Mathematical subjects you wish you learned earlier [closed]

Solution 1:

Category theory and algebraic geometry.

I spent a lot of time in undergrad studying things that were kinda nifty, but way too classical to be of any use/interest beyond "fun math". When I got to grad school, category theory was assumed and made some of my courses much harder than they should've been.

In the words of Ravi Vakil, "algebraic geometry should be learned slowly over a number of years". I currently NEED algebraic geometry, so I don't have this number of years. I wish I would've started that a long time ago. Additionally, both of these topics would've helped me learn the things I was thinking about anyways, in particular commutative algebra.

Solution 2:

Though I'm sure it's not unpopular, I don't think many people learn it early: Group Theory. It's a real nice area with a lot of cool math and some neat applications (like cryptography).

Solution 3:

Lattices and order theory. While these concepts are so ubiquitous, they seem to be banned from mathematics courses. Also, if you know something about order theory, many concepts from category theory turn out to be quite familiar. (E.g. view a poset as a category, then product resp. coproduct become infimum resp. supremum, the slice and coslice category are up and down set, etc.)

Solution 4:

I wish I'd understood the importance of inequalities earlier. I wish I'd carefully gone through the classic book Inequalities by Hardy, Littlewood, and Poyla early on. Another good book is The Cauchy-Schwarz Masterclass.

You can study inequalities as a subject in their own right, often without using advanced math. But they're critical techniques for advanced math.

Solution 5:

I wish I'd learned logic much, much earlier. Obviously young students couldn't handle much depth, but at least a basic introduction to a few concepts would be nice. Just understanding the concept of axioms and deductive rules would put all of math into some perspective. When I finally understood that math was constructed with formal definitions and proofs (or, for example, that there was more than one useful way to axiomatize), I felt I'd been kept in the dark my whole life, doing something (math) that I had absolutely no understanding of.