Abstract Algebra, or Topology

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polymer
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Abstract Algebra, or Topology

Postby polymer » Fri Feb 10, 2012 1:59 am UTC

I enjoy physics, and find mathematical arguments fascinating. Analysis has been fun, and has an enjoyable relationship with physics. Currently I'm re-practicing elementary programming, deciding I might take it more seriously as a job prospect. While practicing, I'm finding little connections between programming and math to be neat! Like recursion for example.

From what little I've read, principles of Analysis seem to be useful in computer science, which surprised me. So I was curious, between the subjects of Abstract Algebra and Topology, which one was more "useful" in the sense that the implications discussed in the respective subject would be most relevant to most/some of physics and/or programming.

If this question is too vague as well, I'd also like to know that(I really have little sense of what exactly either of these respective subjects are).

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Re: Abstract Algebra, or Topology

Postby ++$_ » Fri Feb 10, 2012 4:43 am UTC

The boundaries of "algebra" and "topology" are a little too broad to answer the question definitively, but on the whole I would say that topology is more relevant to computer science than algebra, although both are not very relevant (certainly less relevant than analysis).

So-called combinatorial topology has some applications to data structures, distributed computing, and sensor networks.

As for algebra, linear algebra is obviously relevant to computing in many ways, but I don't think that's what you were getting at. There are some encryption algorithms that make use of field theory, and algebraic structures crop up in graph theory (which is obviously related), but as far as I know the connections are tenuous at best.

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Re: Abstract Algebra, or Topology

Postby doogly » Fri Feb 10, 2012 3:11 pm UTC

I'd take the next steps into algebra, for the main reason that topology without algebra is pretty boring. It's sort of like a more finicky and strange analysis? But algebraic topology is totally sweet.

For programming you can get some fancy things out of algebra, but mostly just in encryption type applications. To my knowledge. So not a programmer. Definitely a physicist though! And physics is pretty much all Lie Groups, which are topological groups. You'd probably start with algebra to talk about them, but eventually everyone falls in love with them.
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gorcee
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Re: Abstract Algebra, or Topology

Postby gorcee » Fri Feb 10, 2012 4:37 pm UTC

They're both useful. I think it would be beneficial to take the abstract algebra course first. It's actually quite useful in computer science-type applications. Ever hear of a Hamming Code, or an Error Correcting Code? They have roots in abstract algebra. Also, things like QAM (Quadrature Amplitude Modulation), which basically dictates how cable modems work, have a basis in abstract algebra as well. There is also quite a bit of algebra when you get to formal numerical analysis. So it's actually quite a pragmatic field in the physical sciences.

But, for both Algebra and Topology, their utility is not in the simple facts and definitions that you'll learn. The true utility from both of those classes is that they give you different ways to look at problems and find solutions, and they give you more tools in your arsenal to attack a challenging problem with. If you can, take both, but I think you'll be better served doing Algebra -> Topology.

Tirian
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Re: Abstract Algebra, or Topology

Postby Tirian » Fri Feb 10, 2012 7:46 pm UTC

Another vote for abstract algebra as well. The applications are even broader than the suggestions here. The Rubik's Cube is another example of a transformation group structure. Essentially any time you combine two "things" to make another sort of "thing" of the same type whose combination process is associative in that formal sense, you have a group. Topology is about analyzing the basic question of "closeness", which is interesting but doesn't come up nearly as often in modeling.

The other nice thing about studying abstract algebra first is that it's often presented as a first glimpse of how you can abstractly analyze a single facet of real numbers, and so it might be more gentle than jumping into topology first. Abstract topics often cover very related themes, and once you study subgroups, group homomorphisms, congruences, quotient groups, and direct sums, you'll be better able to notice how those same themes "echo" in topologies and the likely applications of the subspace topologies, continuous mappings, cones and suspensions, and Cartesian topologies.

fishfry
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Re: Abstract Algebra, or Topology

Postby fishfry » Sat Feb 11, 2012 12:41 am UTC

I'd say abstract algebra would be much more applicable to computer programing than topology.

Algebra is about sets and the operations we can do on their elements. It's about transformations; applying operations to things to get other things. It's really very much like computer programming. Algebra is about collections of objects and the operations and processes we can apply to those objects to get other objects. That's programming.

Topology is about studying ways that you can deform a space without tearing it. It's the study of continuous transformations of space. It's like rubber-sheet geometry. Two shapes are the same if you can bend or stretch one to look like the other; but you can't cut or tear. It's geometry without a notion of distance.

At the level of undergrad math and practical computer programming, abstract algebra is very much applicable, and topology is totally irrelevant.

I would say that at more advanced levels there is a connection. If you're doing theoretical work trying to see if you can automate the validation of programs -- in other words, feeding one program into another program and coming out with the answer "Looks good" or "full of bugs," then you use some notions of formal logic, which actually borrows concepts from topology. Compactness in formal logic is the same as compactness in a topological space.

But that's a stretch. In terms of general programming, I can't think of any relation to topology at all. Well, network people use the word "topology" to mean the configuration of the nodes of a network. But actually that's graph theory, not topology. So it's just a buzzword rather than an application of topology.

polymer
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Re: Abstract Algebra, or Topology

Postby polymer » Sat Feb 11, 2012 1:46 am UTC

Sounds like the concensus is Abstract Algebra then! That was my suspicion, but I thought I'd ask anyways. To give a fuller context as to how applicable it may be to me, my linear algebra course was not very abstract at all, and the part of it that was abstract was done poorly. The implications of an orthonormal basis really didn't make sense until I did fourier analysis, for example. Really all I "got" was the concept of a simple coordination transform, and some simple linear transforms (like rotation in euclidean space). I have little intuition for the advantages and qualities of group theory, and my understanding was that that subject was a very nice generalization of linear algebra for solving problems. So I figured a more mature approach to the subject would excercise elementary skills I could still use.

Regarding topology then, the only way I could take that course is concurrently with Algebra, if I want to graduate next year. The Topology classes are, pointset topology, and differential topology. Don't know how well that leads up to algebraic topology, or if I could just read on algebraic topology later after doing Algebra. But I'm just ranting because I really don't have a good sense of what to expect!

At any rate, thank you for the suggestions, I'll probably be taking the algebra sequence!

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MartianInvader
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Re: Abstract Algebra, or Topology

Postby MartianInvader » Sun Feb 12, 2012 11:50 pm UTC

Point-set topology is sort of a prerequisite for algebraic topology, as is abstract algebra. Actually, the standard joke is that the prereq for algebraic topology is a course in algebraic topology, but once you wrap your head around it and realize how damn cool it is, it's pretty fun.
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