>>12336165There's two main fields: applied and pure math. Any area you focus on can pretty much be separated into these two categories. Pic related pretty much shows you everything and how it relates to other things.
Applied math is math that has direct applications to real-world problems. All math taught in secondary school (and generally college, unless you're doing math in undergrad) is applied math. More advanced subjects here, past algebra and calculus are involved with economics, computer science, physics etc. There is a lot of overlap into fields that require mathematics as an engine to make the ideas behind those subjects work.
Pure math is math that has no concern for application. That's not to say it's useless, rather the main object of development and concern is mathematics itself. You could argue this is just math that hasn't yet found application, as a lot of subjects that are born as pure math fields often end up finding applications. A good example is back in ancient greece, prime numbers were a pure math concept, because there really was no use for them, they were just an interesting mathematical phenomenon. Later on, prime numbers found a huge role in cryptography; prime numbers are essentially the bedrock for this field.
It's important to note PHD level math isn't just about moving numbers around anymore like in algebra. It involves more work in proofs and arguments.
