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
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
136
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
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
93
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
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
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
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
88
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
One of the most interesting examples that happened recently is the Katz-Sarnak conjecture asserting that the average rank of elliptic curves (ordered by some reasonable height) defined over $\mathbb{Q}...
Community wiki
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