Highest scored questions
159,026 questions
1072
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296
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Examples of common false beliefs in mathematics
The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...
531
votes
3
answers
58k
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Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?
Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
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430
votes
16
answers
65k
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Why do roots of polynomials tend to have absolute value close to 1?
While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
- 48.8k
424
votes
93
answers
149k
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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 ...
406
votes
85
answers
189k
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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 ...
401
votes
53
answers
151k
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Widely accepted mathematical results that were later shown to be wrong?
Are there 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 later, possibly ...
399
votes
23
answers
69k
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Thinking and Explaining
How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
394
votes
115
answers
110k
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Not especially famous, long-open problems which anyone can understand
Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list in ...
368
votes
31
answers
80k
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Geometric interpretation of trace
This afternoon I was speaking with some graduate students in the department and we came to the following quandary;
Is there a geometric interpretation of the trace of a matrix?
This question ...
338
votes
16
answers
159k
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What's a mathematician to do?
I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of ...
333
votes
34
answers
96k
views
Why is a topology made up of 'open' sets? [closed]
I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
320
votes
22
answers
104k
views
Why do so many textbooks have so much technical detail and so little enlightenment? [closed]
I think/hope this is okay for MO.
I often find that textbooks provide very little in the way of motivation or context. As a simple example, consider group theory. Every textbook I have seen that ...
298
votes
34
answers
53k
views
What are some reasonable-sounding statements that are independent of ZFC?
Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."
For example, suppose $A$ is an abelian group such ...
296
votes
125
answers
93k
views
What are some examples of colorful language in serious mathematics papers?
The popular MO question "Famous mathematical quotes" has turned
up many examples of witty, insightful, and humorous writing by
mathematicians. Yet, with a few exceptions such as Weyl's "angel of
...
296
votes
8
answers
143k
views
Philosophy behind Mochizuki's work on the ABC conjecture
Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
295
votes
34
answers
40k
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Which journals publish expository work?
I wonder if anyone else has noticed that the market for expository papers in mathematics is very narrow (more so than it used to be, perhaps).
Are there any journals which publish expository work, ...
294
votes
8
answers
22k
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Polynomial representing all nonnegative integers
Lagrange proved that every nonnegative integer is a sum of 4 squares.
Gauss proved that every nonnegative integer is a sum of 3 triangular numbers.
Is there a 2-variable polynomial $f(x,y) \in \...
- 23.8k
283
votes
69
answers
143k
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Awfully sophisticated proof for simple facts
It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ...
282
votes
47
answers
110k
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Examples of unexpected mathematical images
I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...
263
votes
29
answers
89k
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Mathematical games interesting to both you and a 5+-year-old child
Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me...
How to make both of us to do what they want ? I guess ...
256
votes
41
answers
99k
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A single paper everyone should read? [closed]
Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues.
Do ...
256
votes
16
answers
71k
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Why worry about the axiom of choice?
As I understand it, it has been proven that the axiom of choice is independent of the other axioms of set theory. Yet I still see people fuss about whether or not theorem X depends on it, and I don't ...
252
votes
37
answers
178k
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Best algebraic geometry textbook? (other than Hartshorne)
I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best?
It can be a book, preprint, online lecture note, webpage, etc.
One suggestion ...
251
votes
29
answers
168k
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Intuitive crutches for higher dimensional thinking
I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-
An engineer, a physicist, and a mathematician are discussing how to visualise four ...
243
votes
8
answers
31k
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Need advice or assistance for son who is in prison. His interest is scattering theory
The letter below is written by my son. I have been sending him text books and looking for answers on the internet to keep his interest up. He has progressed so far on his own and now he needs ...
239
votes
14
answers
76k
views
Have any long-suspected irrational numbers turned out to be rational?
The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...
238
votes
46
answers
90k
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Most interesting mathematics mistake?
Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...
238
votes
10
answers
44k
views
If $f$ is infinitely differentiable then $f$ coincides with a polynomial
Let $f$ be an infinitely differentiable function on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then does $f$ coincide on $[0,1]$ ...
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236
votes
36
answers
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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 $15$-...
234
votes
16
answers
57k
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What elementary problems can you solve with schemes?
I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-...
231
votes
13
answers
42k
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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 ...
- 65.4k
231
votes
4
answers
16k
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Is $\mathbb R^3$ the square of some topological space?
The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \...
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230
votes
89
answers
45k
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Your favorite surprising connections in mathematics
There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the ...
229
votes
20
answers
39k
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How can a mathematician handle the pressure to discover something new?
Suppose I'm an aspiring mathematician-to-be, who started doing research. Although this is really what I love doing, I found that one disturbing point is that there's always the pressure of discovering ...
228
votes
9
answers
24k
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John Nash's Mathematical Legacy
It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 (reference). My condolences to his family and friends.
Maybe this is an appropriate time to ask a ...
226
votes
11
answers
26k
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Refereeing a Paper [closed]
I've refereed at least a dozen papers in my (short) career so far and I still find the process completely baffling. I'm wondering what is actually expected and what people tend to do...
Some things ...
226
votes
4
answers
16k
views
A game on Noetherian rings
A friend suggested the following combinatorial game. At any time, the state of the game is a (commutative) Noetherian ring $\neq 0$. On a player's turn, that player chooses a nonzero non-unit element ...
- 149k
222
votes
8
answers
35k
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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 ...
- 14.3k
222
votes
0
answers
18k
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Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?
Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
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220
votes
140
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49k
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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 ...
218
votes
67
answers
47k
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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 ...
218
votes
24
answers
48k
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What is torsion in differential geometry intuitively?
Hi,
given a connection on the tangent space of a manifold, one can define its torsion:
$$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$
What is the geometric picture behind this definition&...
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217
votes
28
answers
53k
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The most outrageous (or ridiculous) conjectures in mathematics
The purpose of this question is to collect the most outrageous (or ridiculous) conjectures in mathematics.
An outrageous conjecture is qualified ONLY if:
1) It is most likely false
(Being hopeless is ...
214
votes
40
answers
38k
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Demonstrating that rigour is important
Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with ...
212
votes
52
answers
82k
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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:
...
212
votes
14
answers
60k
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Why doesn't mathematics collapse even though humans quite often make mistakes in their proofs?
To begin with, I am aware of these questions, which seems to be related:
How do I fix someone's published error?, Examples of common false beliefs in mathematics, When have we lost a body of ...
208
votes
72
answers
51k
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What are your favorite instructional counterexamples?
Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical.
In many branches of mathematics, it seems to me that a good counterexample can be ...
201
votes
67
answers
47k
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Examples of eventual counterexamples
Define an "eventual counterexample" to be
$P(a) = T $ for $a < n$
$P(n) = F$
$n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to ...
200
votes
89
answers
54k
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Examples of great mathematical writing
This question is basically from Ravi Vakil's web page, but modified for Math Overflow.
How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to ...
197
votes
94
answers
107k
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Famous mathematical quotes [closed]
Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?
Standard community wiki rules apply: one ...