Examples of dictionaries between two distinct fields of mathematics (or between “differents” structures of math).

Question

I'd like to meet explicit examples of dictionaries between two distinct fields of Mathematics (or between two "different" structures of Mathematics).

I'm not interested in the usual sense dictionary of mathematical terms, ie, in a Handbook with a list of entries to explain the meaning of words that appear in mathematics.

An example of dictionary Mathematica seems to me the Galois theorem that says that certain properties of the root's field of a algebraic polynomial of degree $n$ is equivalent to the properties of the permutation group of the roots of this polynomial. Correct me if I'm wrong.

I've heard talk that there are dictionaries for example between percolation theorems and theorems of complex variables. But I have no idea what can really be a dictionary between percolation theorems and theorems of variables complex variables.

In the end the concept seems very vague. I could answer what is a dictionary of mathematical structures? Please do not respond to the use of category theory. But if not possible, I think a response is satisfactory with interesting examples.

Answer 1

These dictionaries are meant to be somewhat vague for readability. Anyhow:

From algebraic geometry:

• Spaces <--> Rings of Functions

• Map of spaces <--> Map of rings

• Closed subspaces <--> Prime ideals

• Points <--> Maximal ideals

• Infinitesimal neighborhoods <--> Localizations

• Intersections <--> Tensor products

• etc.

From stable homotopy theory (generalizing the Dold-Kan correspondence):

• Spectra <--> Chain complexes

• Suspensions <--> Shifting up

• Loop spaces <--> Shifting down

• Fibers <--> Kernels

• Cofibers <--> Cokernels

• Homotopy groups <--> Homology groups

• Homotopies of maps <--> Chain homotopies

• $K(G,0)$ <--> $G$ in degree zero

• etc.

From knot theory and number theory:

• The integers, Spec Z <--> The 3-sphere $S^3$

• Z/p <--> Knots

• Legendre symbols <-> Linking numbers of knots

• Iwasawa polynomial <--> Alexander polynomial

And then you can cook up various conjectural dictionaries about mirror symmetry (complex <--> symplectic, sheaves <--> Lagrangians, Deformations of complex structure <--> GW invariants), about Langlands ($G$<-->$G^\vee$, etc), and even more specific conjectures (Anabelian schemes <--> Hyperbolic manifolds). I think many of the intriguing conjectures of today rely on such (possibly vague, but tantalizing) "dictionaries" between two distinct plots of mathematical land.