Mathematical discoveries that have occurred by cleverly linking two initially unrelated topics?

Question

I have heard mathematical proofs often require cleverly linking two areas of maths which initially seem disconnected. Could anyone provide an example of this, as I feel at my level of study, many proofs are simple extensions to previous topics?

Many thanks.

Answer 1

This is how Fermat's Last Theorem was ultimately proven. FLT is a three-step version of this, connecting number theory, projective geometry, and complex analysis.

The equation $a^n+b^n=c^n$ is a number theoretic statement, and it can be connected to projective geometry using something called the Frey curve. The Frey curve is a type of elliptic curve whose properties depend on the existence of rational solutions to the equation $a^n+b^n=c^n$. Specifically, if that equation has rational solutions then the solution could be used to construct an elliptic curve that is not modular.

The Modularity theorem (formerly known as the Taniyama–Shimura–Weil conjecture) says that elliptic curves (projective geometry) and modular forms (complex analysis) are more or less the same thing. Unsurprisingly, all modular forms are modular.

Together, these two statements mean that we have the chain implication

$$\exists a,b,c\in\mathbb{N}\;a^n+b^n=c^n \Rightarrow \exists\text{ a non-modular eliptic curve }\Rightarrow\text{ the Modularity theorem is false}$$

What Andrew Wiles actually proved to finish the proof of FLT is the Modularity theorem. Taking the contrapositive of this chain implication tells us that FLT is true.

Other examples that are the topic of ongoing research include the Langlands Program (connecting Galois Theory and automorphic forms) and Geometric Complexity Theory (connecting computational complexity theory and algebraic geometry). It's not a full fledged program (yet?) but this recent paper connects neural networks and something from algebraic geometry known as tropical polynomials. I'm actually currently writing a paper on this very topic.

Answer 2

Questions on elliptic curves $E\colon y^2=x^3+ax+b$ over $\Bbb Q$ are linked to modular forms, which are functions from the upper half plane to the complex numbers with high symmetry $$f(z)=(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right).$$

Both subjects are seemingly completely different. However, every elliptic curve defined over the rational numbers is modular. So these two different worlds are very much linked.

Answer 3

Pedagogically speaking, it's bad form to have your proof ideas appear 'out of nowhere', so it's reasonable that most proofs shown to students are simple extensions of previous topics. Generally, authors work hard to organize them that way.

It's more the revolutions in research and the one-off 'slick' proofs that are the shocking connections between two seemingly unrelated topics.

As an example of each:

1. Descartes' main achievement was showing that geometry and algebra are complementary and can be combined to give techniques that are greater than the sum of their parts.
2. I like Furstenburg's proof of the infinitude of the primes as an example connecting number theory and topology.

Answer 4

At a high level, several Fields Medals have been awarded in recent decades for connections between mathematical physics and topology: Simon Donaldson used gauge theory to study 4-manifolds, and Ed Witten's work has related quantum field theory to low dimensional topology and Morse theory.