Surprising Generalizations

I just learned (thanks to Harry Gindi's answer on MO and to Qiaochu Yuan's blog post on AoPS) that the chinese remainder theorem and Lagrange interpolation are really just two instances of the same thing. Similarly the method of partial fractions can be applied to rationals rather than polynomials. I find that seeing a method applied in different contexts, or just learning a connection that wasn't apparent helps me appreciate a deeper understanding of both.

So I ask, can you help me find more examples of this? Especially ones which you personally found inspiring.

Solution 1:

Galois Connections

Let's be honest, the correspondence between Galois groups and field extension is pretty hott. The first time I saw this I was duly impressed. However, about two years ago, I learned about universal covering spaces. Wow! I swear my understanding of covering spaces doubled when the prof told me that this was a "Galois correspondence for fundamental groups and covering spaces".

Again here is a link!

Solution 2:

I agree! I spend much of my mathematical free time exploring such connections.

Here is a basic one that I constantly ponder. The rules of matrix multiplication encode two things:

• How to compose a linear transformation $A$ with another linear transformation $B$, with respect to a fixed basis.
• How to follow an edge of type $A$ on a graph, and then follow an edge of type $B$ (where $A$ and $B$ are just a disjoint partition of the set of edges).

This means that one can study walks on graphs by studying how a matrix called the adjacency matrix behaves. This leads into all sorts of beautiful mathematics; for example, this is the basic tool behind Google's PageRank algorithm, and it also in some sense motivated Heisenberg's matrix mechanics formulation of quantum mechanics. I often try to recast results in linear algebra in terms of some combinatorial statement about walks on graphs.

Solution 3:

I loved learning about how differential forms and the exterior derivative generalize 3-d vector calculus (div, grad, curl). Differential forms are so elegant in comparison, work in arbitrary dimensions, and give rise to beautiful mathematics (e.g. de Rham cohomology, Hodge theory). And of course, the generalized Stoke's theorem is one of the prettiest equations: $\int_{\partial R} \phi = \int_R d\phi$.