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Results tagged with big-list
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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
111
votes
What recent discoveries have amateur mathematicians made?
After Martin Gardner published what one mathematician claimed to be a complete list of convex pentagons that could tile the plane, amateurs (Richard James III, a computer scientist, and Marjorie Rice, …
66
votes
If you break a stick at two points chosen uniformly, the probability the three resulting sti...
Consider an equilateral triangle with altitude 1. It is not hard to show that if you choose a point randomly in this triangle, the distances to the three sides gives the same distribution of lengths t …
- 6,342
45
votes
Most intricate and most beautiful structures in mathematics
How about the Leech lattice. This is a 24-dimensional packing of unit spheres where each one touches 196560 others. It is the densest 24-dimensional lattice packing (and very likely the densest 24-dim …
41
votes
Demonstrating that rigour is important
The evidence for both quantum mechanics and for general relativity is overwhelming. However, one can prove that without serious modifications, these two theories are incompatible. Hence the (still inc …
29
votes
What are the big problems in probability theory?
Michel Talagrand has a number of open problems (with bounty) listed on his website. I haven't looked at them all, but knowing him, I guarantee you that they are very hard and quite important. These ar …
29
votes
Problems known to be in both NP and coNP, but not known to be in P
One of my favorite problems in NP $\cap$ co-NP is deciding who wins a simple stochastic game. The game is played on a directed graph by two players, call them A and B. This graph contains several type …
28
votes
Noteworthy, but not so famous conjectures resolved recent years
Connes' embedding conjecture (from 1976) about the structure of infinite-dimensional von Neumann algebras was shown to be false in the paper
Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, …
19
votes
Which math paper maximizes the ratio (importance)/(length)?
How about Leonid Levin (1986), Average-Case Complete Problems, SIAM Journal of Computing 15: 285-286? Quite important in complexity theory, and only two pages long, although very, very dense.
17
votes
What are some slogans that express mathematical tricks?
"If you count something two different ways, you get the same result." This is related to the trick of changing the order of integration (or summation) discussed above, but discrete and more general.
…
6
votes
Mathematical ideas named after places
Las Vegas algorithms.
3
votes
What is your favorite "strange" function?
How about the function given by the Banach-Tarski paradox? This maps a ball into two copies of the same size ball, and is composed of isometries on subsets of $\mathbb{R}^3$.
1
vote
Particular problem solved by solving a more general problem.
The generating function proof of Cayley's theorem counting labeled trees (e.g., the theorem that there are $n^{n-2}$ labeled trees on $n$ vertices) is a good example. In the lecture notes I linked to, …