Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How true this is, I am not sure.
So I suppose I want to ask, what sort of work do modern day algebraists do?
What are currently some of the more active areas of modern algebra?
What types of problems do algebraists deal with?
I'm kicking around the idea of pursuing graduate study one day, possibly in some sort of algebraic field, i.e., ring theory or something. What sort of research and problems are open to your average graduate student in algebra (of any sort, not just ring theory)?
This is partially inspired by the question What do modern-day analysts actually do?
Thank you for your responses.
Best Answer
I don't believe what I'm doing is especially active or popular (so hopefully someone else will respond with a better answer), but seeing as no one has answered yet, I'll just mention one of the things algebraists do: invent new algebras.
The process is very easy to describe. It may or may not result in something useful. Take a set $A$ and define a set $F$ of operations on $A$ (maps from $A^n$ into $A$, for various non-negative integer values of $n$). The set $A$ plus the operations $F$ is what we call an algebra, usually denoted $\mathbf{A} = \langle A, F\rangle$. The algebras you already know (e.g., groups, rings, modules) are examples.
In my work, I think about different ways to construct such algebras. Usually I work with finite algebras, often using computer software like GAP or the Universal Algebra Calculator to construct examples and study them. I look at the important features of the algebras and try to understand them better and make general statements about them.
To address your last question, there is the following open problem that I worked on as a graduate student: Given a finite lattice $L$, does there exist a finite algebra $\mathbf{A}$ (as described above) such that $L$ is the congruence lattice of $\mathbf{A}$. This question is at least 50 years old and quite important for our understanding of finite algebras. In 1980 it was discovered (by Palfy and Pudlak) to be equivalent to the following open problem about finite groups: given a finite lattice $L$, can we always find a finite group that has $L$ as an interval in its subgroup lattice? Imho, these are fun problems to work on.