Highest scored questions
159,027 questions
130
votes
10
answers
20k
views
Are there any very hard unknots?
Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any ...
- 29k
129
votes
74
answers
20k
views
Most helpful math resources on the web
What are really helpful math resources out there on the web?
Please don't only post a link but a short description of what it does and why it is helpful.
Please only one resource per answer and let ...
129
votes
19
answers
70k
views
Periods and commas in mathematical writing
I just realized that I am a barbarian when it comes to writing. But I am not entirely sure, so this might be the right place to ask. When typing display-mode formulae do you guys add a period after ...
129
votes
15
answers
51k
views
A learning roadmap for algebraic geometry
Unfortunately this question is relatively general, and also has a lot of sub-questions and branches associated with it; however, I suspect that other students wonder about it and thus hope it may be ...
129
votes
2
answers
16k
views
What are the shapes of rational functions?
I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
- 25.1k
128
votes
13
answers
27k
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{...
- 7,347
128
votes
13
answers
24k
views
Checkmate in $\omega$ moves?
Is there a chess position with a finite number of pieces on the infinite chess board $\mathbb{Z}^2$ such that White to move has a forced win, but Black can stave off mate for at least $n$ moves for ...
- 5,492
128
votes
12
answers
12k
views
Spectral sequences: opening the black box slowly with an example
My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials.
...
- 13.5k
127
votes
63
answers
27k
views
Counterexamples in algebra?
This is certainly related to "What are your favorite instructional counterexamples?", but I thought I would ask a more focused question. We've all seen Counterexamples in analysis and ...
127
votes
23
answers
37k
views
Collection of equivalent forms of Riemann Hypothesis
This forum brings together a broad enough base of mathematicians to collect a "big list" of equivalent forms of the Riemann Hypothesis...just for fun. Also, perhaps, this collection could include ...
127
votes
4
answers
32k
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Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional
A very important theorem in linear algebra that is rarely taught is:
A vector space has the same dimension as its dual if and only if it is finite dimensional.
I have seen a total of one proof of ...
126
votes
67
answers
47k
views
Math puzzles for dinner [closed]
You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a ...
126
votes
15
answers
15k
views
Does Physics need non-analytic smooth functions?
Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), ...
- 23.4k
125
votes
31
answers
16k
views
Papers that debunk common myths in the history of mathematics
What are some good papers that debunk common myths in the history of mathematics?
To give you an idea of what I'm looking for, here are some examples.
Tony Rothman, "Genius and biographers: The ...
125
votes
4
answers
8k
views
What do the stable homotopy groups of spheres say about the combinatorics of finite sets?
The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way:
$\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$
$\mathbb{Z}\...
- 23k
124
votes
40
answers
21k
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 ...
124
votes
15
answers
19k
views
When and how is it appropriate for an undergraduate to email a professor out of the blue?
This may not be appropriate for MathOverflow, as I haven't seen precedent for this type of question. But the answer is certainly of interest to me, and (I think) would be of interest to many other ...
124
votes
17
answers
18k
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 “...
124
votes
37
answers
12k
views
One-step problems in geometry
I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory).
If you ...
123
votes
35
answers
18k
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
...
123
votes
25
answers
18k
views
"Mathematics talk" for five year olds
I am trying to prepare a "mathematics talk" for five year olds from my daughter's elementary school. I have given many mathematics talks in my life but this one feels
very tough to prepare. Could the ...
123
votes
18
answers
14k
views
How do you decide whether a question in abstract algebra is worth studying?
Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...
123
votes
12
answers
29k
views
How to solve $f(f(x)) = \cos(x)$?
I found the following equation on some web page I cannot remember, and found it interesting:
$$f(f(x))=\cos(x)$$
Out of curiosity I tried to solve it, but realized that I do not have a clue how to ...
- 1,571
123
votes
9
answers
14k
views
Breakthroughs in mathematics in 2021
This is somehow a general (and naive) question, but as specialized mathematicians we usually miss important results outside our area of research.
So, generally speaking, which have been important ...
122
votes
41
answers
29k
views
What are some very important papers published in non-top journals?
There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here.
My concern in this question is slightly ...
122
votes
14
answers
36k
views
What are some noteworthy "mic-drop" moments in math?
Oftentimes in math the manner in which a solution to a problem is announced becomes a significant chapter/part of the lore associated with the problem, almost being remembered more than the manner in ...
122
votes
5
answers
27k
views
Is the series $\sum_n|\sin n|^n/n$ convergent?
Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
- 4,535
122
votes
4
answers
39k
views
Is the analysis as taught in universities in fact the analysis of definable numbers?
Ten years ago, when I studied in university, I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...
- 10.1k
122
votes
7
answers
15k
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 ...
122
votes
16
answers
34k
views
How do you keep your research notes organized? [closed]
One of the things I struggle with most in doing research is keeping my notes organized. Since research tends to do a lot of branching, keeping notes in a linear fashion seems useless to me. On the ...
121
votes
33
answers
141k
views
Mathematicians who were late learners?-list [closed]
It is well-known that many great mathematicians were prodigies.
Were there any great mathematicians who started off later in life?
121
votes
15
answers
101k
views
Sum of 'the first k' binomial coefficients for fixed $N$
I am interested in the function $$f(N,k)=\sum_{i=0}^{k} {N \choose i}$$ for fixed $N$ and $0 \leq k \leq N $. Obviously it equals 1 for $k = 0$ and $2^{N}$ for $k = N$, but are there any other ...
- 1,288
121
votes
29
answers
27k
views
Where does a math person go to learn statistical mechanics?
The more math I read, the more I see concepts from statistical mechanics popping up -- all over the place in combinatorics and dynamical systems, but also in geometric situations. So naturally I've ...
121
votes
5
answers
13k
views
What do epimorphisms of (commutative) rings look like?
(Background: In any category, an epimorphism is a morphism $f:X\to Y$ which is "surjective" in the following sense: for any two morphisms $g,h:Y\to Z$, if $g\circ f=h\circ f$, then $g=h$. Roughly, "...
- 24.1k
120
votes
33
answers
15k
views
Examples of theorems misapplied to non-mathematical contexts
For something I'm writing -- I'm interested in examples of bad arguments which involve the application of mathematical theorems in non-mathematical contexts. E.G. folks who make theological arguments ...
120
votes
5
answers
33k
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 ...
- 156k
119
votes
8
answers
35k
views
Zagier's one-sentence proof of a theorem of Fermat
Zagier has a very short proof (MR1041893, JSTOR) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an involution of the set $S= \lbrace (x,y,z) \...
- 6,224
119
votes
6
answers
10k
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What properties make $[0,1]$ a good candidate for defining fundamental groups?
The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in J$. There is an obvious generalization of the ...
- 5,759
118
votes
10
answers
77k
views
What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed to $u^T v$?
In a lot of computational math, operations research, such as algorithm design for optimization problems and the like, authors like to use $$\langle \cdot, \cdot \rangle$$ as opposed to $$(\cdot)^T (\...
118
votes
12
answers
14k
views
How do I fix someone's published error?
Paper A is in the literature, and has been for more than a decade.
An error is discovered in paper A and is substantial in that many
details are affected, although certain fundamental properties
...
118
votes
15
answers
58k
views
Top specialized journals
In geometry/topology, there are (at least) three specialized journals that end up publishing a large fraction of the best papers in the subject -- Geometry and Topology, JDG, and GAFA.
What journals ...
117
votes
22
answers
39k
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.
116
votes
2
answers
31k
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Why is the Hodge Conjecture so important?
The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm ...
- 1,263
116
votes
24
answers
22k
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Tools for collaborative paper-writing
I personally use a revision control system (git) to manage my own paper writing, back things up, and synchronize between different machines. However, I've found most programmer's revision control ...
115
votes
32
answers
21k
views
What notions are used but not clearly defined in modern mathematics?
"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
Felix Klein
What notions are used but not ...
115
votes
36
answers
31k
views
Quick proofs of hard theorems
Mathematics is rife with the fruit of abstraction. Many problems which first are solved via "direct" methods (long and difficult calculations, tricky estimates, and gritty technical theorems) later ...
115
votes
3
answers
5k
views
The number $\pi$ and summation by $SL(2,\mathbb Z)$
Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. (it is the defect in the triangle inequality)
Then, we discovered by heuristic arguments and then verified by computer that
$$\...
- 5,063
114
votes
96
answers
16k
views
What would you want to see at the Museum of Mathematics? [closed]
EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...
114
votes
30
answers
17k
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New grand projects in contemporary math
When I was a graduate student in math (mid-late eighties and early nineties) the arena was dominated by a few grand projects: for instance, Misha Gromov's hyperbolic groups, which spread into many ...
114
votes
34
answers
86k
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Why do we teach calculus students the derivative as a limit?
I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher ...