Let me give you an example of what I mean. Flag algebras are a tool used in extremal graph theory which involve writing inequalities that look like:
(It’s not too important to my question what this inequality means, but let me give you some context. Informally, the things we’re adding and multiplying are probabilities that a random group of vertices in a large graph will induce some specific small subgraph. To make some manipulations rigorously justified, this is not precisely what we mean; instead, they are the limits of such probabilities over a convergent sequence of graphs.)
Aside from being potentially useful in solving math problems I’m curious about, I enjoy using, thinking about, and even looking at statements about flag algebras, because these equations and inequalities just look so cool! Instead of multiplying, adding, and comparing letters and numbers, we get to do the same thing to pictures of things.
So my question is: what are some other topics in mathematics where we get to do the same thing?
Obviously, you can always give any name you like to a variable, like those math problems you see on facebook where cherry plus banana is equal to three times hamburger. I’m not interested in examples like this, because there’s nothing special about those variable names. Instead I’m interested in cases satisfying the following conditions:
- Mathematicians actually working with these objects commonly represent the things they are adding or multiplying or whatever (in general, performing algebraic manipulations on) by pictures.
- The pictures used to represent these objects are actually helpful for understanding what the objects are.
It’s okay if it’s not adding or multiplying specifically we’re doing, as long as we’re manipulating the pictures in ways traditionally reserved for numbers or variables. For example, the things represented by pictures could be elements of some algebraic object (group, ring, etc.)
Noah Snyder’s research statement which can be found online has a really cool example of this:
Also, here’s an example involving Feynman diagrams that I found online: