# Contemporary research in abstract algebra

I am a Physics undergrad, who is considering the prospects of changing to maths. I was wondering if there is research going on in pure abstract algebra nowadays. I am aware of the fields algebraic number theory, and algebraic geometry which applies algebra, but is there research going on in some specific areas of abstract algebra.

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Of course, why would it ever end? The greater the sphere of knowledge, the greater its contact with the unknown... If you have access to a research library and you want to see current directions, find the algebra bookshelves and select books written in the last 5-10 years. –  alancalvitti Oct 11 '12 at 23:25
@alancalvitti: I thought the field was dormant, because I browsed the webpage on research areas of variety of universities, and I coudn't find specific areas in abstract algebra. If yes, please could you tell me some specific areas where research is going on. –  ramanujan_dirac Oct 11 '12 at 23:29
Look up the Langlands program and representation theory for two of the hot topics (which do overlap). –  Chris Janjigian Oct 11 '12 at 23:34
Matter of fact, I was a physics major, changed to a math major, and am now doing research in pure abstract algebra (group theory). So, you can even do it as an undergrad if you work hard! –  Alexander Gruber Oct 11 '12 at 23:38
There's algebra all over the place, in category theory, in computer science, in the generalization of symmetry from groups to inverse semigroups, in data mining and patter recognition in the nature of ultrametric spaces. I just saw a new book at UCSD lib talking about semi-tensor products. It keeps growing. Just a few examples. –  alancalvitti Oct 11 '12 at 23:51

These guys and these guys and these guys seem to think there's still stuff to talk about in algebra. Your university library will probably have the latest editions of them on file. If you want to get a feel for the stuff that's going on right at this second, check those out - even if you don't understand what the articles are about, it's nice to get a glimpse of the buzzwords and general subject matter going on right now. There's also this for reading material, which is even more accessible. You can pick out the algebraic stuff from the list of topics.

To my knowledge the hottest pure algebra topic right now the Langlands program, as mentioned in the comments. Quantum groups and cryptography are applied algebra topics that attract tons of funding (I would take "applied" with a grain of salt here... much of both these topics is as abstract as you would find anywhere else in algebra). There is research happening in just about everything, though.

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Thanks a lot for the answer. Could you also give me some info and links on the research in group theory in particular. To be more precise,I am looking for RESEARCH TOPICS accessible to a physics/maths undergrad. Also, as I do both physics and maths, it would be a long time(after having completed the basics) before I read things like category theory, or representation theory(the pure maths approach that is, I have done some in physics) –  ramanujan_dirac Oct 12 '12 at 0:28
Personally I like finite groups, and a great place to get started with that is Isaac's book Finite Group Theory (it's a graduate text, but I think easily understandable to anybody with one undergrad group theory course). arxiv.org/list/math.GR/new This is the group theory section on arXiv. My advice would be to read some papers at random, whatever sounds interesting, and see if you can find a semi-obscure or new topic about which you might be able to ask an interesting question. You might also ask a professor to help you find a topic. Most have smaller "pet problems" laying around. –  Alexander Gruber Oct 12 '12 at 0:44
A little uninvited advice, since I've been in your same situation... you may want to keep the physics major too, simply to be exposed to other facets of math. I would never have thought I'd like differential geometry before I took string theory, for example. Also, don't be too intimidated by representation theory and the like... it's tough, but with a little grit, everything is readable. –  Alexander Gruber Oct 12 '12 at 0:51
Thanks a lot for the answer and the advice. +1... –  ramanujan_dirac Oct 12 '12 at 10:42
Finite group theory is about properties of groups that only happen in the finite case - the Sylow theorems are an introductory example. Representation theory is about homomorphisms from a group into $GL_n(K)$ for some field $K$ (usually $\mathbb{C}$). It's essentially using ring theoretic techniques to study finite groups. It turns out there are a lot of things you can prove about groups using these methods that you couldn't otherwise, a classic example being that any group whose order has at most $2$ prime divisors is solvable. –  Alexander Gruber Oct 12 '12 at 12:40
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Certainly! I hesitate to draw a well-defined line between 'pure algebra' and 'applications of algebra', but I think these fields can be considered 'pure': commutative algebra, noncommutative ring theory, finite group theory, semigroups, Lie algebras, vertex algebras, universal algebra, category theory, and of course representation theory...

To summarize: there is lots of research in algebra going on. Lots.

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