What kind of work do modern day algebraists do?

Question

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.

Answer 1

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.

Answer 2

'Algebra' is a bit vague - like most parts of math the boundaries are pretty blurry. Lets look at two prominent examples of a specialization among algebra, namely algebraic geometry: Preda Mihailescu from the University of Goettingen and Bernd Sturmfels from U.C. Berkley. Bernd Sturmfels is famous for his work in algebraic geometry. Preda Mihailescu is in the working group in Goettingen that specializes on algebraic geometry and number theory http://www.math.uni-goettingen.de/forschung/index.html (SP 2). However it seems Mr Sturmfels doesn't do much number theory. So they still have the algebraic geometry in common right? Wrong. The flavor of what they consider algebraic geometry changes dramatically depending on the application. For example a number theorist is lost without finite fields, p-adic numbers and so on. But Bernd Sturmfels goes more in the direction of tropical geometry, linear/integer programming (from a casual look at his publications).

Answer 3

For me, the key mechanism of algebra is the ability to reorganize formulae. At it's deeper roots it is a way of thinking about relationships within and between models. I use it to derive and refactor algorithms for software development... but also to gain insights, revise, and refactor processes... even those involving how people work together.

Systems can be described in equations; and those equations can be refactored algebraically; so those systems can be reorganized in similar ways.

Answer 4

Do you play angry birds? It sounds like a dumb question, but I'm teaching HTML5 games programming to kids (CoderDojo, Cork, Ireland).

The game plays full screen, on all screens. You don't know the size of the screen. But you have a variable holding that value once determined. But then programming, you don't know it yet.

You don't know the size of your image,s you can know, but it complicates the code so you can calculate them on the fly, you calculate boundaries from the known origin point.

You are tracking the relative positions of the sprites around the screen, waiting for a collision, by tracking their image start points and calculating relative dimensions to the screen edges and each other.

The purists will argue that its all mechanics or vectors, but these are relatively simple starting points, then you solve for X 50 times a second.

Its all basic stuff from a mathematics point of view, most of the class are under 10 after all, but it gets things primed in their heads, and stops the classic fear of maths.

Sorry its not an answer involving pure abstract algebra, but its an example.