Below is a list of some of my favorite mathematical ideas. Many of these are theorems, but some of them are analogies. Eventually I plan to fill these in with references. For now I'm just using this to keep track of my favorite ideas.
Below is a list of some of my favorite mathematical ideas. Many of these are theorems, but some of them are analogies. Eventually I plan to fill these in with references. For now I'm just using this to keep track of my favorite ideas.
Ostrowski's theorem for number fields.
The number of spanning trees of a graph can be thought of as the "class number number" (as in number fields).
Prime numbers are "knots" in Spec Z.
There's a unique manifold structure on Rn with the Eucliden topology, except when n=4, for which there's infinitely many.
The classification of semi-simple Lie algebras in terms of their Coxeter-Dynkin diagrams.
The Serre-Swan theorem.
The "fundamental theorem of algebra" holds over the octonions.
Compactifications of hyperbolic tilings can be thought of as a generalization of Platonic solids which allow for "higher genus".
Graphs with largest eigenvalue 2 correspond to affine Coxeter graphs.
The entire enterprise of functional analysis of graphs in analogy with Riemannian geometry.
Riemann's rearrangement theorem.
Proving that the fundamental groups of topological groups is Abelian, via abstract nonsense.