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1500 questions
128
votes
13 answers

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 every $n$? This question is motivated by a…
Johan Wästlund
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128
votes
10 answers

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 difficulty. Ever since, I have wondered whether…
gowers
  • 28,729
127
votes
19 answers

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 the formula ends a sentence? Like: This is the…
Jose Capco
  • 2,175
127
votes
2 answers

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 transformations in both domain and range. For degree 1 and…
Bill Thurston
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127
votes
63 answers

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 Counterexamples in topology, so I think it's time for:…
Dylan Wilson
  • 13,183
127
votes
13 answers

Why are modular forms interesting?

Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about modular forms (except they're intrinsically…
Qfwfq
  • 22,715
127
votes
15 answers

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 useful for other people too. I'm interested in…
Akhil Mathew
  • 25,291
126
votes
13 answers

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 $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$. This can be proved, for example, by writing the inverse…
Frank Thorne
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126
votes
67 answers

Math puzzles for dinner

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 party are each receive hats with different prime…
Richard Dore
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125
votes
15 answers

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), there is certainly Taylor expansion. They have a quantity…
Qfwfq
  • 22,715
124
votes
23 answers

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 statements that imply RH or its negation. Here is…
Jon Bannon
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124
votes
4 answers

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 this claim, in Jacobson's "Lectures in Abstract…
Harry Gindi
  • 19,374
123
votes
15 answers

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 undergraduates. Often, while seeing some lecture…
Dylan Wilson
  • 13,183
123
votes
14 answers

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 which the problem was solved. I think that most…
Mark S
  • 2,143
122
votes
37 answers

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 have a problem like this please post it…
Anton Petrunin
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