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180 votes

Tweetable Mathematics

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

Tweetable Mathematics

27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵. Nice try, Euler. link #Counterexamples
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 ...
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".
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.
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-...
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 ...
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 ...
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 ...
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
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, ...
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 ...
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 ...
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 ...
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.
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 ...
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'...
89 votes

Tweetable Mathematics

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

Tweetable Mathematics

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

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