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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.
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Ways to prove the fundamental theorem of algebra
Here is a proof that I think deserves to be recorded here somewhere. Of the proofs already listed it is closest to Pushkar's proof, Lucas Culler's proof, and Gian Maria Dall'Ara's highest-upvoted proo …
- 118k
33
votes
Mathematicians who were late learners?-list
Sophus Lie didn't become interested in mathematics until after university, and before then didn't seem to have shown significant aptitude for it.
- 4,713
99
votes
Your favorite surprising connections in mathematics
From an essay of Arnol'd:
Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and t …
- 21.3k
10
votes
Combinatorial results without known combinatorial proofs
For example, according to Stanley the identity $n \cdot \text{pp}(n) = \sum_{i=1}^{n} \sigma_2(i) \text{pp}(n-i)$ has no known bijective proof, where $\text{pp}(n)$ denotes the number of plane partiti …
- 4,713
27
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Probability in number theory
I learned from Gian-Carlo Rota (Combinatorial snapshots) the following probabilistic motivation for looking at the Riemann zeta function: "subject to technical assumptions," the only probability measu …
- 12.9k
28
votes
Dimension leaps
The Leech lattice.
At least, in the sense that 24 is one of the only dimensions where we know what the densest lattice packing looks like. As usual, John Baez's thoughts. Conway and Sloane is a good …
- 1,446
22
votes
What's the "best" proof of quadratic reciprocity?
I'm a fan of Zolotarev's proof, although the proof by Gauss's lemma does have sentimental value.
- 33.8k
10
votes
Definitions of determinant by unique features
Let $M_n$ be the "affine monoid scheme" of $n \times n$ matrices under multiplication (like an affine group scheme but no inverses).
Claim: Every polynomial monoid homomorphism $M_n \to M_1$ is a no …
- 118k
52
votes
Theorems with unexpected conclusions
I learned this example from Noam Elkies's excellent article The Klein Quartic in Number Theory. Elkies observes that Siegel's 1968 paper Zum Beweise des Starkschen Satzes, in order to prove its main …
- 4,713
12
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Theorems with unexpected conclusions
Here's one I was reminded of recently. Recall that a projective plane is a triple $(P, L, I)$ where $P$ is a set of "points," $L$ is a set of "lines," and $I$ is a subset of $P \times L$ describing t …
- 4,713
34
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Abstract thought vs calculation
The first proof I ever saw of the orthogonality relations for characters of finite groups was computational: it did a lot of matrix computations and manipulations of sums, which I didn't like at all. …
- 4,713
20
votes
What are some examples of colorful language in serious mathematics papers?
I always liked Edward Burger's A Tail of Two Palindromes. It begins as follows:
Upon a preliminary perusal, this parable may appear to be about pairs of palindromes, periods, and pitiful alliteration …
- 4,713
22
votes
Jokes in the sense of Littlewood: examples?
Let $C(x) = \sum_{n \ge 0} \frac{1}{n+1} {2n \choose n} x^n$ be the generating function for the Catalan numbers. Then $C(x) = \frac{1 - \sqrt{1 - 4x}}{2}$. In particular, $C(1) = \frac{1 - \sqrt{-3} …
- 35.5k
22
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Examples of great mathematical writing
Gian-Carlo Rota's On the foundations of combinatorial theory I: Theory of Möbius Functions is an eye-opening gem. The same is true of practically every paper in Gian-Carlo Rota on Combinatorics, so co …
- 4,713
32
votes
Real-world applications of mathematics, by arxiv subject area?
math.CO Combinatorics
Combinatorics finds applications in computer science, especially in the run-time analysis of algorithms. It has also in recent years found applications in physics, at least in …
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