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1500 questions
290
votes
34 answers
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, especially at the "intermediate" level? By…
Pete L. Clark
- 64,763
288
votes
7 answers
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 $f(x,y) \in \mathbf{Q}[x,y]$ such that $f(\mathbf{Z} \times…
Bjorn Poonen
- 23,617
280
votes
47 answers
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 unexpected turn, or let me have a moment of…
Per Alexandersson
- 15,598
271
votes
67 answers
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 example in this M.SE answer (the title of this question…
Mariano Suárez-Álvarez
- 46,795
262
votes
29 answers
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 for everybody who has children, that question…
Alexander Chervov
- 23,944
254
votes
41 answers
A single paper everyone should read?
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 you have such an example?
Let's try to go in the…
Ilya Nikokoshev
- 14,934
250
votes
29 answers
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 dimensions:
Engineer: I never really get it
Physicist: Oh…
Tom Boardman
- 3,190
249
votes
37 answers
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 per answer please. Also, please include an…
sanokun
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249
votes
16 answers
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 see the point. Yes, one can prove some pretty…
Paul Siegel
- 28,772
244
votes
8 answers
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 direction and assistance from a professional in…
Dan Cunningham
- 545
235
votes
36 answers
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 $15$-theorem, the Free-Will theorem—the list goes on and…
Joseph O'Rourke
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235
votes
10 answers
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 $[0,1]$ with some polynomial? If yes then how.
I thought…
C.S.
- 4,735
233
votes
14 answers
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 that $\zeta(3)$ is irrational in 1979.
There are…
I. J. Kennedy
- 1,813
232
votes
16 answers
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-contained problems that scheme theory answers -…
emperordali
- 581
231
votes
13 answers
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 Kolmogorov, i.e., probability measures) are appealing…
Pete L. Clark
- 64,763