Answer by Denis Serre for Results with short, advanced proofs or long,...
Perron-Frobenius theorem.Each of both version answers the question.Weak version : If $A\in{\bf M}_n({\mathbb R})$ has non-negative entries, then the spectral radius $\rho(A)$ is an eigenvalue,...
View ArticleAnswer by Connor W for Results with short, advanced proofs or long,...
Silver's proof that GCH cannot first fail at a singular cardinal of uncountable cofinality is a great example of this. The original one of Silver uses generic ultrapowers, so requires some nontrivial...
View ArticleAnswer by Carl-Fredrik Nyberg Brodda for Results with short, advanced proofs...
I think a good example is the 1955/1958 Adian-Rabin theorem. This says that "given a finite presentation of a group, one can deduce almost nothing about the properties of the group". For example, the...
View ArticleClik here to view.
Answer by Michael Hardy for Results with short, advanced proofs or long,...
results that have very rough proofs from more elementary principles but have much clearer and smoother proofs with some more advanced background. Specifically, what are some examples of results from...
View ArticleAnswer by Yonah Borns-Weil for Results with short, advanced proofs or long,...
The number of spanning trees of an $n$-dimensional hypercube graph is $$2^{2^n-n-1}\prod_{j=1}^nj^{n\choose j}.$$This is a quite tricky graph theory problem to try to prove directly. But it dies almost...
View ArticleAnswer by Michael Greinecker for Results with short, advanced proofs or long,...
The original proof of the Stone-Weierstrass theorem was elementary enough that Stone could present it in a lengthy article in Mathematics Magazine. There is a much shorter proof due to de Branges that...
View ArticleAnswer by Nik Weaver for Results with short, advanced proofs or long,...
The famous example of this in my neck of the woods is Gelfand's slick proof, using Banach algebras (which should really be "Gelfand algebras"), of Wiener's theorem that the reciprocal of a...
View ArticleAnswer by pinaki for Results with short, advanced proofs or long, elementary...
Resolution of singularity of curves (over algebraically closed fields). A somewhat long elementary proof is given e.g. in chapter 7 of Fulton's Algebraic Curves (which he made available online for...
View ArticleAnswer by Vergilius for Results with short, advanced proofs or long,...
Brouwer's theorem is immediately obtained as a 'by-product' of the development of integration on manifolds. The 'elementary' alternative is to prove and use Sperner's Lemma, which (to me at least)...
View ArticleAnswer by fr_andres for Results with short, advanced proofs or long,...
A famous example is the Abel-Ruffini theorem.In this video, Fields medallist Richard Borcherds introduces Galois Theory and its background, mentioning that the proof by Ruffini was quite cumbersome and...
View ArticleAnswer by Timothy Chow for Results with short, advanced proofs or long,...
The associativity of the group law on an elliptic curve can be proved in an elementary way by explicitly manipulating algebraic expressions, but this is not very enlightening. By using more advanced...
View ArticleResults with short, advanced proofs or long, elementary proofs
Recently I was preparing an undergrad-level proof of (a form of) the Jordan Curve Theorem, and I had forgotten just how much work is involved in it. The proof stored my head was just using Alexander...
View Article
