So I’m getting ready to start my graduate studies (hopefully) in the next year. I find that really enjoy algebra. I’ve liked group theory, galois theory, representation theory of groups, etc. But I find that I like learning about and playing with wackier or less popular algebraic structures like monoids and semigroups as well. Playing by different rules is refreshing and gives interesting insights (in my opinion) why things like groups or fields are so nice.
Everywhere I look, it seems like professors are group theorists, ring theorists, algebraic geometers. I have yet to see a professor at an R1 call themselves a loop theorist, mention quasigroups or quandles, etc. Is there just not enough general interest in these other structures?
I know for example that there are very compelling reasons to care about groups (encoding information about symmetry, acting on sets, applications in physics, Maschke’s theorem fails spectacularly for monoid representations, etc.) and polynomials over a ring are a ring so that’s also natural. Rings look like numbers we’re familiar with and you can do a lot of cool stuff like making field extensions, etc.
The advice I’ve gotten from a couple of different professors is to build a research program in something most mathematicians care about, without being super hot (like deep learning in CS right now). You can get funding and you maximize your chances of saying something worthwhile/interesting about the field.
Should I focus on mainstream algebra research programs in graduate school or should I try to blend in my weirder interests. I’m quite happy to have a mathematical hobby, if that’s how algebra is in the US/Europe right now.