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1500 questions
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4 answers

What is Chern-Simons theory?

What is Chern-Simons theory? I have read the wikipedia entry, but it's pretty physics-y and I wasn't really able to get any sense for what Chern-Simons theory really is in terms of mathematics. Chern-Simons theory is supposed to be some kind of…
Kevin H. Lin
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44
votes
3 answers

When does iterating $z \mapsto z^2 + c$ have an exact solution?

If one iterates the map $z \mapsto z^2 + c$ there is obviously a simple formula for the sequence one gets if $c=0$. Less obviously, there is also a simple formula when $c = -2$ (use the identity $2 \cos(2x) = (2\cos(x))^2 - 2)$. Are there any other…
Richard Borcherds
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44
votes
3 answers

Publishing a Simple Paper as an Undergraduate

First off I apologize if this question does not belong here, I would be happy to hear about any better locations to post this on. I am a (first year) undergraduate mathematics student, and I recently discovered some interesting properties hidden in…
user918212
  • 1,025
44
votes
12 answers

How to explain to an engineer what algebraic geometry is?

This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most classical complex algebraic geometry for…
Qfwfq
  • 22,715
44
votes
4 answers

A curious process with positive integers

Let $k > 1$ be an integer, and $A$ be a multiset initially containing all positive integers. We perform the following operation repeatedly: extract the $k$ smallest elements of $A$ and add their sum back to $A$. Let $x_i$ be the element added on…
Mikhail Tikhomirov
  • 5,382
44
votes
6 answers

Explaining the main ideas of proof before giving details

I'll be the first to admit that this is a risky question to try to get away with on math overflow, but I'm going to give it a shot anyway. Roughly speaking, the question is this: Is it good to try to explain the main ideas of a complicated proof…
SBK
  • 1,141
44
votes
4 answers

Is there a universal countable group? (a countable group containing every countable group as a subgroup)

This recent MO question, answered now several times over, inquired whether an infinite group can contain every finite group as a subgroup. The answer is yes by a variety of means. So let us raise the stakes: Is there a countable group containing (a…
Joel David Hamkins
  • 224,022
44
votes
1 answer

Conjecturally unsafe RSA primes $p=27a^2+27a+7$

We got strong numerical evidence that primes of the form $p=27a^2+27a+7$ are unsafe for cryptographic purposes since they can be found in the factorization. Consider the following generic factoring algorithm for factoring $n$ which is divisible by…
joro
  • 24,174
44
votes
5 answers

Finding a 1-form adapted to a smooth flow

Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say that a smooth $1$-form $\theta$ on $M$ is adapted…
Terry Tao
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44
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2 answers

Bijection from the plane to itself that sends circles to squares

Let me apologize in advance as this is possibly an extremely stupid question: can one prove or disprove the existence of a bijection from the plane to itself, such that the image of any circle becomes a square? Or, more generally, are there any…
Tom Solberg
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44
votes
17 answers

What is your favorite proof of Tychonoff's Theorem?

Here is mine. It's taken from page 11 of "An Introduction To Abstract Harmonic Analysis", 1953, by…
Pierre-Yves Gaillard
  • 4,336
44
votes
7 answers

The missing link: an inequality

I've been working on a project and proved a few relevant results, but got stuck on one tricky problem: Conjecture. If $2\leq n\in\mathbb{N}$ and $0
T. Amdeberhan
  • 41,802
44
votes
4 answers

Did Gaston Julia ever get to see a computer-generated image of his eponymous set?

I learned from Wikipedia that Gaston Julia died in 1978. Is it known if he ever got to see a computer-generated image of the set named after him?
T. Donaldson
  • 457
44
votes
1 answer

Microwaving Cubes

First a little background. Microwaves do not heat uniformly. To help overcome this, your food is rotated, however this is not usually sufficient to produce totally uniform heating. Informally, this is the question: Is there a way of moving our food…
Mark Bell
  • 3,125
44
votes
5 answers

Does pi contain 1000 consecutive zeroes (in base 10)?

The motivation for this question comes from the novel Contact by Carl Sagan. Actually, I haven't read the book myself. However, I heard that one of the characters (possibly one of those aliens at the end) says that if humans compute enough digits…
Tony Huynh
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