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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4$\begingroup$ 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. $\endgroup$– Zev ChonolesCommented Apr 22, 2011 at 23:10
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2$\begingroup$ The two generic answers are "algebraic number theory" and "algebraic geometry". $\endgroup$– MarkCommented Apr 23, 2011 at 0:19
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10$\begingroup$ I'd much rather hear about your thesis. $\endgroup$– Theo Johnson-FreydCommented Apr 23, 2011 at 0:31
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6 Answers
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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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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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The answer that I am going to give is implicitly contained in a few answers already given, but it is a bit too implicit, to my taste, so let me give it out and loud: Gröbner bases. When you solve a system of linear equations, you use Gaussian elimination, when you solve a system of polynomial equations of higher degrees, you use Gröbner bases, and it is very clear that solving systems of polynomial equations is something that people have to do for all sorts of applications.
That "very clear" is not just a belief held by a pure mathematician: on a few occasions that I talked about something mathematical to people doing research in some real world questions of statistics, biology, engineering, Gröbner bases would be the only aspect of somewhat advanced algebra, not just commutative algebra, that they would have ever heard of. You can see some relevant bits of software solving applied problems in various areas here: http://www.risc-software.at/en/.
I can't resist from also saying that in some areas of pure maths, for a long time, saying the words "Gröbner bases" was a bit of faux pas, something that a true pure mathematician should rather leave as a discussion topic to people concerned with applications, something as silly and naive and so not worth mentioning as using a calculator to multiply two numbers. However, besides being a useful tool for computations, Gröbner bases and their generalisations also give methods to construct resolutions (starting from work of Anick in 1980s), and in particular to prove that a certain algebra (or an operad) is Koszul etc. So it certainly is something worth being aware of, really.
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Google "Gunnar Carlsson" and "Rob Ghrist" and "Bernd Sturmfels" and "John Canny", and...
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$\begingroup$ +1 although I'm not sure how the first two apply commutative algebra (beyond coefficient rings for homology)? $\endgroup$ Commented Apr 23, 2011 at 0:36
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$\begingroup$ @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. $\endgroup$ Commented Apr 23, 2011 at 1:18
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$\begingroup$ @Igor Rivin, thanks for the answer; of course, this is a legitimate point of view. $\endgroup$– user9072Commented Apr 23, 2011 at 1:43
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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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For applications in physics (string theory) see http://link.springer.com/chapter/10.1007%2F978-1-4614-5292-8_2 (Some Applications of Commutative Algebra to String Theory, by P.S. Aspinwall) and http://arxiv.org/abs/hep-th/0703279 (Topological D-Branes and Commutative Algebra, by the same author).