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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.
1
vote
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 …
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 …
4
votes
Conditions equivalent to finiteness
Here are two fun ones off the top of my head. I don't know if you want these to be in separate answers:
A set $X$ is finite iff the free vector space $k[X]$ is finite-dimensional, for any field $k$. …
8
votes
Suggestions for good notation
Forgive me for bumping such an old big-list question, but I can't resist. In linear algebra texts you will sometimes find the notation $[v]_B$ for the vector of coordinates of a vector $v \in V$ with …
39
votes
Mathematical conjectures on which applications depend
The use of RSA for public-key encryption is widely believed to rely on the assumption that factoring is hard. Actually it relies on a stronger assumption than this, namely that the RSA problem is hard …
20
votes
Why is the definition of the higher homotopy groups the "right one"?
There are many things to say here. Here's one. Suppose you want to classify all spaces up to (weak) homotopy equivalence, or equivalently all CW complexes up to homotopy equivalence. The zeroth step i …
38
votes
Accepted
Linear algebra in terms of abstract nonsense?
To my mind there are two classes of interesting categorical facts here, loosely speaking "additive" facts and "multiplicative" facts. Some additive facts:
Finite-dimensional vector spaces over $k$ h …
- 118k
73
votes
Sophisticated treatments of topics in school mathematics
It's common in calculus classes and textbooks to state that the antiderivative of $\frac{1}{x}$ is $\log |x| + C$, where $C$ is a constant. This is incorrect: $C$ need only be a locally constant funct …
86
votes
Sophisticated treatments of topics in school mathematics
The angle addition formula $\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha) \tan(\beta)}$ for tangent gives one of the simplest nontrivial examples of a formal group law, nam …
6
votes
What are examples when the equality of some invariants is good enough in algebraic topology?
Here is the easiest generalization of the fact you cite about Eilenberg-MacLane spaces. Spaces $X$ with exactly two nontrivial homotopy groups $\pi_n(X), \pi_m(X), 2 \le n < m$ are classified (up to ( …
21
votes
Solving algebraic problems with topology
There are cool examples already for finite group: Dijkgraaf and Witten recover and generalize a combinatorial formula due to Burnside using the TFT formalism.
It's probably worth elaborating on t …
16
votes
Accepted
Examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differe...
The following is really an adjunction between $2$-categories but I am going to ignore that subtlety. This blog post discusses everything in more detail and with a few more examples.
Consider on the o …
13
votes
Math books for advanced high school students
The list I give undergraduates and strong high schoolers is here.
9
votes
What are your favorite concrete examples of limits or colimits that you would compute during...
It seems that he has been faced with the following situation (to which I myself have never been faced; that is why I am asking here, in the hope that someone else already has been in the same situa …
138
votes
Accepted
What is entropy, really?
Here is a simple story one can tell about the entropy
$$H = -\sum_{i=1}^n p_i \log p_i$$
of a discrete probability distribution. Suppose you wanted to describe how surprised you are upon learning …