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1500 questions
230
votes
4 answers
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 $X \times X$ were homeomorphic to $\mathbb{R}$, then $X$…
Richard Dore
- 5,205
230
votes
89 answers
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 two is unexpected. The first example that comes to…
Victor Miller
- 4,626
229
votes
46 answers
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 that Euclid's parallel axiom is really necessary…
Ilya Nikokoshev
- 14,934
228
votes
9 answers
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 question about John Nash's work which has been on…
Paul Siegel
- 28,772
225
votes
11 answers
Refereeing a Paper
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 I've wondered about:
1) Should you summarize the…
Rbega
- 2,279
224
votes
20 answers
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 something new. When I'm doing mathematical…
user103598
222
votes
4 answers
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 of the ring, and replaces the ring with its…
Will Sawin
- 135,926
220
votes
140 answers
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 further basic or central examples and I think…
Gil Kalai
- 24,218
218
votes
8 answers
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 around 10 times) in my life.
But for some reason I…
aglearner
- 13,995
215
votes
67 answers
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 that a powerful and unexpected technique is…
gowers
- 28,729
213
votes
0 answers
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 $Q(x)$ are also $0,1$ polynomials.
Is this…
Sil
- 2,181
212
votes
24 answers
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—what does torsion measure intuitively?
Jan Weidner
- 12,846
211
votes
40 answers
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 statements that have proofs than we are with…
gowers
- 28,729
209
votes
51 answers
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:
Ahlfors, Complex Analysis, using Liouville's…
Anweshi
- 7,272
208
votes
26 answers
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 NOT enough.)
2) It is not known to be false
3) It…
Gil Kalai
- 24,218