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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.

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It is not clear to me that there is a useful distinction between algebraists and algebraic geometers, algebraic number theorists, algebraic topologists, ... –  Qiaochu Yuan Aug 11 '12 at 2:35
@QiaochuYuan I think we are getting to a point where the main distinction is motivation. –  Mike B Aug 11 '12 at 5:01
@QiaochuYuan You are missing Hwang's point, she was probably hinting towards "pure" algebraists (Hopf algebras, Quantum groups, ring theory, group theory, generally non-commutative "stuff") –  Chindea Filip Aug 11 '12 at 6:37
@Chindea: it is also not clear to me that quantum groups count as "pure" algebra. –  Qiaochu Yuan Aug 11 '12 at 6:38
@Yuan Well, there are some people "out there" having something of Midas' "golden touch". Still, it's left to the OP to agree with the lack of a "useful" distinction between completely different areas of mathematics, as you suggest –  Chindea Filip Aug 11 '12 at 7:04

2 Answers 2

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.

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A pure ring theorist will often have thought quite a bit about Kothe's conjecture. They will have thought about stuff that has the morpheme "nil" in it. Is the polynomial ring of a nil ring nil? Nilpotent maybe? When you hear these theorems and problems for the first time in a single talk, you might have trouble distinguishing between open questions and solved or even trivial problems soon after that. They all sound rather similar. In general, there are loads of simple-sounding problems like Kothe's conjecture in ring theory that are difficult. Many of them have probably never been asked.

Some of ring theorists, I believe these are mainly from Iran, will have considered some kind of graph defined by ring-theoretic stuff. Take a ring and call the zero divisors vertices. Throw in an edge between $x$ and $y$ whenever $xy=0$. You get a graph that you can do all kinds of things with. You can ask which rings induce a graph with this or that property.

Often a noncommutative ring theorist will be looking for some kinds of left-right symmetries. If you define a left Xical ring-theoretic thing and a right Xical ring-theoretic thing, are they the same ring-theoretic thing? The Jacobson radical may have been the inspiration for this.

Another thing is that, as in a lot of mathematics, algebraists will be trying to classify their objets. Rings in general don't seem reasonably classifiable, which leaves room for attempts at partial classification. In ring/algebra theory these will often aim at generalizing Wedderburn's theorem.

This is a very narrow part of what algebraists do. Algebra comes in so many flavors. The commutative-noncommutative boundary is especially strong I think. Also, some algebraists will think a lot about universal algebra, varieties and pseudo-varieties, some won't. Some will be deeply in love in categories, some will say meh.

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