Question

What is the most beautiful/brilliant mathematical solution you have ever come across?

Answer 1

The following two felt really clever to me. One is a definition and the other is an application.

1. The definition of a Cauchy sequence and the definition of convergence of a sequence. This leads to how we can define real numbers as limits of Cauchy sequences. What felt really clever about this was the way the conditions were placed on these definitions. I don’t really know how to explain what I’m trying to say, but they felt like ingenious solutions.

2. Using orthogonal projections in an inner product space to find solutions to minimisation problems. The idea being that we can reformulate minimisation problems in terms of the distance between two points. One point being in a vector space and the other being in some subspace of that vector space. The example that I though was really cool was that you can use a result regarding orthogonal projections to find a polynomial of degree at most 5 that approximates sin(x) as best as possible on some given interval. The example is in the book Linear Algebra Done Right, page 199.

Answer 2

Galois theory is pretty awe-inspiring in general. Loosely, the ‘fundamental theorem’ of Galois theory provides a one-to-one correspondence between field extensions and their automorphism groups, subject to some conditions. This allows you to study field theory with group theory and vice-versa, which can be quite powerful. It takes a second course in abstract algebra to fully appreciate the theoretical depth of this subject, and it is even more amazing because Galois invented it when he was 17!

Complex analysis is also full of surprising and unexpected results. My personal favourite is that the integral of a holomorphic function is invariant over homotopic curves. For some reason which I can’t fully articulate, I found this very bizarre and unintuitive.

Answer 3

Said it before and I’ll say it again - the proof that bounded variation functions are differentiable a.e via the rising sun lemma.

Answer 4

The complex analysis thing when you learn that most ugliness on real analysis functions completely disappears in the complex field and you get wonderful regularity properties.

Answer 5

There are so many, but I love the proof of quadratic reciprocity (I think it's Gauss's third proof of it) based on counting how many integer points sit inside a certain triangle.

Answer 6

The Cantor diagonal argument showing there is no bijection between R and N is stunning.

Answer 7

"one line" proof of Fermat's sum of 2 squares theorem, which can be made into a very short visual proof of sum of 2 squares theorem. Highly elegant. Can even be expanded into Jacobi's sum of 2 squares theorem.

Ramanujan's number, and relatedly the proof of Euler's lucky numbers.

For another pi formula, I love Ramanujan's and Chudnovsky's formula for pi. Just insane, and to obtain such formula you need knowledge from many different fields.

There are many other beautiful results, but I try to keep them to stuff anyone can appreciate without any background.

Answer 8

1. the classification of semisimple lie algebras and coxeter groups (and understanding diagram folding and triality is quite fun)

2. the fact that stochastic calculus actually works out and gives a workable theory always amazed me

3. the realization that basic algebraic number theory is basically covering theory

Answer 9

Lagrange multiplier method for constrained optimization.

İt's just neat and interestingly simple

Answer 10

I recommend reading "Proofs from the book". It is a collection of fantasticly elegant and beautiful proofs in all areas of math.