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My background is about third-year mathematics major, including linear algebra and abstract algebra. I'm now studying Atiyah's commutative algebra, but it lacks some concrete, easy examples and it's too brief to fully understand this subject.

In fact I think I don't like algebra, and I want to study other areas. But I heard that algebra is basic, so you need to study to some graduate level(commutative algebra, algebraic geometry, etc..). Is this true?

Anyway, doing only algebra is a little boring. Can you recommend some interesting branch which is suitable for a third-year student based on differential geometry or differential equations? I was interested in those classes.

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Yes. ${}{}{}{}$ – Mariano Suárez-Alvarez May 23 '11 at 22:05
Many of the theorems in commutative algebra have a geometric interpretation. In fact, if you like differential geometry, you might like algebraic geometry, too: there are a lot of similarities. – Jesse Madnick Nov 22 '12 at 1:15
up vote 7 down vote accepted

For commutative algebra texts, see Reference request: introduction to commutative algebra.

For an interesting application of algebra, try reading about differential Galois theory and the existence of elementary solutions for differential equations; see for instance How can you prove that a function has no closed form integral?; my answer there contains a list of papers and books.

Another major use of commutative algebra besides algebraic geometry is number theory. There are many excellents books on algebraic number theory; see a list here.

PS: I assume you have seen fields and Galois theory. If not, then this is definitely the way to go.

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I would argue that algebra is one of the most basic of maths, necessary to all graduate math students. So in response to you question: is it true that algebra is necessary? - I say absolutely (I say also that topology and analysis are necessary... but that's another topic).

If you've only taken a semester of linear and a semester on groups/rings/fields, then I would recommend looking into Galois Theory. Or, interestingly, I would recommend looking into quantum mechanics/pde, as algebra can have interesting implications there.

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Maybe off topic, but why do you think topology is necessary? – Eric Naslund May 23 '11 at 17:40
@Eric: This may just be a product of my background in pure, rather than applied, math. My quals are in Topology, Algebra, Analysis, Complex, and Manifolds, and it may have colored my perceptions. At my undergrad institution, I think the quals were in topology, algebra, and analysis, too (even though it's a very applied school). In some sense, I think almost all math has ties in these. I think of topology as a generalization of geo and set theory, sort of containing the useful parts of set theory to most mathematicians. Topology also plays nicely with higher algebra and analysis. Hard to say. – mixedmath May 24 '11 at 13:47
I don't think that Galois theory is necessary. One semester on groups/rings/fields is enough if you don't want to specialize in algebra. – AlexE Sep 15 '11 at 14:46

If you say you liked differential equations but find algebra boring, I'd say you should read Michio Kuga's Galois' Dream: Group Theory and Differential Equations, which sounds like exactly what you're looking for.

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One more immediate action you might consider in your current coursework is to supplement your Atiyah text with other texts on commutative algebra. For example, look at some of the recommendations posted here on the subject.

I've often found I've had to do that in a number of courses, and it's always handy to have reference texts to refer to for more concrete examples, or different ways of approaching a proof, or helping to motivate a new concept. Some texts aim to be precise and concise, so they may lack expository material such as examples. Other texts aim to help students make connections between new concepts and material they already know, but these may lack rigorous precision. Each has its purpose.

At any rate, don't feel you need to limit yourself to Atiyah! (Feel free to springboard a bit for additional explication or examples elsewhere, till you develop a sense of owning the material if that makes sense.) That's not to say you should abandon Atiyah! But exploring a bit elsewhere may help you return to Atiyah with a fresh outlook.

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@Will, amWhy, I noticed that you cross-edit quite a lot in the past few hours. I think that you're a bit overdoing it right now. Please try to keep the bumping of old questions (especially at these less-active hours) to a minimum! (I'm not a fan of such comments, but I feel that I have to post one!) – Asaf Karagila Nov 22 '12 at 0:39
@asaf it's partly due to a light-hearted "grammar war" - no intent to disrupt! – amWhy Nov 22 '12 at 1:04

It is true that all parts of mathematics use some algebra. But you don't have to study commutative algebra to learn the definition of a module. You have already taken courses in linear algebra and abstract algebra. This should be enough if you don't want to learn algebraic geometry or algebraic number theory.

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