180
votes

### Tweetable Mathematics

Four color theorem: any planar graph can be colored with 4 colors. Only proof by computer. SAD.

Community wiki

179
votes

### Nonequivalent definitions in Mathematics

Perhaps the mother of all examples is "natural number". You can start an internet flame war by asking whether zero is a natural number.

Community wiki

164
votes

### Do you know important theorems that remain unknown?

Monotony is a superfluous hypothesis in the
Monotone convergence theorem for Lebesgue integral.
In fact the following is true.
Theorem - Let $(X, \tau, \mu)$ be a measurable space, $f_n : X \...

Community wiki

152
votes

### Nonequivalent definitions in Mathematics

Linear functions:
In high-school algebra (sometimes called "pre-calculus"), we are taught that linear functions are those of the form $y=mx+b$, because they are graphed by a straight line in the ...

Community wiki

137
votes

### Short exact sequences every mathematician should know

There is one obvious sequence that underlies all vector analysis and a lot that builds up on it, no matter if its applied analysis, PDE, physics or the original foundations of algebraic topology. Yet ...

Community wiki

137
votes

### Suggestions for special lectures at next ICM

How about a lecture on proof assistants/formal proofs?
Most mathematicians are still skeptical of the value of proof assistants, and it's certainly true that proof assistants are still very difficult ...

Community wiki

132
votes

### The most outrageous (or ridiculous) conjectures in mathematics

A long-standing conjecture in Number Theory is that for each positive integer $n$ there is no stretch of $n$ consecutive integers containing more primes than the stretch from 2 to $n+1$. Just looking ...

Community wiki

132
votes

### What are examples of (collections of) papers which "close" a field?

In this classic article, Steinitz closed not just one, but all fields.

Community wiki

125
votes

### The most outrageous (or ridiculous) conjectures in mathematics

W. Hugh Woodin, at a 1992 seminar in Berkeley at which I was present, proposed a new and ridiculously strong large cardinal concept, now called the Berkeley cardinals, and challenged the seminar ...

Community wiki

122
votes

### Tweetable Mathematics

27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵. Nice try, Euler. link
#Counterexamples

Community wiki

122
votes

### Every mathematician has only a few tricks

$$
\sum_{i=1}^m\sum_{j=1}^n a_{i,j}=\sum_{j=1}^n\sum_{i=1}^m a_{i,j}
$$
(and its variants for other measure spaces).
I still get misty-eyed whenever I read something that capitalizes on this trick in ...

Community wiki

121
votes

### Nonequivalent definitions in Mathematics

Not a word but a piece of notation: Sometimes I have seen $\subset$ used to mean "is a proper subset of" while other times I have seen it used to mean "is a subset of".

Community wiki

117
votes

### Tweetable Mathematics

The sentence "The first positive integer that cannot be specified in a 140 character tweet" doesn't specify a well defined integer.

Community wiki

116
votes

### LaTeX tricks that save time in typesetting

Related to the question. I showed the following web-pages to a table-neighbor at a conference and literally got the "You just saved 3 days of my life" reaction.
Doi2Bib
ISBN2Bib
Arxiv2Bib
These web-...

Community wiki

113
votes

### PhD dissertations that solve an established open problem

I find George Dantzig's story particularly impressive and inspiring.
While he was a graduate student at UC Berkeley, near the beginning of
a class for which Dantzig was late, professor Jerzy ...

Community wiki

106
votes

### Nontrivially fillable gaps in published proofs of major theorems

In 1970, I. N. Baker published a proof of a basic result in holomorphic dynamics:
a transcendental entire function cannot have more than one completely invariant domain.
A completely invariant ...

Community wiki

105
votes

### Mathematicians with both “very abstract” and “very applied” achievements

John von Neumann was the first person to come to my mind.
He published over 150 papers in his life: about 60 in pure mathematics, 60 in applied mathematics, 20 in physics, and the remainder on ...

Community wiki

104
votes

Accepted

### Tweetable Mathematics

Every rational r is xyz(x+y+z) for some rational x,y,z.
Proof: Euler (1749) found x(r),y(r),z(r). Nobody knows how.
I have a guess.
https://people.math.harvard.edu/~elkies/euler_14t.pdf

Community wiki

94
votes

### What programming language should a professional mathematician know?

Python, so they can use Sage.
From their website:
SageMath is a free open-source mathematics software
system licensed under the GPL. It builds on top of many existing
open-source packages: NumPy,
...

Community wiki

94
votes

### What are some noteworthy "mic-drop" moments in math?

The best known lower bound for the minimal length of superpermutations was originally posted anonymously to 4chan.
The story is told at Mystery Math Whiz and Novelist Advance Permutation Problem, and ...

Community wiki

92
votes

### Examples of math hoaxes/interesting jokes published on April Fool's day?

I enjoyed the hexasphere by
A. V. Akopyan, J. Crowder, H. Edelsbrunner, R. Guseinov
from last year:
http://pub.ist.ac.at/~edels/hexasphere/
In the link, the sphere is animated, so you can look at it ...

Community wiki

92
votes

### When has discrete understanding preceded continuous?

I would say that a lot of topology was discrete before it was continuous.
The Euler characteristic was first observed (in 1752) as an invariant of
polyhedra. Around 1900 Poincaré first calculated ...

Community wiki

91
votes

### Tweetable Mathematics

Not deep, but if [0,1]² is cut in N triangles of equal area, N is even. If not, extend 2-adic valuation on Q to R, tricolor plane and apply Sperner.

Community wiki

91
votes

### Prominent non-mathematical work of mathematicians

Samuel Eilenberg, one of the key mathematicians of the XX Century (co-created category theory, systematized homological algebra, opened new roads in topology, etc), was a good example:
he had (at ...

Community wiki

89
votes

### The most outrageous (or ridiculous) conjectures in mathematics

$P=NP$
Let me tick the list:
Most likely false, because, as Scott Aaronson said "If $P = NP$, then the world would be a profoundly different place than we usually assume it to be."
Yes, it'...

Community wiki

89
votes

### Tweetable Mathematics

Chebyshev said, and I say it again,
there is always a prime between n and 2n.
#BertrandsPostulate

Community wiki

89
votes

### Every mathematician has only a few tricks

A very useful generic trick:
If you can't prove it, make it simpler and prove that instead.
An even more useful generic trick:
If you can't prove it, make it more complicated and prove that instead!

Community wiki

85
votes

### Tweetable Mathematics

Integral of exp(-x²) dx over R = Γ(1/2) = √π.
Proof: square the Gaussian integral and use polar coordinates!

Community wiki

82
votes

### Suggestions for special lectures at next ICM

I suggest lectures on big and transformative ideas. For example, it would be great to have a lecture by Tim Gowers about the future of mathematics publishing, and getting away from the issues with our ...

Community wiki

81
votes

### Theorems that are essentially impossible to guess by empirical observation

Bootstrap percolation is a two-dimensional two-state cellular automaton with a von Neumann 5-square ("plus") neighborhood where a "white" cell become "black" if it has at ...

Community wiki

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