answers
35k views

Pascal triangle and prime numbers

Back in the days when I was in high school, I developed a big interest about number theory specifically prime numbers and prefect numbers, I used to stay awake all night long with a bunch of sketch ...
Alix Axel's user avatar
  • 299
22 votes
4 answers
48k views

complexity of eigenvalue decomposition

what is the computational complexity of eigenvalue decomposition for a unitary matrix? is O(n^3) a correct answer?
Majid's user avatar
  • 221
60 votes
7 answers
25k views

Is the Jaccard distance a distance?

Wikipedia defines the Jaccard distance between sets A and B as $$J_\delta(A,B)=1-\frac{|A\cap B|}{|A\cup B|}.$$ There's also a book claiming that this is a metric. However, I couldn't find any ...
rgrig's user avatar
  • 1,335
43 votes
10 answers
45k views

Is square of Delta function defined somewhere?

I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere. In the beginning, this question might look strange. But by restricting the space of the test functions, ...
215 votes
67 answers
45k views

Proofs that require fundamentally new ways of thinking

I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is ...
31 votes
8 answers
16k views

Why are polynomials so useful in mathematics?

This is perhaps unanswerable, or perhaps I am too algebraically ignorant to phrase it cogently, but: Is there some identifiable reason that polynomials over $\mathbb{C}$, $\mathbb{R}$, $\mathbb{...
112 votes
19 answers
41k views

What is the definition of "canonical"?

I just received a referee report criticizing that I would too often use the word "canonical". I have a certain understanding of what "canonical" should stand for, but the report ...
158 votes
14 answers
39k views

What is an integrable system?

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "...
Gil Kalai's user avatar
  • 24.2k
67 votes
6 answers
17k 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 ...
Morteza Azad's user avatar
50 votes
11 answers
10k views

Publishing papers that became classics before they were submitted

Sometimes the following happens: a result is proven, but the author never submits a paper for publication. In some cases, a preprint appears. In some cases, the proof is so short that it can be ...
69 votes
28 answers
7k views

Results from abstract algebra which look wrong (but are true)

There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...
24 votes
8 answers
20k views

Interesting Applications of the Classical Stokes Theorem?

When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y ...
137 votes
28 answers
18k views

Which mathematical definitions should be formalised in Lean?

The question. Which mathematical objects would you like to see formally defined in the Lean Theorem Prover? Examples. In the current stable version of the Lean Theorem Prover, topological groups ...
119 votes
38 answers
19k views

Noteworthy, but not so famous conjectures resolved recent years

Conjectures play important role in development of mathematics. Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch ...
17 votes
9 answers
24k views

Why isn't likelihood a probability density function?

I've been trying to get my head around why a likelihood isn't a probability density function. My understanding says that for an event $X$ and a model parameter $m$: $P(X\mid m)$ is a probability ...
brabster's user avatar
  • 273
11 votes
0 answers
2k views

Eric T. Sawyer's proof of Fourier restriction conjecture

Some days ago Eric T. Sawyer uploaded a paper to arxiv claiming a proof of the Fourier restriction conjecture https://arxiv.org/pdf/2311.03145.pdf. If complete and correct this work will be a landmark ...
a curious fellow's user avatar
116 votes
5 answers
32k 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 $193{,}707{,}721\times 761{,}838{,}257{,}287$. There doesn't seem to be ...
David E Speyer's user avatar
163 votes
9 answers
28k views

Endless controversy about the correctness of significant papers

In principle, a mathematical paper should be complete and correct. New statements should be supported by appropriate proofs. But this is only theory. Because we often cannot enter into the smallest ...
22 votes
3 answers
3k views

Is the sum of the reciprocals of the products of pairs of coprime positive integers and their sums equal to 2?

Does the following hold?: $$ \sum_{a, b \in \mathbb{N}^+, \ \gcd(a,b) = 1} \frac{1}{ab(a+b)} \ = \ 2 $$ Numerical computations suggest this may hold, but on the other hand it would be quite ...
Stefan Kohl's user avatar
  • 19.5k
126 votes
13 answers
26k views

Should the formula for the inverse of a 2x2 matrix be obvious?

As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{...
Frank Thorne's user avatar
  • 7,199
113 votes
22 answers
37k views

What's the "best" proof of quadratic reciprocity?

For my purposes, you may want to interpret "best" as "clearest and easiest to understand for undergrads in a first number theory course," but don't feel too constrained.
4 votes
4 answers
35k views

Is it alright for STD error bars to be below zero?

I have some statistical data from which I want to graph the means and use the standard deviations as error bars. However this produces a graph with some of the error bars passing below zero. A ...
hoju's user avatar
  • 177
51 votes
10 answers
25k views

Mathematically interesting screensavers

A screensaver is a computer program that fills a computer screen with a moving pattern that eluminates each pixel for approximately the same proportion of time. Originally designed to prevent burn-in ...
121 votes
17 answers
17k views

Pressure to defend the relevance of one's area of mathematics

I am a set theorist. Since I began to study this subject, I became increasingly aware of negative attitudes about it. These were expressed both from an internal and an external perspective. By the “...
54 votes
16 answers
15k views

Why do we need random variables?

In this MathStackExchange post the question in the title was asked without much outcome, I feel. Edit: As Douglas Zare kindly observes, there is one more answer in MathStackExchange now. I am not ...
Filippo Alberto Edoardo's user avatar
58 votes
8 answers
35k views

Inverse gamma function?

This is an analysis question I remember thinking about in high school. Reading some of the other topics here reminded me of this, and I'd like to hear other people's solutions to this. We have the ...
jeremy's user avatar
  • 2,139
87 votes
12 answers
11k views

Why do we make such big deal about the 'unsolvability' of the quintic?

The unsolvability of a general quintic equation in terms of the basic arithmetic operations and $n$th roots (i.e. the Abel–Ruffini theorem) is considered a major result in the mathematical canon. I ...
Arthur's user avatar
  • 1,379
27 votes
1 answer
2k views

Is every real number in [0,1] a product of three (or more) Cantor set's numbers?

It is well known that every number $x$ in the unit interval $[0,1]$ is the arithmetic mean of two elements of the (triadic) Cantor set $C$. The way to see it I like the most: the Cantor set is the ...
Pietro Majer's user avatar
  • 56.5k
84 votes
11 answers
12k 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,...
17 votes
3 answers
23k views

Which hard mathematical problems do you have to solve to earn bitcoins ?

A virtual currency called bitcoins has been in the news recently. It is said that in order to "mine" bitcoins, you have to solve hard mathematical problems. Now, there are two kinds of mathematical ...
Chandan Singh Dalawat's user avatar
27 votes
29 answers
29k views

Alternative undergraduate analysis texts

Other than the standard baby Rudin, Spivak, and Stein-Shakarchi, are there other alternative and comprehensive analysis texts at the undergraduate level? For example something that has general results ...
47 votes
5 answers
28k views

Why was John Nash's 1950 Game Theory paper such a big deal?

I'm trying to understand why John Nash's 1950 2-page paper that was published in PNAS was such a big deal. Unless I'm mistaken, the 1928 paper by John von Neumann demonstrated that all n-player non-...
TSGM's user avatar
  • 563