From here on out Noncommutative Geometry will be denoted NCG. I will admit upfront that this is based on the article NCG for Pedestrians. I may or may not have found this article from the n-Category Cafe. Just trying to cover my bases. I am far from an expert (in fact this is my first encounter with it), so any person with some experience that finds incorrect information, please point that out.
For all you non-mathematicians, I thank you, and will attempt to keep this as nontechnical as possible, but I know your thoughts at this point. Uh, commutative, wasn’t that thing we learned that said ? How could that ever not happen? And geometry was that stuff with triangles and circles we did in 10th grade. So NCG is like saying when you add a triangle to a circle it isn’t the same as adding a circle to a triangle.
Let’s start there then. What is NCG? Like all fields of math, it is usually extremely difficult to give a well-defined definition. Yet at the same it is often the case that you can point to an explicit paper on ___ and go, “Oh yeah, that’s ___.” I don’t remember who said this, but someone said that mathematics has two pillars: algebra and analysis (of course topology is the heavens and not of the lowly corporal world). Essentially NCG falls on the side of algebra. When doing NCG, we are trying to interpret (noncommutative) algebraic structures in a geometric way.
So I guess I should try to convince those out there with no experience in this that there is such a thing as noncommutative algebra. We are actually most concerned with rings. So a ring is a set S, with two operations, call them + and * (this is rather abstract, + and * DO NOT have to be the standard addition and multiplication, heck S doesn’t have to contain numbers so addition and multiplication might not even make sense) such that the following properties are satisfied:
+ is associative: ; + is commutative: ; there is an + identity: ; + has an inverses: ;
* is associative: ;
There is a distributive law: and .
Things to note: + is a lot more restricted than *. We don’t need an identity or inverses in *. Most importantly, when we say noncommutative we mean in the * operation, since + is required to be commutative.
The example that should convince you that these things exist. At my high school we were taught about matrices, but I’m not sure about the general public on that. Anyway, check for yourself that if you take, say 2 by 2 matrices, that with matrix addition you satisfy the first set of properties, and with matrix multiplication you satisfy associativity, but you do not satisfy commutativity. This seems to be an extremely important example considering you have something most people have heard of not satisfying commutativity, so it is more common than you might think.
So I may have gone a little off topic, and I didn’t even get to the article. I might as well jump into the article next considering there are probably two types of people reading this: those that are like come on this is basic why aren’t you introducing terms like fiber bundle, affine schemes, and Weyl quantization? and the other type is going I have no idea what just happened, but I think it hurt…