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Hi. I'm preparing a thesis in commutative algebra, and when I say this to my friends they always ask me what are the applications to "real-world", and I don't know what to answer. This let me think that I'm studying something useless. I'm studying on the Matsumura and on the Herzog-Bruns. Any of you know some applications of this abstract algebra to the real-world?

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I personally do not know of any uses (however, I am positive I will find out about some of them in the answers to this question). But why is it so terrible to be studying something simply because you find it fascinating? Something doesn't have to save lives or build bridges to be worthwhile. –  Zev Chonoles Apr 22 '11 at 23:10
Seconding Igor. –  Zev Chonoles Apr 23 '11 at 0:09
The two generic answers are "algebraic number theory" and "algebraic geometry". –  Mark Apr 23 '11 at 0:19
I'd much rather hear about your thesis. –  Theo Johnson-Freyd Apr 23 '11 at 0:31

4 Answers 4

The book "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by Cox, Little & O'Shea, contains some "real world" applications, specifically chapter 6 (of the 3rd edition) is titled "Robotics and Automatic Geometric Theorem Proving".

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Google "Gunnar Carlsson" and "Rob Ghrist" and "Berndt Sturmfels" and "John Canny", and...

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+1 although I'm not sure how the first two apply commutative algebra (beyond coefficient rings for homology)? –  Mark Grant Apr 23 '11 at 0:36
Not that it is very important, but still I am wondering, since you said this should all be CW, why isn't your answer already? –  quid Apr 23 '11 at 1:11
@unknown (google): a fair point, but I choose to be chicken (or is it egg?) @Mark Grant: philistine that I am, I view any subject beset by many projective resolution as part of (or at least almost a part of) commutative algebra. –  Igor Rivin Apr 23 '11 at 1:18
@Igor Rivin, thanks for the answer; of course, this is a legitimate point of view. –  quid Apr 23 '11 at 1:43

Commutative Algebra and Algebraic Geometry are of relevance to Statistics and in recent years there was quite a lot of activity on this.

See e.g. http://en.wikipedia.org/wiki/Algebraic_statistics (and scroll down, the beginning is perhaps also interesting for your purpose, but what I mean is rather at the end of the page).

For example there is this book L. Pachter and B. Sturmfels. Algebraic Statistics for Computational Biology from 2005.

And there is a fairly recent (I believe) Activity Group of SIAM (Society for Applied and Industrial Mathematics) for Algebraic Geometry (which perhaps is close enough CA), about to hold its first conference http://www.siam.org/meetings/ag11/ (looking up the planery speakers should yield further details; there is a considerable intersection with names I. Rivin gives).

Another topic at the borderline of commutative algebra and number theory is Elliptic Curve Cryptography see http://en.wikipedia.org/wiki/Elliptic_curve_cryptography and also other cryptographic problems, but in part they feeel perhaps too number theoretic for you.

Finally, not really your question, but apparently the motivation: to convince your friends, depending on the background of your friends, I suggest to explain them the (simple) congruence arithmetic behind the final digit of the ISBN numbers. This was the only thing that I found that I felt had some real impact on the opinion of some of my friends on the usefulnes of pure mathematics.

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Counting (partially) magic squares (and in fact combinatorics and commutative algebra have had really fruitful interactions).

One should also look at this question.

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