Mathematics students across the country take courses such as “abstract algebra” and “modern algebra.” The content of these courses is incredibly standardized at this point. Generally, students cover group theory with emphasis on finite groups, perhaps “up to” Sylow’s theorems, then they move on to some linear algebra over a general, fixed field (rather than or ), then they cover some elementary ring and ideal theory, and finally some courses will delve into field theory and Galois theory, culminating with such classic results as the insolubility of the general quintic. As with most mathematics courses, after having taken elementary algebra students may get the feeling that the subject has been mostly exhausted, but this is simply not the truth. In fact, even some basic questions in finite group theory are still unresolved.
However, this is not to say that the primary topic of interest is finite group theory, for instance. Truthfully, we do have an excellent understanding of finite groups. This is why, in so many related subjects, we seek to translate hard questions into group theoretic terms, rendering them simple by comparison. There are any topics which use algebra to study other domains, e.g. algebraic topology, algebraic geometry, and algebraic number theory. In fact, it seems few topics in modern mathematics have gone untouched by algebraic influence.
But, what of “pure algebra,” as dubious as the idea of such a classification may be? Well, one major research area I feel most would say fits this description is the theory of Lie groups (and Lie algebras), which are groups which have additional structure, namely that of a smooth manifold. Many of the groups you are familiar with are Lie groups, for instance , , and . This document will give you some citations of fairly recent developments. And, if you are really interested, it is frankly not difficult to pick up the basics of the subject if you have a decent background in algebra and differential geometry.
A fairly recent development in finite group theory was the classification of finite simple groups. This was a massive collaborative effort spanning decades, which ended in 2004 and which took tens of thousands of pages contained in hundreds of articles and produced by over a hundred authors. While one needs to be an expert (with a lot of time) to understand the classification theorem’s proof, its statement is very simple.
Theorem. Every finite simple group is isomorphic to one of the following: a cyclic group of prime order, an alternating group of degree at least , a simple Lie group, or the 26 sporadic simple groups.
But, this was clearly a monumental task. Are there smaller problems in “classical” group theory which have only recently been solved? Why, yes, there are! There is still some work to be done for finite groups, but frankly the number of mathematicians working in that area has been and will continue to be dropping quite quickly. If we turn to groups of infinite order, however, we can find several interesting, difficult problems. For instance, there have been several recent developments with respect to the so-called “inverse Galois group problem,” though none that I know of have been particularly groundbreaking (full disclosure: I am not a group theorist). A good note on the subject was written by Zywina, and the same mathematician authored a proof in 2012 that the fundamental group solves the inverse Galois problem for .
But, what of ring, ideal, and module theory? Is this still an area of research? Certainly, commutative algebra and related subjects are now used by many, many mathematicians, and there have been too many developments from too many areas to even begin to discuss any in detail here. Nonetheless, here is a link to a somewhat recent MSRI workshop on the subject, where you may be able to find some guidance.
Even linear algebra, which many see as now becoming more computational and applied, is still an area of current research in pure mathematics. Moreover, derivatives of linear algebra such as representation theory remain very active. In fact, representation theory has played a major role in the famous Langlands program.
And, indeed, Galois theory too is an active area of research, though it is becoming increasingly uncommon for that terminology to be used. A good sampling of what mathematicians care about related to Galois theory can be found at this page for a program held at the University of Pennsylvania in 2006.
So, while it seems everyone’s research involves some algebra these days, from geometry to number theory to combinatorics, there are still plenty of “pure” algebraists out there. Hopefully, this post has given the reader some idea as to what it is these people do.
Note: My work involves a lot of algebra, but I am not an algebraist.